The ziqqurath of exact sequences of n-groupoids

UNIVERSIT ` A DEGLI STUDI DI MILANO F acolt` a di Scienze Matematic he Fisic he e Naturali Dipartimen to di Matematica “F. Enriques” Corso di Dottorato di Ricerca in Matematica, XX ciclo T esi di Dottorato di Ricerca The Ziqqura th of Exa ct Sequences of n -Gr oupoids MA T \ 02 Relatori Candidato Prof. Enrico M. Vitale Dott. Giusepp e Metere Prof. Stefano Kasangian Co ordinatore di Dottorato Prof. An tonio Lan teri Anno Accademico 2006–2007 Ai miei genitori Lalla e Sa verio, c he hanno sempre creduto in me. A V alen tina, senza di lei questa T esi non sarebb e mai stata scritta. Ac kno wledgmen ts I w ould lik e to thank my sup ervisors Stefano Kasangian and Enrico Vitale, for their encouragement and the pleasant time sp ent together w orking at this pro ject. I would also like to thank George Janelidze and Marco Grandis for their helpful suggestions and remarks. I ac knowledge the members of the Category Seminar of the Univ ersity of Milano for man y fruitful discussions, and I am esp ecially grateful to Sandra Man tov ani for her tireless supp ort. I am in debt with m y wife V alentina Guarino, who fought in v ain against m y Ital-English while revising this thesis. Con ten ts 1 In tro duction 1 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 F urther dev elopments . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Con ven tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Case study: dimension one . . . . . . . . . . . . . . . . . . . 8 1.5.1 Bro wns’result . . . . . . . . . . . . . . . . . . . . . . . 8 1.5.2 Bro wn’s result revisited . . . . . . . . . . . . . . . . . 10 1.6 Case study: dimension tw o . . . . . . . . . . . . . . . . . . . 11 1.6.1 Homotop y ﬁb ers . . . . . . . . . . . . . . . . . . . . . 11 1.6.2 2-Exact sequences . . . . . . . . . . . . . . . . . . . . 12 1.6.3 Lo wering the dimension: ﬁrst step . . . . . . . . . . . 13 1.6.4 Lo wering the dimension: second step . . . . . . . . . . 14 2 Basics on sesqui-categories 16 2.1 Sesqui-categories . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Morphisms of sesqui-categories . . . . . . . . . . . . . . . . . 20 2.3 2-Natural transformation of sesqui-functors . . . . . . . . . . 22 2.4 Sesqui-categories and 2-categories . . . . . . . . . . . . . . . . 24 2.5 Finite pro ducts in a sesqui-category . . . . . . . . . . . . . . 25 2.6 Pro duct in terchange rules . . . . . . . . . . . . . . . . . . . . 26 2.7 h -Pullbac ks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Strict n -categories 32 3.1 n Cat : the data . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 n Cat : the underlying category . . . . . . . . . . . . . . . . . 37 3.3 n Cat : the hom-categories . . . . . . . . . . . . . . . . . . . . 41 3.3.1 V ertical comp osition . . . . . . . . . . . . . . . . . . . 41 3.3.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 n Cat : the sesqui-categorical structure . . . . . . . . . . . . . 48 3.4.1 Deﬁning reduced left-comp osition . . . . . . . . . . . . 48 3.4.2 Left-comp osition axioms . . . . . . . . . . . . . . . . . 52 3.4.3 Deﬁning reduced righ t-comp osition . . . . . . . . . . . 53 CONTENTS vii 3.4.4 Righ t-comp osition axioms . . . . . . . . . . . . . . . . 56 3.4.5 Whisk ering axiom . . . . . . . . . . . . . . . . . . . . 58 3.5 Pro ducts in n Cat . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.1 2-univ ersality of categorical pro ducts . . . . . . . . . . 58 3.6 The standard h -pullbac k in n Cat . . . . . . . . . . . . . . . . 63 3.6.1 Comp osition . . . . . . . . . . . . . . . . . . . . . . . 64 3.6.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.3 Pro jections and ε . . . . . . . . . . . . . . . . . . . . . 73 3.6.4 Univ ersal prop erty . . . . . . . . . . . . . . . . . . . . 73 3.6.5 Pullbac ks and h -pullbacks . . . . . . . . . . . . . . . . 77 4 n -Group oids and exact sequences 78 4.1 n -Equiv alences . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.1 In verses . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1.2 Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 n -Group oids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 The sesqui-functor π 0 . . . . . . . . . . . . . . . . . . . . . . 83 4.3.1 π 0 on ob jects . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2 π 0 on morphisms . . . . . . . . . . . . . . . . . . . . . 86 4.3.3 The underlying functor . . . . . . . . . . . . . . . . . 87 4.3.4 π 0 on 2-morphisms . . . . . . . . . . . . . . . . . . . . 88 4.3.5 π 0 comm utes with (ﬁnite) pro ducts . . . . . . . . . . . 92 4.3.6 π 0 preserv es equiv alences . . . . . . . . . . . . . . . . 93 4.3.7 A remark on π 0 . . . . . . . . . . . . . . . . . . . . . . 93 4.4 The discretizer . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1 D ( n ) on ob jects and morphisms . . . . . . . . . . . . . 95 4.4.2 D ( n ) on 2-morphisms . . . . . . . . . . . . . . . . . . . 96 4.4.3 A remark on D ( n ) . . . . . . . . . . . . . . . . . . . . 97 4.5 The adjunction π ( n ) 0 a D ( n ) . . . . . . . . . . . . . . . . . . . 97 4.5.1 In lo w dimension . . . . . . . . . . . . . . . . . . . . . 97 4.5.2 The general setting . . . . . . . . . . . . . . . . . . . . 97 4.6 n -Discrete h -pullbac ks . . . . . . . . . . . . . . . . . . . . . . 104 4.7 Exact sequences of n-group oids . . . . . . . . . . . . . . . . . 106 4.7.1 P ointedness and h -ﬁb ers . . . . . . . . . . . . . . . . . 106 4.7.2 Equiv alences and h -surjectiv e morphisms of n-group oids 106 4.7.3 Exact sequences . . . . . . . . . . . . . . . . . . . . . 107 4.7.4 π 0 preserv es exactness . . . . . . . . . . . . . . . . . . 108 4.8 The sesqui-functor π 1 . . . . . . . . . . . . . . . . . . . . . . 113 5 3 -Morphisms of n -categories 116 5.1 What structure for n Cat ? . . . . . . . . . . . . . . . . . . . . 116 5.2 Lax n -mo diﬁcation . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 n Cat ( C , D ): the underlying category . . . . . . . . . . . . . . 124 5.4 n Cat ( C , D ): the hom-categories . . . . . . . . . . . . . . . . . 125 CONTENTS viii 5.4.1 Comp osition . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.5 n Cat ( C , D ): the sesqui-categorical structure . . . . . . . . . . 128 5.5.1 Reduced left-comp osition . . . . . . . . . . . . . . . . 128 5.5.2 Reduced righ t-comp osition . . . . . . . . . . . . . . . 132 5.5.3 Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.6 0-whisk ering of 3-morphisms . . . . . . . . . . . . . . . . . . . 137 5.6.1 Reduced left-comp osition . . . . . . . . . . . . . . . . 138 5.6.2 Reduced righ t-comp osition . . . . . . . . . . . . . . . 140 5.6.3 Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.7 Dimension raising 0-comp osition of 2-morphisms . . . . . . . 146 5.7.1 Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 153 6 h -Pullbac ks revisited and the long exact sequence 158 6.1 2-dimensional h -pullbac ks in n Cat . . . . . . . . . . . . . . . 158 6.2 Ω and a second deﬁnition of π 1 . . . . . . . . . . . . . . . . . 166 6.2.1 0-comp osition in P c 0 ,c 0 0 ( C ) . . . . . . . . . . . . . . . . 172 6.2.2 0-units in P c 0 ,c 0 0 ( C ) . . . . . . . . . . . . . . . . . . . . 175 6.2.3 Comparison isomorphism S . . . . . . . . . . . . . . . 175 6.2.4 Bac k to the Theorem . . . . . . . . . . . . . . . . . . . 178 6.2.5 Final remark on S . . . . . . . . . . . . . . . . . . . . 182 6.3 Monoidal structure on Ω( C ) . . . . . . . . . . . . . . . . . . . 184 6.4 Ω and π 1 preserv e exactness . . . . . . . . . . . . . . . . . . . 185 6.5 Fibration sequence of a n -functor and the Ziqqurath of exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5.1 Connecting morphism ∇ . . . . . . . . . . . . . . . . . 188 6.5.2 Connecting 2-morphism σ . . . . . . . . . . . . . . . . 192 6.5.3 The ﬁbration sequence of F . . . . . . . . . . . . . . . 196 6.5.4 The Ziqqurath of a morphism of (p oin ted) n -group oids 196 A n -Group oids, comparing deﬁnitions 200 A.1 n Cat : the globular approach . . . . . . . . . . . . . . . . . . 200 A.2 The group oid condition . . . . . . . . . . . . . . . . . . . . . 205 A.3 System of adjoin t inv erses . . . . . . . . . . . . . . . . . . . . 208 Bibliograph y 212 Chapter 1 In tro duction Higher Dimensional Cate gories are sho wing relev ant implications in Algebraic T op ology (classiﬁcation of homotopy n -t yp es, op erads, cob ordism), and in Algebraic Geometry (Grothedieck’s n -stacks, non-ab elian cohomology), not to mention recen t applications in Mathematical Physics (TQFT, higher order gauge theory) and Computer Science. Nev ertheless basic algebraic to ols, in order to further dev elop the theory , are far from b eing established. A step forward tow ards this direction would b e ha ving an essential understanding of the notion of exactness for Higher Dimensional Categories, and of the limits in volv ed in deﬁning this notion. 1.1 Summary In 1970 Ronald Brown published a paper [ Bro70 ] on an uprising area of mathematical researc h: the theory of group oids. According to the category-theorist, a gr oup oid is a category with all of its morphisms inv ertible w.r.t. comp osition [ ML98 , Hig05 ]. In fact the notion of group oid was introduced earlier as a generalization of the notion of gr oup , where the binary op eration is only partially deﬁned. In studying connections with algebraic top ology and non-ab elian cohomology , Bro wn show ed that, given a ﬁbration F of p ointed group oids and its (strict) k ernel K s : K s K / / B F / / C it w as p ossible to obtain a 6-term exact sequence π 1 K s π 1 K / / π 1 B π 1 F / / π 1 C δ / / π 0 K s π 0 K / / π 0 B π 0 F / / π 0 C (1.1) of groups and p oin ted sets, where π 0 is the functor giving set of isomorphism classes of ob ject, and π 1 the group of endomorphisms of the p oin t. 1.1 Summary 2 Still, the condition of F b eing a ﬁbration was motiv ated b y “the analogy [. . . ] with top ological situations” [ Bro70 ]: this inﬂuenced the kind of limit considered, i.e. a (strict) kernel. Ho wev er it suﬃces to consider the homotopy kernel instead of the strict k ernel to remov e the need of restricting to ﬁbrations, so that construction ab o ve still holds for a generic group oid morphism. These ideas developed further in [ HKK02 , DKV04 ], where the authors gener- alized Brown’s result to 2-group oids. The 2-group oids considered are w eakly in vertible 2-categories. Duskin, Kieb oom and Vitale show ed in [ HKK02 ] that, giv en a morphism of 2-group oids G : C → D , it is still p ossible to get a 6-term sequence as in (1.1) of (strict) categorical groups and p oin ted group oids, where π 0 is the classifying-functor, and π 1 giv es the cat-group of endomorphisms of the p oin t. This sequence is exact in a suitable sense (see [ Vit02 ]). F urther, since π 0 ◦ π 1 = π 1 ◦ π 0 , applying π 1 and π 0 again w e get t wo 6-term exact sequences that can b e pasted together in a 9-term exact sequence, where the left-most three terms are ab elian groups, the central three are groups and the righ t-most three are just p ointed sets. The purp ose of this thesis is to extend these results to a n-dimensional con text. The setting is the sesqui-category [ Str96 ] of n -group oids, strict n -functors and lax n -transformations. W e consider a notion of n -group oid equiv alen t to that of [ KV91 ], i.e. a w eakly in vertible n -category , but our approach is gen uinely recursive. Being more precise, n -categories and n -functors are given by means of the standard enric hmen t in the category of ( n − 1)-categories and ( n − 1)-functors, w.r.t its cartesian bi-closed structure. Diﬀeren tly , n -transformations considered come in a lax v ersion, b eing a direct generalization of those of [ Bor94 ] for 2-categories. In fact a notion of strict natural n -transformation is also sk etched, but that has sho wn inadequate in dev eloping the theory for all morphisms, and not just for ﬁbrations. The lax n -transformations in tro duced are equiv alen t to that considered b y Crans in [ Cra95 ], y et inductiv e deﬁnition allo ws to deal with such morphisms directly , without the complications of the theory of pasting schemes [ Joh89 ] (as in [Cra95]). The n -group oids we consider are n -categories which are lo cally ( n − 1)- group oids and whose 1-cells are equiv alences. In this setting we introduce a straigh tforward generalization of π 0 (lo wer “ ∗ ” stays for p ointed): π ( n ) 0 : n Gp d ∗ → ( n − 1) Gpd ∗ F urther, standard h -pullbac ks [ Mat76a ] are introduced, together with their 1-dimensional, i.e. ordinary , univ ersal prop ert y . This allo ws to deal with 1.1 Summary 3 h -k ernels, and to deﬁne a lo op -functor Ω : n Cat → n Cat , C 7→ Ω( C ) / /   I [ ∗ ]   I [ ∗ ] / / C 9 A z z z z z z z z whic h gives the functor π ( n ) 1 : n Gp d ∗ → ( n − 1) Gpd ∗ , with π ( n ) 1 ( C ) = π ( n ) 0 (Ω( C )) T o extend the last to a (contra-v ariant) sesqui-functor, it is necessary to consider h -pullbac ks with their 2-dimensional univ ersal prop erty , this in- v olving 3-morphisms b et ween n -natural transformations, dimension-raising horizon tal comp osition of 2-morphisms and the necessary algebra for those. T o this end, an appropriate notion of sesqui 2 category has b een in tro duced. Finally , w e set up a notion of exactness for a triple ( K , ϕ, F ) in n Gp d ∗ K K / / 0 B F / / C ϕ   and w e prov e the Main Resul t F or an y natural num ber n , 1. Sesqui-functors π ( n ) 0 and π ( n ) 1 preserv e exactness, the last up to reversing the directions of 2-morphisms. 2. Sesqui-functors π ( n ) 0 and π ( n ) 1 comm ute, i.e. π ( n ) 1 ◦ π ( n − 1) 0 = π ( n ) 0 ◦ π ( n − 1) 1 . 3. Giv en a morphism of p oin ted n -group oids F : B → C , with h -k ernel K , there exists a canonical morphism ∆ of p oin ted ( n − 1)-group oids, and 2-morphism δ , such that the sequence b elow is exact: π 1 K π 1 K / / 0 < < π 1 B π 1 F / / 0 " " π 1 C ∆ / / 0 < < π 1 ϕ   δ   π 0 K π 0 K / / 0 " " π 0 B π 0 F / / π 0 C π 0 ϕ   1.2 F urther developments 4 Applying π ( n − 1) 0 and π ( n − 1) 1 , w e get t w o six-term sequences, exact b y (1) ab o ve. Those can b e pasted b y (2) ab o ve, in a nine-term exact sequence of ( n − 2)-group oids (cells to b e pasted are dotted in the diagram): · / /   · / / π 2 1 ϕ   B B ·   / / · / / B B · / / π 1 π 0 ϕ   · π 1 δ   · / / B B · / / π 0 π 1 ϕ     · B B / / · / /   · / / π 2 0 ϕ   · π 0 δ   Iterating the pro cess we ﬁnally obtain a 9 · n exact sequence of p ointed sets. F urthermore, since the sesqui-functors π 0 force h -groups structures, the ﬁrst 9 · ( n − 1) terms are structured as an exact sequence of groups, ab elian up to the 9 · ( n − 2) th . Similarly , the last-but-one step pro duces a 9 · ( n − 1) term sequence of p ointed group oids, 9 · ( n − 2) of which are indeed categorical groups [ Vit02 ], ﬁnally 9 · ( n − 3) are comm utative. Arranging these sequences from top (dimension n ) to b ottom (dimension 0) w e unv eil the shap e of a Ziqqur ath 1 , in which each level is an exact sequence of k -group oids ( k = 0 , . . . , n ). In the relations b et ween contiguous levels are nested classiﬁcation prop erties of n -group oids and their morphisms, man y of them are still to b e in vestigated. 1.2 F urther dev elopmen ts The sesqui-categorical setting presented here yields a fruitful p ersp ectiv e in the study of n -dimensional categorical structures. In fact this is a general fact, and it p ermeates categorical in vestigations from its very b eginning: “categories are two steps aw a y from naturality , the concept they w ere designed to formalize. [. . . ] F rom the v ery study of the established practice of routinely sp ecifying morphisms along with eac h mathematical structure, we were presented, in the 1940’s, with an extra dimension: morphisms b etw een morphisms. W e w ere naturally led by naturality to ob jects, arrows and 2-cells. [Str96]” 1 Ziqquraths (or Ziggurats) were a t yp e of step pyramid temples common to the inhabi- tan ts of ancient Mesop otamia [Sto97, Opp77]. 1.3 Conventions 5 Moreo ver the inductive approach p ermits to deal with a sesqui-categorical en vironment in an y dimension, carrying along the constructions while as- cending the dimensional ladder. Tw o main p oints are currently b eing inv estigated by the author. First, the native sesqui-categorical setting oﬀers the chance to study w eak structures inductively , reducing most of coherence issues to (inductively nested) planar diagrams. This w ould mak e possible to describ e some lax n -dimensional structures more easily in a pure diagrammatic wa y , in terms of cells and of comp ositions giv en explicitly . Namely , it would b e in teresting to consider an inductiv ely deﬁned sesqui-categories of n -group oids with weak units, in order to compare it with the sp ecial connected 3-dimensional case of [JK07]. Similarly for other semi-strict versions, as in [Pao07]. Second, an application to ab elian chain complexes is announced. In fact, most authors prefer globally deﬁned versions of n -categorical structures, as it makes the internalization pro cess easier. In this w ay is usually pro ved the equiv alence b etw een ω -categories in Ab (the category of ab elian groups) and non-negativ ely graded chain complexes of ab elian groups. Y et it is p ossible to deal with this equiv alence in our context to o. In fact, the category of length- n ab elian c hain complexes is equiv alen t to the category of ab elian group ob jects in n Cat ([ Lei04 ], Example 1.4.11). More in terestingly this equiv alence extends to homotopies and natural n -transformation. No w, ab elian group ob jects in n Cat b eha ve w ell with resp ect to the h - structure of n Cat , similarly to what is shown for monoid ob jects in a similar situation b y Grandis in [ Gra97 ]. That is the reason wh y it is worth studying in this new setting ho w the theory extends to. 1.3 Con v en tions The purp ose of this section is to mak e life easier to the reader, stating the notational conv en tions (and other) adopted. Nevertheless exceptions to these con ven tions are not rare, although alwa ys p oin ted out. Doubled capitals as A , B , C are used for n -categories and n -group oids, cap- itals as F , G, H for their morphisms, low er case Greek letters α, β , γ for their 2-morphism. Finally capital Greek letters as Σ , Λ are reserved to 3- morphisms. Ob jects of a n -category ( n -group oid) are denoted by low er case Latin letters, subscripted by the num ber “0”. Ob jects hav e often the same letter as the 1.3 Conventions 6 big doubled capital denoting the whole structure, using primes (or other mo diﬁers) for diﬀeren t ob jects, e.g. a 0 , a 0 0 , a 00 0 are ob jects of A . Cells follow a similar con ven tion, where the n umber represent the dimension of the cell, as the 1-cell b 1 : b 0 / / b 0 0 of B . 2-Cells are often represen ted by double arro ws ( b 2 : b 1 + 3 b 0 1 ). As in higher dimension it would not b e quite practical, for representing a k -cell we lab el the arro w itself with the num ber k : b k : b k − 1 k / / b 0 k − 1 . In order to av oid confusion with the name morphisms (reserved for the sorts of the environmen t sesqui-category) the sorts of our n -categories ( n - group oids) are alw ays named c el ls . A ma jor exception to these rules is the Chapter on Sesqui-Cate gories . Indeed it uses its o wn notational conv en tions, explained therein. Comp ositions are dealt with diﬀerent symbols in order to distinguish the (in- ternal) comp ositions of cells from the (external) comp ositions of morphisms. W e adopt for cells-comp osition the empty circle sup erscripted by a n umber that represents the dimension of the in tersection cell. Example: c h ◦ m c k means that the h -cell c h and the k -cell c k are comp osed along their common b oundary , that is a m -cell c m . In this case we will use the terms m -domain and m -co domain. Let us p oin t out that sup erscripted m is often omitted, sp ecially when it is 0. F or the comp ositions of morphisms we use the ﬁlled circle sup erscripted b y a num ber that represents the dimension of the intersection morphisms. In the present work w e will use only 0-comp ositions and 1-comp ositions of morphism, hence the symbols • 0 and • 1 . They come with a low er-scripted L or R if they are left or right whiskering, resp ectiv ely . Moreov er dimension raising 0-comp osition of 2-morphisms is denoted ∗ . All comp osition-sym b ols are omitted when clear from the context. The comp ositions of cells morphisms will b e written in algebraic order, e.g. for c 1 : c 0 → c 0 0 and c 0 1 : c 0 0 → c 00 0 w e will write c 1 ◦ 0 c 0 1 : c 0 → c 00 0 . The other order is considered as evaluation , so parentheses will b e used, e.g. for F : A → B and G : B → C we will write G ( F ( − )) : A → C . 1.4 Synopsis 7 1.4 Synopsis The thesis is organized as follo ws: the rest of the introduction is dedicated to analyze lo w-dimensional cases, that inspired this generalization; the second chapter gives the sesqui-categorical-theoretical framework: dif- feren t c haracterizations are compared, ﬁnite pro ducts and h -pullbac ks are in tro duced with their universal prop erties; strict n -categories are deﬁned in the third c hapter, together with their mor- phisms ( n -functor) and their 2-morphisms (lax n -transformations); moreo v er ﬁnite pro ducts and standard h -pullbac ks are constructed explicitly; the fourth chapter in tro duces n -group oids, h -surjectiv e morphisms and equiv- alences: these are necessary to formulate a notion of exactness for the sesqui-categories of p ointed n -group oids; moreov er we extend to them some classical result as the adjunction discrete/iso-classes functor, and one p oint susp ension; in order to deal with 3-morphisms of n -categories, a new framew ork is de- ﬁned in the ﬁfth chapter, namely sesqui 2 - c ate gories ; lax n -mo diﬁcations are in tro duced thereafter, together with other whiskering and comp ositions; in fact a dimension raising 0-comp osition of 2-morphisms is giv en and man y useful algebraic prop erties are pro ved; with the mac hinery developed in the previous ones, the sixth c hapter presents the main result: the construction of (a Ziqqur ath of ) exact sequences in any lo wer dimension from a given morphism of n -group oids; this is ac hieved in few steps, the starting p oin t b eing a 2-dimensional prop erty of pullbacks that h -pullbac ks in n Cat are prov ed to satisfy; ﬁnally the app endix con tains a comparison with the globular approac h, the group oid condition and the c hoice of inv erses. 1.5 Case study: dimension one 8 1.5 Case study: dimension one In this section w e recall, for reader’s conv enience, the construction set up in [Bro70]. 1.5.1 Bro wns’result Let us supp ose w e are given a functor F : B → C b et ween tw o group oids. F or ev ery ﬁxed ob ject b 0 in B , F induces a map St F ( b 0 ) : St B ( b 0 ) → St C ( F b 0 ) where St B ( b 0 ) = S b 0 0 ∈ B 0 B 1 ( b 0 , b 0 0 ). Supp ose now that for ev ery b 0 , the map St F ( b 0 ) is surjective. Such F is called star-surje ctive , or ﬁbr ation . W e can consider its (strict) k ernel w.r.t. an ob ject b of B : K s K / / B F / / C (1.2) Here the group oid K s is just the strict ﬁb er o ver the ob ject F b of C , i.e. the group oid with ob jects b 0 of B suc h that F b 0 = F b , and arrows b 1 of B suc h that F b 1 = 1 b . Finally K is the natural inclusion. Diagram (1.2) ab ov e can b e r estricte d to automorphisms groups ov er ﬁxed ob jects, th us giving the exact sequence 1 / / K s ( b, b ) K b,b / / B ( b, b ) F b,b / / C ( F b, F b ) Pr o of. Exactness in K s ( b, b ) for K injectiv e, in B ( b, b ) for K s = F − 1 ( F b ). F urthermore (1.2) gives also an exact sequence of pointed sets, when we apply the isomorphism-classes-functor π 0 : Gp d → Set that sends a group oid in the set of classes of isomorphic ob jects. W e obtain the diagram π 0 K s π 0 K / / π 0 B π 0 F / / π 0 C exact in π 0 B . 1.5 Case study: dimension one 9 Pr o of. Clearly Im ( π 0 K ) ⊆ Ker ( π 0 F ). Supp ose then π 0 F ( { b 0 } ) = { F b 0 } = { F b } , where brack ets means iso-class. Then the hom-set C ( F b, F b 0 ) is nonempt y , containing an element c 1 , say . Star-surjectivity in b 0 implies that there is a an arrow b 1 : b 0 → b 0 suc h that F ( b 1 ) = c 1 − 1 , but this means { b 0 } = { b 0 } . Since F b 0 = F b the pro of is complete. Finally w e deﬁne a morphism of p ointed sets δ : C ( F b, F b ) / / π 0 K s in the following w a y: given the arrow c 1 : F b → F b star-surjectivit y yields a b 1 : b → b 0 suc h that F b 1 = c 1 . Then we let δ ( c 1 ) = { b 0 } . Clearly this map is w ell deﬁned, since for a diﬀeren t lifting of c 1 , its co domain is isomorphic to b 0 . Moreov er it is ob viously p oin ted by the identit y . No w the new sequence connected by δ is ev erywhere exact 1 / / K s ( b, b ) K b,b / / B ( b, b ) F b,b / / C ( F b, F b ) δ / / π 0 K s π 0 K / / π 0 B π 0 F / / π 0 C (1.3) where the last three terms are p oin ted sets, the other are groups. Pr o of. It remains to prov e the exactness in C ( F b, F b ) and in π 0 K s . As for the ﬁrst, let b 1 : b → b b e given. Then among the liftings of F b 1 there is b 1 itself, hence δ ( F b 1 ) = { b } . Conv ersely let c 1 : F b → F b in the k ernel of δ . This means that for a lifting b 1 : b → b 0 of c 1 , { b 0 } = { b } in the ﬁb er, i.e. there is a b 0 1 : b 0 → b suc h that F ( b 0 1 ) = 1 b . Then F ( b 1 ◦ b 0 1 ) = F ( b 1 ) ◦ F ( b 0 1 ) = F ( b 1 ) ◦ 1 b = F ( b 1 ) = c 1 , with b 1 ◦ b 0 1 ∈ B ( b, b ). F or the second, let a c 1 : F b → F b b e giv en. Then δ ( c 1 ) = { b 0 } for a lifting b 1 : b → b 0 of c 1 . Hence { b 0 } = { b } in π 0 B . Conv ersely let { b 0 } in the k ernel of π 0 K . This means { b 0 } = { b } in π 0 B , i.e. there exists a b 1 : b → b 0 in B , whic h implies δ ( F b 1 ) = { b 0 } . A ﬁrst attempt in extending this to higher dimensional structures has b een done b y Hardie, Kamps and Kiebo om in [ HKK02 ], where they obtain a similar result for ﬁbrations of bigroup oids. Nev ertheless the necessit y to consider only ﬁbrations is not just a limitation in the c hoice of morphisms, but it in tro duces also serious diﬃculties in trying to further extend the result to n -group oids. On this lines Duskin, Kieb o om, and Vitale prop osed in [ DKV04 ] the dif- feren t setting given by considering homotopy k ernels instead of kernels. Consequen tly a new notion of exactness was introduced. 1.5 Case study: dimension one 10 1.5.2 Bro wn’s result revisited In order to fully understand the generalization of [ DKV04 ], we start by con- sidering the one dimensional construction w.r.t. homotopy kernels instead of strict kernels. Notice that here we provide only constructions since pro ofs are consequence of our general result. Let us consider a functor F : B → C b et ween tw o group oids. W e deﬁne the h -ﬁb er o ver a chosen ob ject F b of C K K / / [ F b ] B F / / C ϕ   where • [ F b ] is the constan t functor; • K is the comma-group oid with ob jects the pairs ( b 0 , c 1 : F b / / F b 0 ). An arro w ( b 0 , c 1 ) → ( b 0 0 , c 0 1 ) is a pair ( b 1 , =) where b 1 : b 0 , b 0 0 , and the “=” sta ys for the equality c 1 = F b 1 ◦ c 0 1 ; • K : K → B is the faithful functor deﬁned by K  ( b 0 , c 1 )  = b 0 , K  ( b 1 , =)  = b 1 ; • ϕ : [ F b ] + 3 K F is the natural isomorphism with comp onen ts ϕ ( b 0 ,c 1 ) = c 1 : F b → F b 0 . Again w e can get an exact sequence of groups and p ointed sets 1 / / K ( b, b ) K b,b / / B ( b, b ) F b,b / / C ( F b, F b ) δ / / π 0 K π 0 K / / π 0 B π 0 F / / π 0 C (1.4) where the connecting map δ is deﬁned in a natural wa y δ ( c 1 : F b / / F b ) = ( b, c 1 ) No w Brown’s result can b e seen as a corollary , and this giv es a conceptual insigh t ab out the relation b etw een these tw o diﬀeren t settings. In fact, there exists a fully faithful functor I b : K s → K ; this is giv en explicitly b y letting I b ( b 0 ) = ( b 0 , 1 F b ), indeed it is pro vided by the universal prop erty deﬁning the homotop y kernel as a comparison functor. Clearly F is a ﬁbration of group oids if, and only if, for each ob ject b of B the functor I b is essentially surjective on ob jects. Then when F is a ﬁbration, one can replace K ( b, b ) and π 0 K b y K s ( b, b ) and π 0 K s , and obtain Brown’s exact sequence (1.3) from (1.4). 1.6 Case study: dimension two 11 1.6 Case study: dimension t w o In [ DKV04 ] the authors prov e a similar result for morphisms of 2-group oids, i.e. weakly in v ertible strict 2-categories, and they claim that it easily extends to bi-group oids. In the present work w e will keep close to the ﬁrst setting in order to generalize it to n -group oids (w eakly inv ertible strict n -categories). 1.6.1 Homotop y ﬁbers Let us supp ose w e are given a morphism (2-functor) of 2-group oids F : B → C that is a 2-functor b etw een t w o 2-categories in which ev ery arro w is an equiv alence and ev ery 2-cell is an isomorphism. If w e ﬁx an ob ject b of B , the homotop y ﬁb er F = F F,F b of F ov er F b is the follo wing 2-group oid: • ob jects are pairs ( b 0 , F b 0 c 1 / / F b ); • an arro w ( b 0 , c 1 ) → ( b 0 0 , c 0 1 ) is a pair ( b 1 , c 2 ) as in the diagram b elo w b 0 b 1 / / b 0 0 F b 0 F b 1 / / c 1   4 4 4 4 4 4 4 4 4 4 4 4 4 4 F b 0 0 c 0 1                F b c 2 3 ; o o o o o o o o o o • a 2-cell ( b 1 , c 2 ) ⇒ ( b 0 1 , c 0 2 ) is a pair ( b 2 , ≡ ) as in the diagram b elo w F b 1 F b 2 + 3 F b 0 1 F b 1 ◦ c 0 1 F b 2 ◦ id c 0 1 + 3 F b 0 1 ◦ c 0 1 c 1 c 0 2 ? G                             c 2 W _ 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1.6 Case study: dimension two 12 The homotop y ﬁb er comes with a 2-functor that embeds it in B K : F F,F b → B whic h sends the 2-cell ( b 2 , ≡ ) : ( b 1 , c 2 ) ⇒ ( b 0 1 , c 0 2 ) : ( b 0 , c 1 ) → ( b 0 0 , c 0 1 ) to b 2 : b 1 ⇒ b 0 1 . 1.6.2 2 -Exact sequences Before going further with our description, we urge to introduce a notion of exactness suitable for a 2-dimensional con text. A notion of 2-exactness has b een introduced by Vitale in [ Vit02 ], in order to study some classical exact sequences of ab elian groups asso ciated with a morphism of commutativ e unital rings from sequences of p ointed group oids and categorical groups. W e rep ort the deﬁnition in the con text of p ointed group oids, as for categorical groups it applies plainly with no c hanges 2 . Let us consider the 2-category of p ointed group oids Gp d ∗ , where the mor- phisms are functors that preserves the base p oints, 2-morphisms are natural isomorphisms whose comp onent at the base p oint is the identit y on the p oin t. F or a given morphism F B → C , we deﬁne its h -k ernel as the triple ( K , K : K → B , ϕ : [ ∗ ] ⇒ K F ) ([ ∗ ] denoting the constant 0-functor) satisfying the follo wing universal prop erty Univ ersal Prop ert y 1.1 ( h -k ernels) . F or any other triple ( K 0 , K 0 , ϕ 0 ) ther e exists a unique L : K 0 → K such that K 0 = LK . K K % % L L L L L L L L L L L L [ ∗ ]   B F / / C K 0 L D D K 0 9 9 s s s s s s s s s s s [ ∗ ] F F ϕ             ϕ 0 N V & & & & & & & & & & , , , , , , , , , , , , This univ ersal prop erty deﬁnes the ( h -)kernel up to isomorphism. 2 As a guiding analogy , do consider that exactness in the category of groups may b e deﬁned on the underlying p ointed sets. 1.6 Case study: dimension two 13 R emark 1.2 . Let us notice that last univ ersal prop erty uses only whiskerings of morphisms with a 2-morphism, and do es not use the full horizon tal comp osition of 2-morphisms a v ailable in a 2-category . Hence a step forward to wards a full generalization of Brown’s result to weak n -structures can b e accomplished b y dev eloping a theory that deals with these 1 . 5-univ ersal prop erties (a.k.a. sesqui -universal, a.k.a. h -univ ersal). R emark 1.3 . In dimension 1, our h -k ernel satisﬁes also a universal prop ert y of a bi-limit, and this is indeed a p oint of view well considered in [ DKV04 ]. Nev ertheless, last R emark motiv ate the c hoice to restrict our attention to the h -limits considered. Finally w e are able to give the following Deﬁnition 1.4. A triple ( E , ϕ, F ) A E / / [ ∗ ] B F / / C ϕ   in Gp d ∗ is c al le d exact if the c omp arison with the 2-kernel is ful l and essential ly surje ctive on obje cts. Let us notice that in the ab ov e deﬁnition the 2-kernel can b e replaced b y the h -k ernel, since fullness and essential surjectivit y are preserv ed by equiv alences. W e leav e to the conscious reader the deep ening of the theory of such exactness for the 2-dimensional context of p oin ted group oids and categorical groups, w.r.t. cohomology , extensions, chain-complexes, ([ Vit02 , DKV04 , BV02 , Rou03, KV00, CGV06, KMV06, dRMMV05, GdR06, GDR05, GIdR04]). 1.6.3 Lo w ering the dimension: ﬁrst step Bac k to [ DKV04 ], let us consider a morphism of 2-group oids F : B → C and its homotopy kernel K → B . They deﬁne indeed an exact sequence in the sesqui-category 2 Gp d ∗ , but this p oin t of view is not analyzed explicitly in [DKV04]. Instead the authors consider the diagram K K / / B F / / C and they apply to that hom-of-the-p oint functor [ − ] 1 ( ∗ , ∗ ) : 2 Gp d ∗ → Gp d ∗ the classifying functor C ` : 2 Gp d ∗ → Gp d ∗ . The ﬁrst yields a 2-exact sequence K 1 ( ∗ , ∗ ) K ∗ , ∗ 1 / / B 1 ( ∗ , ∗ ) F ∗ , ∗ 1 / / C 1 ( ∗ , ∗ ) 1.6 Case study: dimension two 14 The second assigns to a 2-groupoid, the group oid with the same set of ob jects, and whose arrows are 2-isomorphism classes of arrows [ B ´ en67 ], th us pro viding the 2-exact sequence C ` K C `K / / C ` B C `F / / C ` C Moreo ver it is p ossible to deﬁne a connecting functor δ : C 1 ( ∗ , ∗ ) → C ` K suc h that the 6-term sequence obtained this wa y is ev erywhere 2-exact. The functor δ is deﬁned as follows: • ( on obje cts ) given an ob ject c 1 : ∗ → ∗ in the domain, δ ( c 1 ) = ( ∗ , c 1 ); • ( on arr ows ) given an arrow c 2 : c 1 ⇒ c 0 1 in the domain, δ ( c 2 ) = { (1 ∗ , c 2 ) } , where brac kets denote C ` asses (it is well deﬁned). 1.6.4 Lo w ering the dimension: second step The 6-term exact sequence K 1 ( ∗ , ∗ ) K ∗ , ∗ 1 / / B 1 ( ∗ , ∗ ) F ∗ , ∗ 1 / / C 1 ( ∗ , ∗ ) δ / / C ` K C `K / / C ` B C `F / / C ` C with ob vious transformations is suc h that the left-most three terms underly a strict monoidal structure given by the (former) 0-comp osition. Moreov er, since we started with (weakly) inv ertible strict 2-categories, they are indeed categorical groups. No w, if we denote b y π 1 the functor Gp d ∗ → Set ∗ that assigns to a p ointed group oid the p oin ted set of the isomorphism classes of its ob jects, it is p ossible to show that it preserv es exactness, i.e. it sends 2-exact sequences of p oin ted group oids (categorical groups), to exact-sequences of p ointed sets (groups). The same can b e said of the functor π 0 . Moreo ver π 1 ( C ` ( − )) = π 0 ([ − ] 1 ( ∗ , ∗ )), hence we get a 9-term exact se- quence π 1 ( K 1 ( ∗ , ∗ )) / / π 1 ( B 1 ( ∗ , ∗ )) / / π 1 ( C 1 ( ∗ , ∗ )) q q d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d π 1 ( C ` K ) = π 0 ( K 1 ( ∗ , ∗ )) / / π 1 ( C ` B ) = π 0 ( B 1 ( ∗ , ∗ )) / / π 1 ( C ` C ) = π 0 ( C 1 ( ∗ , ∗ )) q q d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d π 0 ( C ` K ) / / π 0 ( C ` B ) / / π 0 ( C ` C ) 1.6 Case study: dimension two 15 where the three left-most terms are ab elian groups, the three cen tral terms are groups, the three right-most terms are p oin ted sets. The reason wh y the three leftmost terms are ab elian follo ws from a general fact of strict n -categories for homs o v er an identit y cell (see [ Sim98 ]), that is another variazione on the classical Eckmann-Hilton argument. Chapter 2 Basics on sesqui-categories 2.1 Sesqui-categories The notion of sesqui-category is due to Ross Street [ Str96 ]. The term sesqui comes from the latin semis-que , that means (one and) a half. Hence a sesqui-category is something in-b etw een a category and a 2-category . More precisely Deﬁnition 2.1. A sesqui-c ate gory C is a c ate gory bC c with a lifting of the hom-functor to Cat , such that the fol lowing diagr am of c ate gories and functors c ommutes, ob j b eing the functor that for gets the morphisms: Cat ob j   bC c op × bC c C ( − , − ) 8 8 p p p p p p p p p p p p p p p p p p p p bC c ( − , − ) / / Set (2.1) Obje cts and morphisms of bC c ar e also obje cts and 1-c el ls of C , while mor- phisms of C ( A, B ) ’s (with A and B running in ob j ( bC c ) ) ar e the 2-c el ls of C . W e ﬁrst observe that the deﬁnition ab o v e induces a 2-graph structure on C , whose underlying graph underlies the category C . Besides, the functor C ( − , − ) pro vides hom-sets of the category bC c with a category structure, whose comp osition is termed vertic al c omp osition (or 1-comp osition) of 2- cells. Finally , condition expressed by diagram (2.1) on the lifting C ( − , − ) giv es a reduced horizontal comp osition, or whiskering (or 0-comp osition), compatible with 1-cell comp osition and with the 2-graph structure of C . In fact, for A 0 a / / A and B b / / B 0 in bC c , the functor C ( a, b ) : C ( A, B ) / / C ( A 0 , B 0 ) 2.1 Sesqui-c ate gories 17 giv es explicitly such a comp osition: for a 2-cell α : f + 3 g : A / / B , it whisk ers the diagram A 0 a / / A f   g @ @ B b / / B 0 α   to get the 2-cell A 0 a • f • b ' ' a • g • b 7 7 B 0 a • α • b   where a • α • b is just a concise form for C ( a, b )( α ). By functorialit y of whiskering, the op eration may also b e giv en in a left-and- right fashion. In fact it suﬃces to identify a • L α = a • α • 1 B , α • R b = 1 A • α • b This fact can b e made precise, and giv es a more tractable deﬁnition, b y the follo wing characterization (see, for example [Gra94, Ste94]): Prop osition 2.2. L et C b e a r eﬂexive 2-gr aph C 2 s / / t / / C 1 e o o s / / t / / C 0 e o o whose underlying gr aph bC c = C 1 s / / t / / C 0 e o o has a c ate gory structur e. Then C is a sesqui-c ate gory pr e cisely when the fol lowing c onditions hold: 1. for every p air of obje cts A, B of C 0 , the gr aph C ( A, B ) has a c ate gory structur e, c al le d the hom-c ate gory of A, B . 2. (p artial) r e duc e d horizontal c omp ositions ar e deﬁne d, i.e. for every A 0 , A, B and B 0 obje cts of C 0 , c omp osition in bC c extends to binary op er ations • L : : bC c ( A 0 , A ) × C ( A, B ) / / C ( A 0 , B ) (2.2) • R : C ( A, B ) × bC c ( B , B 0 ) / / C ( A, B 0 ) , (2.3) that satisfy e quations b elow, whenever the c omp osites ar e deﬁne d: A 00 a 0 / / A 0 a / / A f   g @ @ B b / / B 0 b 0 / / A 00 α   2.1 Sesqui-c ate gories 18 ( L 1) 1 A • L α = α ( R 1) α • R 1 B = α ( L 2) a 0 a • L α = a 0 • L ( a • L α ) ( R 2) α • R bb 0 = ( α • R b ) • R b 0 ( L 3) a • L 1 f = 1 af ( R 3) 1 f • R b = 1 f b ( L 4) a • L ( α · β ) = ( a • L α ) · ( a • L β ) ( R 4) ( α · β ) • R b = ( α • R b ) · ( β • R b ) ( LR 5) ( a • L α ) • R b = a • L ( α • R b ) In these e quations, 1 A and 1 B ar e identity 1-c el ls, while 1 f , 1 af and 1 f b ar e identity 2-c el ls, and · is the (vertic al) c omp osition inside the hom-c ate gories. Axiom (LR5) wil l b e also c al le d whiskering axiom . Pr o of. Let C b e a sesqui-category . The fact that C ( A, B ) are categories is clear from the deﬁnition, hence 1 is satisﬁed. Now, deﬁne for chosen A, A 0 , B and B 0 − 1 • L − 2 = C ( − 1 , 1 B )( − 2 ) − 1 • R − 2 = C (1 A , − 2 )( − 1 ) Then for reduced left comp osition axioms, w e hav e: (L1) 1 A • L α = C (1 A , 1 B )( α ) = α b y functoriality w.r.t. units of C ( − , − ). (L2) ( a 0 a ) • L α = C ( a 0 a, 1 B )( α ) = C (( a 0 , 1 B )( a, 1 B ))( α ) = C ( a 0 , 1 B )( C ( a, 1 B )( α )) b y functoriality w.r.t. comp osition of C ( − , − ). Notice the contra v ariance on the ﬁrst comp onen t. (L3) a • L 1 f = C ( a, 1 B )(1 f ) = 1 C ( a, 1 B )( f ) = 1 af b y functoriality w.r.t. units of C ( a, 1 B ). (L4) a • L ( α · β ) = C ( a, 1 B )( α · β ) = C ( a, 1 B )( α ) · C ( a, 1 B )( β ) = ( a • L α ) · ( a • L β ) b y functoriality w.r.t. comp osition of C ( a, 1 B ). Analogous pro ofs hold for (R1) to (R4). Finally (LR5) ( a • L α ) • R b = C (1 A 0 , b )( C ( a, 1 B )( α )) = C ((1 A 0 , b )( a, 1 B ))( α ) = C ( a, b )( α ) = 2.1 Sesqui-c ate gories 19 = C (( a, 1 B 0 )(1 A , b ))( α ) = C ( a, 1 B 0 )( C (1 A , b )( α )) = a • L ( α • R b ) b y functoriality w.r.t. comp osition of C ( − , − ). Con versely , supposing w e are giv en a reﬂexive 2-graph C underlying a category bC c , hom-categories (1) and left/righ t-comp ositions satisfying conditions ab o ve (2). W e sho w C is a sesqui-category . T o this end w e deﬁne a functor bC c op × bC c C ( − , − ) / / Cat On ob jects, this is giv en by condition (1); on arrows, for ( a, b ) : ( A, B ) / / ( A 0 , B 0 ) ( i.e. a : A 0 / / A and b : B / / B 0 ), equation (2.5) helps us to deﬁne C ( a, b ) : C ( A, B ) / / C ( A 0 , B 0 ) b y C ( a, b )( − ) = a • L − • R b These assignmen ts give indeed functors and are functorial. (i) C ( a, b ) is functor w.r.t. units C ( a, b )(1 f ) = a • L 1 f • R b = 1 af b = 1 C ( a,b )( f ) b y (L3) and (R3) (ii) C ( a, b ) is functor w.r.t. comp osition C ( a, b )( α · β ) = a • L ( α · β ) • R b = (( a • L α ) · ( a • L β )) • R b = = ( a • L α • R b ) · ( a • L β • R b ) = C ( a, b )( α ) · C ( a, b )( β ) b y (L4) and (R4). (iii) C ( − , − ) is functor w.r.t. units C (1 A , 1 B )( α ) = 1 A • L α • R 1 B = α b y (L1) and (R1). (vi) C ( − , − ) is functor w.r.t. comp osition C ( a 0 a, bb 0 )( α ) = ( a 0 a ) • L α • R ( bb 0 ) = (( a 0 • L ( a • L α )) • R b ) • R b 0 = = ( a 0 • L C ( a, b )( α )) • R b 0 = C ( a 0 , b 0 )( C ( a, b )( α )) b y (L2) and (R2). That C ( − , − ) mak es (2.1) commute is immediate from its deﬁnition. 2.2 Morphisms of sesqui-c ate gories 20 Notice that reduced horizon tal left/right comp osition will b e often denoted simply b y • , when this do es not cause ambiguit y . 2.2 Morphisms of sesqui-categories Morphisms b et ween sesqui-categories are termed sesqui-functors. More precisely a sesqui-functor F : C / / D is a 2-graph morphism suc h that • bF c : bC c / / bD c is a functor, • for ev ery A, B in C 0 , F A,B : C ( A, B ) / / D ( F ( A ) , F ( B )) are functors comp onen t of a natural transformation F bC c op × bC c C ( − , − ) & & M M M M M M M M M M bF c op ×bF c   Cat bD c op × bD c D ( − , − ) 8 8 q q q q q q q q q q F             (2.4) that lifts b F c : bC c ( − , − ) + 3 ( bF c op × bF c ) · bD c ( − , − ). R emark 2.3 . Notice that every functor b etw een categories gives rise to such a natural transformation as b F c for bF c . F rom this p oint of view, the last condition ma y b e re-formulated saying that a sesqui-functor is the lifting of a functor b etwe en the underlying c ate gories . W e can translate the deﬁnition of sesqui-functor in terms of left/right com- p ositions: Prop osition 2.4. L et C and D b e sesqui-c ate gories, and let F : C / / D b e a 2-gr aphs homomorphism, whose underlying gr aph homomorphism bF c : bC c / / bD c is a functor. Then F is a sesqui-functor pr e cisely w hen the fol lowing c onditions hold: 1. for every p air of obje cts A, B of C 0 , the gr aph homomorphism F A,B : C ( A, B ) / / D ( F ( A ) , F ( B )) is a functor, c al le d the hom-functor at A, B . 2.2 Morphisms of sesqui-c ate gories 21 2. (p artial) horizontal r e duc e d c omp ositions ar e pr eserve d, i.e. for every diagr am A 0 a / / A f   g @ @ B b / / B 0 α   in C 0 , e quations b elow hold: ( L 6) F ( a • L α ) = F ( a ) • L C ( α ) ( R 6) F ( α • R b ) = F ( α ) • R F ( b ) Pr o of. Let F : C / / D b e a sesqui-functor. Then F is a fortiori a homo- morphism of 2-graphs, underlying a functor bF c . F urthermore, for ev ery c hoice of A and B in C 0 , the F A,B are functors to o. What remains to prov e is that F preserv es left/right-compositions in the sense of (L6) and (R6), and this follo ws easily from naturality of F . In fact, for (L6) F ( a • L α ) = F ( C ( a, 1 B )( α )) b y deﬁnition = D ( F ( a ) , F (1 B ))( F ( α )) by naturality = D ( F ( a ) , 1 F ( B ) )( F ( α )) by functoriality = F ( a ) • L F ( α ) by deﬁnition (R6) from a similar calculation. Con versely , supp ose we are giv en tw o sesqui-categories C and D , together with a 2-graph homomorphism F satisfying conditions (1) and (2) ab o ve. W e will pro ve naturality of F , i.e. for ev ery A 0 a / / A and B b / / B 0 in C , the follo wing is a commutativ e diagram in Cat : C ( A, B ) C ( a,b ) / / F A,B   C ( A 0 , B 0 ) F A 0 ,B 0   D ( F ( A ) , F ( B )) D ( F ( a ) , F ( b )) / / D ( F ( A 0 ) , F ( B 0 )) That this diagram comm utes on ob jects ( i.e. on 1-cells of C and D ) is clear from the fact that bF c is a functor and that left/right-compositions extend 1-cell-comp ositions. Finally , for a 2-cell α as ab ov e, F ( C ( a, b )( α )) = F ( a • L α • R b ) = F ( a • L α ) • R F ( b ) = F ( a ) • L F ( α ) • R F ( b ) = D ( F ( a ) , F ( b ))( F ( α )) follo ws from (L6) and (R6). 2.3 2-Natur al tr ansformation of sesqui-functors 22 R emark 2.5 . In the follo wing w e will need the notion of 2 -c ontr avariant sesqui-functor . This is simply a sesqui-functor as ab o ve, such that the functors comp onen t F A,B are usual con trav ariant functors. Of course, c haracterization ab ov e still holds, mutatis mutanda : e.g. if α : f ⇒ g : A → B , then F ( α ) : F ( g ) ⇒ F ( f ) : F ( A ) → F ( B ) . 2.3 2-Natural transformation of sesqui-functors Deﬁnition 2.6 (strict sesqui-transformations) . L et two p ar al lel sesqui- functors F , G : C → D b e given, and let b e given a 2-gr aph tr ansformation ∆ : F ⇒ G whose underlying 1-tr ansformation b ∆ c : bF c ⇒ bG c is a natur al tr ansformation of functors. Then ∆ is a (strict) natur al tr ans- formation of sesqui-functors when, for every α : f ⇒ g : A → B in C , F ( α ) • R ∆ B = ∆ A • L G ( α ) F ( A ) ∆ A / / F ( f )   F ( g )   G ( A ) G ( f )   G ( g )   F ( B ) ∆ B / / G ( B ) F ( α ) + 3 G ( α ) + 3 Notice that while vertical comp osition of (strict) natural transformation of sesqui-functors can b e easily deﬁned, the same is not true for horizontal com- p osition. Therefore the category Sesqui CA T of sesqui-categories, regardless of size issues, is indeed a sesqui-category itself. The notion of (strict) natural transformation of sesqui-functors is essentially of a categorical nature. Namely the “functor” b−c : Sesqui CA T → CA T is also a “sesqui-functor”, when w e consider the 2-category CA T as a sesqui- category . 2.3 2-Natur al tr ansformation of sesqui-functors 23 Therefore those are just usual natural transformations that b eha v e nice with resp ect to reduced left and righ t comp ositions. F or the same reason the notions of adjunction and equiv alence of sesqui-categories (w.r.t. strict trans- formations) are straightforw ard generalization of their categorical analogues. Generalizing sesqui-categories (Chapter 5) we will need a further notion of sesqui-transformation, whose deﬁnition follo ws Deﬁnition 2.7 (lax sesqui-transformations) . L et two p ar al lel sesqui-functors F , G : C → D b e given, and let b e given a 2-gr aph tr ansformation Γ : F ⇒ G . Then a lax natur al tr ansformation Γ : G ⇒ G is given by the fol lowing data: • F or every obje ct A of C , an arr ow Γ A : F ( A ) → G ( A ) • ( naturalit y w.r.t. 1-cells) F or every arr ow f : A → B of C , a 2-c el l Γ f : Γ A • G ( f ) ⇒ F ( f ) • Γ B F ( A ) Γ A / / F ( f )   G ( A ) G ( f )   Γ f w  v v v v v v v v v v v v v v v v v v v v F ( B ) Γ B / / G ( B ) • (naturalit y w.r.t. 2-cells) F or every 2-c el l α : f ⇒ g : A → B in C , an e quation Γ A • G ( f ) Γ A • L G ( α ) + 3 Γ f   Γ A • G ( g ) Γ g   F ( f ) • Γ B F ( α ) • R + 3 F ( g ) • Γ B Those data have to satisfy the fol lowing functoriality axioms: 2.4 Sesqui-c ate gories and 2-c ate gories 24 • F or every obje ct A of C F ( A ) Γ A / / F (1 A )   G ( A ) G (1 A )   Γ 1 A x  y y y y y y y y y y y y y y y y y y F ( A ) Γ A / / G ( A ) = F ( A ) Γ A / / 1 F ( A ) G ( A ) 1 G ( A ) 1 Γ A y y y y y y y y y y y y y y y y y y y y F ( A ) Γ A / / G ( A ) • F or every c omp osable p air A f / / B h / / C in C F ( A ) Γ A / / F ( f )   G ( A ) G ( f )   Γ f s { o o o o o o o o o o o o o o o o F ( B ) Γ B / / F ( h )   G ( B ) G ( h )   Γ h s { o o o o o o o o o o o o o o o o F ( C ) Γ C / / G ( C ) = F ( A ) Γ A / / F ( f h )   G ( A ) G ( f h )   Γ f h {                      F ( C ) Γ C / / G ( C ) R emark 2.8 . In general, a lax sesqui-transformation is not a natural trans- formation of the functors underlying domain and co-domain sesqui-functors. 2.4 Sesqui-categories and 2-categories That a sesqui-category induces a category structure on the underlying graph is clear from the v ery deﬁnition of sesqui-categories. Hence, the question that naturally arises concerns when a sesqui-category is also a 2-category . In fact, given a sesqui-category C , this underlies a 2-category precisely when, for ev ery diagram of the kind • f   g B B • h   k B B • α   β   the follo wing equation is satisﬁed: ( f • L β ) · ( α • R k ) = ( α • R h ) · ( g • L β ) (2.5) In this situation, the t wo composites are denoted α • β , and termed horizontal c omp osition of α and β . 2.5 Finite pr o ducts in a sesqui-c ate gory 25 In other terms, it is p ossible to show that suc h a comp osition deﬁnes a family of functors C ( A, B ) × C ( B , C ) • A,B ,C / / C ( A, C ) indexed b y triples ( A, B , C ) of ob jects of C , satisfying 2-categorical axioms. There follows an interc hange la w for horizontal and vertical comp osition holds: for every four 2-cells α, β , γ , δ •   / / F F •   / / F F • α   γ   β   δ   ( α · γ ) • ( β · δ ) = ( α • β ) · ( γ • δ ) Ev en when equation (2.5) do es not hold, some pasting op erations of 2-cells are still a v ailable. W e sho w this with an example. Consider the diagram: • a / / d   • b / / g   • c   • e / / • f / / • α {  ~ ~ ~ ~ ~ ~ ~ ~ β {  ~ ~ ~ ~ ~ ~ ~ ~ Since intersection b etw een α and β is one dimensional, it is unambiguous to deﬁne the p asting ( α | β ) = ( a • β ) · ( α • f ) In the presen t work, we will not go any further into this sub ject. 2.5 Finite pro ducts in a sesqui-category In the sesqui-categorical context we will refer to binary pro ducts according to the follo wing 2-dimensional universal prop erty Deﬁnition 2.9. L et C b e a sesqui-c ate gory, A and B two obje cts of C . A pr o duct of A and B is a triple ( A × B , π A : A × B → A, π B : A × B → B ) satisfying the fol lowing Universal pr op erty F or every obje ct Q of C and 2-c el ls α : a + 3 a 0 : Q / / A , β : b + 3 b 0 : Q / / B ther e exists a unique 2-c el l γ : q + 3 q 0 : Q / / A × B with γ • π A = α and γ • π B = β . We wil l write γ = h α, β i 2.6 Pr o duct inter change rules 26 The situation ma y b e visualized on the diagram b elow Q             A A × B π A o o π B / / B α  ' F F F F F F γ + 3 β x  y y y y y y Suc h a pro duct satisﬁes also the universal prop erty deﬁning categorical pro ducts. It suﬃces to choose α = 1 a and β = 1 b : the unique γ of the prop ert y satisﬁes γ • π A = 1 a and γ • π B = 1 b . Hence, taking domains and co domains, w e get q π A = a, q π B = b, q 0 π A = a, q 0 π B = b This giv es 1 q • π A = 1 q π A b y axiom (R3) = 1 a 1 q • π B = 1 q π B b y axiom (L3) = 1 b Finally , uniqueness forces γ = 1 q , and in turn, there exists a unique q (= q 0 ) suc h that q π A = a and q π B = b . Deﬁnition 2.10. L et C b e a sesqui-c ate gory. A terminal obje ct is an obje ct I of C satisfying the fol lowing universal pr op erty (UP) for every other obje ct X of C , ther e exists a unique 2-c el l ξ : x ⇒ x 0 : X → I With a calculation similar to that of pro ducts, this universal prop ert y is equiv alen t to the existence of a unique ! X : X → I , henceforth ξ is indeed the iden tity 2-cell on ! X . Pro ducts and terminals deﬁned this wa y are determined up to isomorphism. F urthermore ﬁnite pro ducts and canonical isomorphisms are deﬁned as in the categorical case. 2.6 Pro duct in terchange rules In the previous section we were concerned with prop erties of pro ducts in a sesqui-category that sp ecialize in classical (viz. categorical) ones. 2.6 Pr o duct inter change rules 27 No w we fo cus our attention on pro ducts of 2-cells. What we recapture is the idea of indep endence of the comp onents of a product, and a sort of comm utativity that arises. Consider the 2-cells α : f ⇒ g : A → B and β : h ⇒ k : C → D in a sesqui-category C . A 2-cell α × β : f × h ⇒ g × k : A × C → B × D is uniquely determined b y the universal prop erty: α × β = h π A • α, π C • β i . Notice that this induces a kind of commutativ e horizontal comp osition of 2-cells, pro vided they are on diﬀerent pro duct-comp onents. In fact, w e need the following Lemma 2.11. F or α and β as ab ove, ((1 A × β ) • ( f × 1 D )) · ((1 A × k ) • ( α × 1 D )) = ((1 A × h ) • ( α × 1 D )) · ((1 A × β ) • ( g × 1 D )) Pr o of. By universalit y of pro ducts, they are b oth equal to α × β , b ecause they hav e the same comp osite with pro jections. In fact we pro ve just the left side, the righ t side b eing analogous. ((1 A × β ) • ( f × 1 D )) · ((1 A × k ) • ( α × 1 D )) • π D = = ((1 A × β ) • ( f × 1 D ) • π D ) · (((1 A × k ) • ( α × 1 D )) • π D ) = ((1 A × β ) • π D ) · ((1 A × k ) • ( α × 1 D ) • π D ) = ((1 A × β ) • π D ) · ((1 A × k ) • π D ) = ( π C • β ) · ( π C k ) = π C • β ((1 A × β ) • ( f × 1 D )) · ((1 A × k ) • ( α × 1 D )) • π B = = ((1 A × β ) • ( f × 1 D ) • π B ) · (((1 A × k ) • ( α × 1 D )) • π B ) = ((1 A × β ) • π A f ) · ((1 A × k ) • ( α × 1 D ) • π B ) = ( π A f ) · ((1 A × k ) π A • α ) = ( π A f ) · ( π A • α ) = π A • α 2.6 Pr o duct inter change rules 28 A × C 1 A × k ' ' 1 A × h   1 A × k                 A × D g × 1 D ) ) f × 1 D   A × D f × 1 D                 B × D α × 1 D z  | | | | | | | | 1 A × β z  | | | | | | | | A × C 1 A × k   1 A × h   A × D g × 1 D   f × 1 D   B × D α × 1 D k s 1 A × β k s A × C 1 A × k   1 A × h w w 1 A × h   < < < < < < < < < < < < < < A × D g × 1 D   < < < < < < < < < < < < < < A × D g × 1 D   f × 1 D w w B × D α × 1 D ] e B B B B B B B B 1 A × β ] e B B B B B B B B A × C 1 A × k   1 A × h y y f × 1 C   g × 1 C % % A × D g × 1 D   < < < < < < < < < < < < < < B × C 1 B × h                 B × D α × 1 C z  | | | | | | | | 1 A × β ] e B B B B B B B B 1 1   m m   Q Q - - M M q q A × C 1 A × k % % 1 A × h   f × 1 C y y g × 1 C   B × C 1 B × k   < < < < < < < < < < < < < < A × D f × 1 D                 B × D α × 1 C ] e B B B B B B B B 1 A × β z  | | | | | | | | A × C f × 1 C w w g × 1 C   f × 1 C   < < < < < < < < < < < < < < B × C 1 B × k   < < < < < < < < < < < < < < B × C 1 B × k   1 B × h u u B × D α × 1 C ] e B B B B B B B B 1 B × β ] e B B B B B B B B A × C g × 1 C   f × 1 C   B × C 1 B × k   1 B × h   B × D α × 1 C k s 1 B × β k s A × C g × 1 C ' ' f × 1 C   g × 1 C                 B × C 1 B × k ) ) 1 B × h   B × C 1 B × h                 B × D α × 1 C z  | | | | | | | | 1 B × β z  | | | | | | | | L emma 2.11 allo ws us to deﬁne a horizontal comp osition of this kind of 2-cells (1 A × β ) • ( α × 1 D ) = α × β = ( α × 1 C ) • (1 B × β ) 2.7 h -Pul lb acks 29 and to pro ve diagram equalities, such as the one ab ov e. These kind of diagrammatic equations will b e called pr o duct inter change rules . 2.7 h -Pullbac ks W e introduce here a notion of standard h -pullback suitable for our purp oses. This notion has b een formalized by Michael Mather in [ Mat76b ], for generic categories of spaces, with (even tually p ointed) top ological spaces in mind. It has b een further generalized to h -categories 1 b y Marco Grandis in [ Gra94 ]. W e (ab)use the term h -pullbac k, instead of that of c omma-squar e b ecause w e will w ork mainly in a n-group oidal con text, with 2-morphisms b eing weakly in vertible. Deﬁnition 2.12. Consider the fol lowing diagr am in a sesqui-c ate gory C C g   A f / / B A n h -pul lb ack of f and g is a four-tuple ( P ( f , g ) , p, q , ε ) P q / / p   C g   A f / / B ε ; C ~ ~ ~ ~ ~ ~ ~ ~ wher e P = P ( f , g ) , that satisﬁes the fol lowing Universal Pr op erty F or any other four-tuple ( X , m, n, λ ) as in X n / / m   C g   A f / / B λ : B } } } } } } } } ther e exists a unique ` : X → P such that 1. `p = m 2. `q = n 1 A h -category is a w eaker notion than that of a sesqui-category , see [Gra94]. 2.7 h -Pul lb acks 30 3. ` • L ε = λ Lemma 2.13. Univ ersal Prop ert y 2.12 deﬁnes h -pul lb acks up to isomor- phisms. Pr o of. Let ( P , p, q , ε ) b e a h -pullbac k, according to deﬁnition ab ov e, and let ( P 0 , p 0 , q 0 , ε 0 ) b e another four-tuple satisfying the universal prop erty . Then, since the ﬁrst is a h -pullbac k, there exists ` : P 0 → P `p = p 0 , `q = q 0 , `ε = ε 0 and, since the second is a h -pullbac k, there exists ` 0 : P → P 0 ` 0 p 0 = p, ` 0 q 0 = q , ` 0 ε 0 = ε No w, applying the universal prop ert y of the ﬁrst one to itself, ` 0 ` and id P satisfy the same equations, and by uniqueness are equal. Similarly applying the universal prop erty of the second one to itself, `` 0 and id P 0 satisfy the same equations, and by uniqueness they are equal to o. Hence ` and ` 0 are isomorphisms. Lemma 2.14 (Pullbac k of h -pro jections.) . In the sesqui-c ate gory C , let b e given the diagr am b elow, wher e the left-hand squar e is c ommutative and the right-hand squar e ε is a h -pul lb ack R s / / r   P q / / p   D g   A e / / B f / / ε : B ~ ~ ~ ~ ~ ~ ~ ~ C then the c omp osition s • L ε is a h -pul lb ack if, and only if, the left hand squar e is a pul lb ack. Pr o of. W e will sho w that the four-tuple ( R, r, sq , s • L ε ) satisﬁes the univ ersal prop ert y 6.1. Let the four-tuple ( X , y , z , ξ ) b e given as in the diagram b elow X z / / y   D g   A ξ 3 ; n n n n n n n n n n n n n n e / / B f / / C Since P is an h -pullback, there exists a unique ` : X → P such that ( i ) `p = y e, ( ii ) `q = z , ( iii ) ` • L ε = ξ . Y et since R is a pullback, condition ( i ) is equiv alent to: 2.7 h -Pul lb acks 31 there exists a unique x : X → R such that ( iv ) xr = y , ( v ) xs = `. Substituting, there exists a unique x : X → R such that ( i ) 0 xr ( iv ) = y ( ii ) 0 xsq ( v ) = `q ( ii ) = z ( iii ) 0 x • L ( s • L ε ) ( L 2) = xs • L ε ( v ) = ` • L ε ( iii ) = ξ R emark 2.15 . This Lemma still holds in a mere h -category ([ Gra94 ] L emma 2.2. ). Chapter 3 Strict n -categories W e giv e an inductiv e deﬁnition of the sesqui-category n Cat , whose ob jects are (strict and small) n -categories, morphisms are (strict) n -functors and 2-morphisms are (lax) n -transformations. F urthermore, n Cat has sesqui- categorical ﬁnite pro ducts. In the next three sections, we recall a standard inductive construction of n Cat , well known in literature, recalled for instance in [ Str87 ]. This is in fact a notion based up on a more general and inﬂuential theory of enrichmen t, dev elop ed by Gregory Maxwell Kelly (see [Kel05]). This new persp ective arises in the sesqui-categorical structure dev elop ed thereafter. More precisely our notion of lax natural n -transformation generalizes the ordinary 2-dimensional version, as recalled for example in [ Bor94 ]. This coincides with the inductiv e deﬁnition given in the in ternal ab elian case in [ Bou90 ]. Moreov er it is equiv alent to the global deﬁnition giv en in [ Cra95 ] (see Deﬁnition 9.1 , L emma 9.2 for a comparison), closely related to that of m -fold homotopies of ω -group oids in [BH87]. F or n = 0, n Cat is Set , the sesqui-category of sets and maps with trivial ( i.e. identit y) transformations. Cartesian product pro vides the required sesqui-categorical pro duct. F or n = 1, the 2-category Cat of categories, functors and natural transfor- mations has a underlying sesqui-categorical structure, when we consider only reduced horizontal comp osition of natural transformations with functors. Categorical pro duct giv es again the required sesqui-categorical pro duct. 3.1 n Cat : the data 33 3.1 n Cat: the data n -categories F or given in teger n > 1, a (strict) n-category C consists of the follo wing data: • a set of ob jects C 0 ; • for ev ery pair of ob jects c 0 , c 0 0 of C 0 , a ( n − 1)category C 1 ( c 0 , c 0 0 ) called hom ( n − 1) c ate gory ov er c 0 and c 0 0 , and sometimes written [ c 0 , c 0 0 ] in order to simplify notation; • for ev ery ob ject c 0 of C 0 , a morphism of ( n − 1)categories I ( n − 1) u 0 ( c 0 ) / / C 1 ( c 0 , c 0 ) called the 0 -identity of c 0 ; • for ev ery triple of ob jects c 0 , c 0 0 , c 00 0 of C 0 , a morphism of ( n − 1)categories C 1 ( c 0 , c 0 0 ) × ( n − 1) C 1 ( c 0 0 , c 00 0 ) ◦ 0 c 0 ,c 0 0 ,c 00 0 / / C 1 ( c 0 , c 00 0 ) called 0 -c omp osition , following the dimension-intersection conv ention. Here, × ( n − 1) sta ys for the binary pro duct in ( n − 1) Cat , while I ( n − 1) is the 0-ary pro duct in ( n − 1) Cat . Subscripts will b e usually omitted, unless this causes confusion. All these data m ust satisfy the following axioms, expressed by commutativ e diagrams in ( n − 1) Cat : • ( asso ciativity axiom ) for ev ery four-tuple of ob jects c 0 , c 0 0 , c 00 0 , c 000 0 of C 0 ( C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 )) × C 1 ( c 00 0 , c 000 0 ) α ∼ / / ◦ 0 c 0 ,c 0 0 ,c 00 0 × id   C 1 ( c 0 , c 0 0 ) × ( C 1 ( c 0 0 , c 00 0 ) × C 1 ( c 00 0 , c 000 0 )) id ×◦ 0 c 0 0 ,c 00 0 .c 000 0   C 1 ( c 0 , c 00 0 ) × C 1 ( c 00 0 , c 000 0 ) ◦ 0 c 0 ,c 00 0 .c 000 0 ) ) R R R R R R R R R R R R R C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 000 0 ) ◦ 0 c 0 ,c 0 0 ,c 000 0 u u l l l l l l l l l l l l l C 1 ( c 0 , c 000 0 ) (3.1) where α = α C 1 ( c 0 ,c 0 0 ) , C 1 ( c 0 0 ,c 00 0 ) , C 1 ( c 00 0 ,c 000 0 ) 3.1 n Cat : the data 34 is the usual asso ciator giv en by universal prop erty of pro duct; • ( left and right unit axioms ) for ev ery pair of ob jects c 0 , c 0 0 of C 0 I × C 1 ( c 0 , c 0 0 ) u 0 ( c 0 ) × id   C 1 ( c 0 , c 0 0 ) λ ∼ o o id ρ ∼ / / C 1 ( c 0 , c 0 0 ) × I id × u 0 ( c 0 )   C 1 ( c 0 , c 0 ) × C 1 ( c 0 , c 0 0 ) ◦ 0 c 0 ,c 0 ,c 0 0 / / C 1 ( c 0 , c 0 0 ) C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 0 0 ) ◦ 0 c 0 ,c 0 0 ,c 0 0 o o (3.2) where λ = λ C 1 ( c 0 ,c 0 0 ) and ρ = ρ C 1 ( c 0 ,c 0 0 ) are the usual left and righ t unit isomorphisms giv en by the univ ersal prop erty of the pro duct. Morphisms of n -categories F or a given integer n > 1, and giv en n -categories C and D , a (strict) n-functor F : C / / D is a pair ( F 0 , F 1 ) where: • F 0 : C 0 / / D 0 is a map; • for ev ery pair of ob jects c 0 , c 0 0 of C 0 F c 0 ,c 0 0 1 : C 1 ( c 0 , c 0 0 ) / / D 1 ( F 0 c 0 , F 0 c 0 0 ) is a morphism of ( n − 1)categories. These data must satisfy the following axioms, expressed by commutativ e diagrams in ( n − 1) Cat : • ( functoriality w.r.t. c omp osition ) for every triple of ob jects c 0 , c 0 0 , c 00 0 of C 0 C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) C ◦ 0 / / F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1   C 1 ( c 0 , c 00 0 ) F c 0 ,c 00 0 1   D 1 ( F 0 c 0 , F 0 c 0 0 ) × D 1 ( F 0 c 0 0 , F 0 c 00 0 ) D ◦ 0 / / D 1 ( F 0 c 0 , F 0 c 00 0 ) (3.3) 3.1 n Cat : the data 35 • ( functoriality w.r.t. units ) for every triple of ob ject c 0 of C 0 I C u 0 ( c 0 ) / / D u 0 ( F 0 c 0 ) ) ) R R R R R R R R R R R R R R R R R C 1 ( c 0 , c 0 ) F c 0 ,c 0 1   D 1 ( F 0 c 0 , F 0 c 0 ) (3.4) 2-Morphisms of n -categories F or a giv en in teger n > 1, and given n -functors F , G : C / / D , a lax natural n -transformation C F & & G 8 8 D α   is a pair α = ( α 0 , α 1 ), where • α 0 : C 0 / / ‘ c 0 ∈ C 0 [ D 1 ( F 0 c 0 , G 0 c 0 )] 0 is a map such that, for every c 0 in C 0 , α 0 ( c 0 ) : F 0 ( c 0 ) / / G 0 ( c 0 ) ; • ( n-natur ality ) for every pair c 0 , c 0 0 of C , α c 0 ,c 0 0 1 is a 2-morphism of ( n − 1) Cat , as in the diagram b elo w C 1 ( c 0 , c 0 0 ) F c 0 ,c 0 0 1 y y s s s s s s s s s s G c 0 ,c 0 0 1 % % K K K K K K K K K K D 1 ( F 0 c 0 , F 0 c 0 0 ) −◦ 0 α 0 c 0 0 % % K K K K K K K K K K D 1 ( G 0 c 0 , G 0 c 0 0 ) α 0 c 0 ◦ 0 − y y s s s s s s s s s s D 1 ( F 0 c 0 , G 0 c 0 0 ) α c 0 ,c 0 0 1 k s In order to k eep notation lighter we will often write α c 0 instead of α 0 ( c 0 ). These data must satisfy functoriality axioms expressed by the follo wing equations of diagrams in ( n − 1) Cat : • ( functoriality w.r.t. c omp osition ) for every triple of ob jects c 0 , c 0 0 , c 00 0 of C 0 , 3.1 n Cat : the data 36 C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) id × F c 0 0 ,c 00 0 1 s s h h h h h h h h h h h h h h h h h h h id × G c 0 0 ,c 00 0 1                 F c 0 ,c 0 0 1 × id 7 7 7 7 7 7 7   7 7 7 7 7 7 7 G c 0 ,c 0 0 1 × id + + V V V V V V V V V V V V V V V V V V V ≡ C 1 ( c 0 , c 0 0 ) × D 1 ( F 0 c 0 0 , F 0 c 00 0 ) id × ( −◦ α 0 c 00 0 )   D 1 ( G 0 c 0 , G 0 c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) ( α 0 c 0 ◦− ) × id   C 1 ( c 0 , c 0 0 ) × D 1 ( G 0 c 0 0 , G 0 c 00 0 ) id × ( α 0 c 0 0 ◦− ) w w n n n n n n n n n n n n D 1 ( F 0 c 0 , F 0 c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) ( −◦ α 0 c 0 0 ) × id ' ' P P P P P P P P P P P P C 1 ( c 0 , c 0 0 ) × D 1 ( F 0 c 0 0 , G 0 c 00 0 ) F c 0 ,c 0 0 1 × id   D 1 ( F 0 c 0 , G 0 c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) id × G c 0 0 ,c 00 0 1   D 1 ( F 0 c 0 , F 0 c 0 0 ) × D 1 ( F 0 c 0 0 , G 0 c 00 0 ) ◦ 0 + + V V V V V V V V V V V V V V V V V V V D 1 ( F 0 c 0 , G 0 c 0 0 ) × D 1 ( G 0 c 0 0 , G 0 c 00 0 ) ◦ 0 s s h h h h h h h h h h h h h h h h h h h D 1 ( F 0 c 0 , G 0 c 00 0 ) id × α c 0 0 ,c 00 0 1 c k P P P P P P P P P P P P α c 0 ,c 0 0 1 × id s { n n n n n n n n n n n n (3.5) = C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) ◦ 0   C 1 ( c 0 , c 00 0 ) F c 0 ,c 00 0 1 w w p p p p p p p p p p p G c 0 ,c 00 0 1 ' ' O O O O O O O O O O O D 1 ( F 0 c 0 , F 0 c 00 0 ) −◦ α 0 c 00 0 ' ' N N N N N N N N N N N D 1 ( G 0 c 0 , G 0 c 00 0 ) α 0 c 0 ◦− w w o o o o o o o o o o o D 1 ( F 0 c 0 , G 0 c 00 0 ) α c 0 ,c 00 0 1 k s • ( functoriality w.r.t. units ) for every ob ject c 0 of C 0 , I u 0 ( c 0 )   C 1 ( c 0 , c 0 ) F c 0 ,c 0 1 y y t t t t t t t t t G c 0 ,c 0 1 % % K K K K K K K K K D 1 ( F 0 c 0 , F 0 c 0 ) −◦ α 0 c 0 % % J J J J J J J J J D 1 ( G 0 c 0 , G 0 c 0 ) α 0 c 0 ◦− y y s s s s s s s s s D 1 ( F 0 c 0 , G 0 c 0 ) α c 0 ,c 0 1 k s = I [ α 0 c 0 ]   [ α 0 c 0 ]   D 1 ( F 0 c 0 , G 0 c 0 ) id k s (3.6) R emark 3.1 . In deﬁning transformations, we used expressions suc h as − ◦ α c 0 0 or α c 0 ◦ − to denote comp osition (n-1)functors. In fact, giv en a n-category C 3.2 n Cat : the underlying c ate gory 37 and a 1-cell c 1 : c 0 → c 0 0 , it is alwa ys p ossible to deﬁne a pair of (n-1)functors for ev ery other chosen ob jects ¯ c 0 of C : [ − ◦ 0 c 1 ] ¯ c 0 : C 1 (¯ c 0 , c 0 ) → C 1 (¯ c 0 , c 0 0 ) [ c 1 ◦ 0 − ] ¯ c 0 : C 1 ( c 0 0 , ¯ c 0 → C 1 ( c 0 , ¯ c 0 ) As a matter of fact, they are just restrictions of 0-comp osition functors in C . The assignmen t is (mutually) natural in ¯ c 0 . In fact, whisk ering makes the following diagram commute, for ¯ c 1 : ¯ ¯ c 0 → ¯ c 0 : C 1 (¯ c 0 , c 0 ) −◦ c 1 / / ¯ c 1 ◦−   C 1 (¯ c 0 , c 0 0 ) ¯ c 1 ◦−   C 1 ( ¯ ¯ c 0 , c 0 ) −◦ c 1 / / C 1 ( ¯ ¯ c 0 , c 0 0 ) A natural n -transformation of n -functors α : F ⇒ G : C → D is called strict when for ev ery pair of ob jects c 0 , c 0 0 of C , α c 0 ,c 0 0 1 is an iden tity . 3.2 n Cat: the underlying category n Cat has the underlying category denoted b n Cat c , whose description fol- lo ws. Notice that in this section w e will assume n b e an integer greater than 1. Giv en n -functors C F / / D G / / E their comp osition F • 0 G (or simply F G ) is the morphism with [ F • 0 G ] 0 = F 0 G 0 in Set , and, for ev ery pair of ob jects c 0 , c 0 0 of C 0 , [ F • 0 G ] c 0 ,c 0 0 1 = F c 0 ,c 0 0 1 • 0 G F 0 c 0 ,F 0 c 0 0 1 in ( n − 1) Cat . The pair ([ F • 0 G ] 0 , [ F • 0 G ] 1 ) deﬁnes indeed a morphism of ( n − 1) Cat . This is clear b y pasting the commutativ e diagrams b elo w, for every triple of ob jects c 0 , c 0 0 , c 00 0 of C 0 . They ensure functorialit y w.r.t. comp osition and units of (3.3) and (3.6). [ c 0 , c 0 0 ] × [ c 0 0 , c 00 0 ] C ◦ 0 / / F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1   [ c 0 , c 00 0 ] F c 0 ,c 00 0 1   [ F c 0 , F c 0 0 ] × [ F c 0 0 , F 0 c 00 0 ] D ◦ 0 / / G F c 0 ,F c 0 0 1 × G F c 0 0 ,F c 00 0 1   [ F c 0 , F 0 c 00 0 ] G F c 0 ,F c 00 0 1   [ G ( F c 0 ) , G ( F c 0 0 )] × [ G ( F c 0 0 ) , G ( F 0 c 00 0 )] E ◦ 0 / / [ G ( F c 0 ) , G ( F 0 c 00 0 )] I C u 0 ( c 0 ) / / D u 0 ( F c 0 ) S S S S S S S S ) ) S S S S S E u 0 ( G ( F c 0 )) # # G G G G G G G G G G G G G G G G G G G G G G G [ c 0 , c 0 ] F c 0 ,c 0 1   [ F c 0 , F c 0 ] G F c 0 ,F c 0 1   [ G ( F c 0 ) , G ( F c 0 )] 3.2 n Cat : the underlying c ate gory 38 F urthermore, for ev ery n -category C , an identit y functor C id C / / C is deﬁned b y the pair ([ id C ] 0 ] , [ id C ] 1 ]), where [ id C ] 0 = id C 0 in Set , and, for ev ery pair of ob jects c 0 , c 0 0 of C [ id C ] c 0 ,c 0 0 1 = id C 1 ( c 0 ,c 0 0 ) in ( n − 1) Cat . Notice that ([ id C ] 0 ] , [ id C ] 1 ]) satisﬁes trivially functoriality diagrams (3.3) and (3.6). Prop osition 3.2. (smal l and strict) n -c ate gories and (strict) n -functors deﬁne a c ate gory: b n Cat c Pr o of. Firstly , for every pair C , D of (small) n -categories, H om ( C , D ) = { n − functors F : C → D } is a set, since D is small. Hence we will show that comp osition is asso ciative and that identities are neutral. Concerning asso ciativit y , let a comp osable triple of morphisms b e giv en: C F / / D G / / E H / / F W e w ant to prov e ( F G ) H = F ( GH ). On ob jects, let us consider the follo wing equalities in the category Set : [( F G ) H ] 0 = [ F G ] 0 H 0 = = ( F 0 G 0 ) H 0 = = F 0 ( G 0 H 0 ) = = F 0 [ GH ] 0 = [ F ( GH )] 0 Besides, for ev ery pair of ob jects c 0 , c 0 0 of C , ( n − 1)-asso ciativity implies: [( F G ) H ] c 0 ,c 0 0 1 = [ F G ] c 0 ,c 0 0 1 H G ( F c 0 ) ,G ( F c 0 0 ) 1 = =  F c 0 ,c 0 0 1 G F c 0 ,F c 0 0 1  H G ( F c 0 ) ,G ( F c 0 0 ) 1 = = F c 0 ,c 0 0 1  G F c 0 ,F c 0 0 1 H G ( F c 0 ) ,G ( F c 0 0 ) 1  = = F c 0 ,c 0 0 1 [ GH ] F c 0 ,F c 0 0 1 = [ F ( GH )] c 0 ,c 0 0 1 3.2 n Cat : the underlying c ate gory 39 T urning no w to identities, let us consider the situation: C id C / / C F / / D id D / / D W e w ant to prov e id C F = F = F id D . On ob jects, [ id C F ] 0 = [ id C ] 0 F 0 = = id C 0 F 0 = = F 0 = F 0 id D 0 = = F 0 [ id D ] 0 = [ F 0 id D ] 0 and, for ev ery pair of ob jects c 0 , c 0 0 of C , neutral ( n − 1)-identities imply: [ id C F ] c 0 ,c 0 0 1 = [ id C ] c 0 ,c 0 0 1 F c 0 ,c 0 0 1 = = id C 1 ( c 0 ,c 0 0 ) F c 0 ,c 0 0 1 = = F c 0 ,c 0 0 1 = F c 0 ,c 0 0 1 id D 1 ( F c 0 ,F c 0 0 ) = = F c 0 ,c 0 0 1 [ id D ] F c 0 ,F c 0 0 1 = [ F 0 id D ] c 0 ,c 0 0 1 Prop osition 3.3. The c ate gory b n Cat c has ﬁnite pr o ducts. Pr o of. W e will sho w that b Cat c has a terminal ob ject and binary pro ducts. Giv en n -categories C and D , their (standard) pro duct is deﬁned as follows: [ C × D ] 0 = C 0 × D 0 and, for ev ery pair ( c 0 , d 0 ), ( c 0 0 , d 0 0 ) in [ C × D ] 0 , [ C × D ] 1  ( c 0 , d 0 ) , ( c 0 0 , d 0 0 )  = C 1 ( c 0 , c 0 0 ) × D 1 ( d 0 , d 0 0 ) Comp osition is deﬁned by means of univ ersalit y of pro ducts in ( n − 1) Cat : for every triple ( c 0 , d 0 ), ( c 0 0 , d 0 0 ) and ( c 00 0 , d 00 0 ) in [ C × D ] 0 , the dotted arrow in the diagram b elo w gives comp osition: [ C × D ] 1 (( c 0 , d 0 ) , ( c 0 0 , d 0 0 )) × [ C × D ] 1 (( c 0 0 , d 0 0 ) , ( c 00 0 , d 00 0 )) id C × D ◦ / / [ C × D ] 1 (( c 0 , d 0 ) , ( c 00 0 , d 00 0 )) id  C 1 ( c 0 , c 0 0 ) × D 1 ( d 0 , d 0 0 )  ×  C 1 ( c 0 0 , c 00 0 ) × D 1 ( d 0 0 , d 00 0 )  τ    C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 )  ×  D 1 ( d 0 , d 0 0 ) × D 1 ( d 0 0 , d 00 0 )  C ◦× D ◦ / / C 1 ( c 0 , c 00 0 ) × D 1 ( d 0 , d 00 0 ) 3.2 n Cat : the underlying c ate gory 40 where the t wist isomorphism τ = τ C 1 ( c 0 ,c 0 0 ) , D 1 ( d 0 ,d 0 0 ) , C 1 ( c 0 0 ,c 00 0 ) , D 1 ( d 0 0 ,d 00 0 ) is giv en b y pro ducts prop erties in ( n − 1) Cat . Iden tities are deﬁned in the same wa y . F or ev ery ob ject ( c 0 , d 0 ) in C × D , b y the dotted arro w in the diagram b elow: I C × D u 0 (( C 0 ,d 0 )) / / iso   [ C × D ] 1 (( c 0 , d 0 ) , ( c 0 , d 0 )) id I × I C u 0 ( c 0 ) × D u 0 ( d 0 ) / / C 1 ( c 0 , c 0 ) × D 1 ( d 0 , d 0 ) Pro duct pro jections C C × D Π C o o Π D / / D are giv en resp ectively by pro jections [Π C ] 0 = π C 0 × D 0 C 0 , [Π D ] 0 = π C 0 × D 0 D 0 and b y the following comp ositions in ( n − 1) Cat , for ev ery pair ( c 0 , d 0 ) and ( c 0 0 , d 0 0 ): [ C × D ] 1 (( c 0 , d 0 ) , ( c 0 0 , d 0 0 )) [Π C ] ( c 0 ,d 0 ) , ( c 0 0 ,d 0 0 ) 1 / / id × ! ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q C 1 ( c 0 , c 0 0 ) C 1 ( c 0 , c 0 0 ) × I iso : : u u u u u u u u u , [ C × D ] 1 (( c 0 , d 0 ) , ( c 0 0 , d 0 0 )) [Π D ] ( c 0 ,d 0 ) , ( c 0 0 ,d 0 0 ) 1 / / id × ! ( ( R R R R R R R R R R R R R D 1 ( d 0 , d 0 0 ) D 1 ( d 0 , d 0 0 ) × I iso : : t t t t t t t t t It is a matter of rep eated use of the universal prop ert y of pro ducts in ( n − 1) Cat to prov e that all these data deﬁne a n -category and tw o n - categories morphisms, and that ( C × D , Π C , Π D ) is a pro duct in n Cat . Concerning the terminal n -category , a standard construction follows. A terminal I ( n ) is giv en by the pair [ I ( n ) ] 0 = {∗} , [ I ( n ) ] ∗ , ∗ 0 = I ( n − 1) ; with comp osition I ( n − 1) × I ( n − 1) ∼ / / I ( n − 1) . 3.3 n Cat : the hom-c ate gories 41 Classical constructions of categorical limits help in deﬁning n -ary pro ducts and canonical isomorphisms α A , B , C : ( A × B ) × C ∼ / / A × ( B × C ) (3.7) ρ A : A ∼ / / A × I ( n ) (3.8) λ A : A ∼ / / I ( n ) × A (3.9) τ A , B , C , D : ( A × B ) × ( C × D ) ∼ / / ( A × C ) × ( B × D ) (3.10) 3.3 n Cat: the hom-categories In this section we describ e, hom-categories n Cat ( C , D ), once n -categories C and D are ﬁxed. 3.3.1 V ertical composition Giv en the diagram: C G A A F / / E   D α   ω   ; one deﬁnes a (vertical, or 1-)comp osition ω • 1 α : E + 3 G in the following w ay: • for ev ery ob ject c 0 in C , [ ω α ] 0 ( c 0 ) = ω 0 c 0 ◦ 0 α 0 c 0 : E c 0 / / Gc 0 • for ev ery pair of ob jects c 0 , c 0 0 in C , the diagram b elo w describ es [ ω • 1 α ] c 0 ,c 0 0 1 =  ω c 0 ◦ α c 0 ,c 0 0 1  • 1  ω c 0 ,c 0 0 1 ◦ α c 0 0  3.3 n Cat : the hom-c ate gories 42 C 1 ( c 0 , c 0 0 ) E 1 v v m m m m m m m m m m m m F 1   G 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q D 1 ( E c 0 , E c 0 0 ) −◦ ω c 0 0   D 1 ( Gc 0 , Gc 0 0 ) αc 0 ◦−   D 1 ( F c 0 , F c 0 0 ) ω c 0 ◦− v v m m m m m m m m m m m m −◦ αc 0 0 ( ( Q Q Q Q Q Q Q Q Q Q Q Q D 1 ( E c 0 , F c 0 0 ) −◦ αc 0 0 ( ( Q Q Q Q Q Q Q Q Q Q Q Q ≡ D 1 ( F c 0 , Gc 0 0 ) ω c 0 ◦− v v m m m m m m m m m m m m D 1 ( E c 0 , Gc 0 0 ) α c 0 ,c 0 0 1 l t b b b b b b b b b b b b b b b b b b b b b b ω c 0 ,c 0 0 1 j r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T o pro ve that these data give indeed a 2-morphism, unit functorialit y (3.6) and comp osition functorialit y (3.5) equations must hold. T o this end, c hose an ob ject c 0 of C and consider the following chain of diagrams equalities: I u ( c 0 )   [ c 0 , c 0 ] E 1 x x p p p p p p p p p p F 1   G 1 & & N N N N N N N N N N [ E c 0 , E c 0 ] −◦ ω c 0   [ Gc 0 , Gc 0 ] αc 0 ◦−   [ F c 0 , F c 0 ] ω c 0 ◦− x x p p p p p p p p p p −◦ αc 0 & & N N N N N N N N N N [ E c 0 , F c 0 ] −◦ αc 0 & & N N N N N N N N N N ≡ [ F c 0 , Gc 0 ] ω c 0 ◦− x x p p p p p p p p p p [ E c 0 , Gc 0 ] α c 0 ,c 0 1 m u c c c c c c c c c c c c c c c c c c ω c 0 ,c 0 1 i q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( i ) = 3.3 n Cat : the hom-c ate gories 43 = I u ( c 0 )   ω c 0                              αc 0   2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 [ F c 0 , F c 0 ] ω c 0 ◦− y y s s s s s s s s s s s s −◦ αc 0 % % K K K K K K K K K K K K id c k P P P P P P P P id s { n n n n n n n n [ E c 0 , F c 0 ] −◦ αc 0 % % K K K K K K K K K K K K ≡ [ F c 0 , Gc 0 ] ω c 0 ◦− y y s s s s s s s s s s s s [ E c 0 , Gc 0 ] ( ii ) = I  [ ω α ] c 0     [ ω α ] c 0    [ E c 0 , Gc 0 ] id k s where ( i ) follows for units functoriality of ω and α , while ( ii ) from functori- alit y of constant functors. This pro ves unit functoriality . Concerning comp osition functoriality , take three ob jects c 0 , c 0 0 and c 00 0 in C , and consider the follo wing diagram: 3.3 n Cat : the hom-c ate gories 44 [ c 0 ,c 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ E c 0 0 ,E c 00 0 ] [ c 0 ,c 0 0 ] × [ F c 0 0 ,F c 00 0 ] [ F c 0 ,F c 0 0 ] × [ c 0 0 ,c 00 0 ] [ Gc 0 ,Gc 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ E c 0 0 ,F c 00 0 ] [ c 0 ,c 0 0 ] × [ F c 0 0 ,Gc 00 0 ] [ E c 0 ,F c 0 0 ] × [ F c 0 0 ,Gc 00 0 ] [ E c 0 ,F c 0 0 ] × [ c 0 0 ,c 00 0 ] [ F c 0 ,Gc 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ E c 0 0 ,Gc 00 0 ] [ E c 0 ,E c 0 0 ] × [ E c 0 0 ,Gc 00 0 ] [ E c 0 ,Gc 00 0 ] [ E c 0 ,Gc 0 0 ] × [ Gc 0 0 ,Gc 00 0 ] [ E c 0 ,Gc 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ Gc 0 0 ,Gc 00 0 ] [ E c 0 ,E c 0 0 ] × [ c 0 0 ,c 00 0 ] id × E 1 v v m m m m m m m m m m m m m m m m m m m m m m m m m id × F 1                             id × G 1                                 E 1 × id ' ' ' ' ' ' ' ' ' ' ' ' ' ' '   ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' F 1 × id : : : : : : : : : : : : :   : : : : : : : : : : : : : G 1 × id ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q id × ( −◦ ω c 00 0 )                            id × ( ω c 0 0 ◦− ) z z z z z z z z z z | | z z z z z z z z z z id × ( −◦ αc 00 0 )                             id × ( αc 0 0 ◦− )                          ( −◦ ω c 0 0 ) × id < < < < < < < < < <   < < < < < < < < < < ( ω c 0 ◦− ) × id & & & & & & & & & & & & &   & & & & & & & & & & & & & ( −◦ αc 0 0 ) × id D D D D D D D D D D " " D D D D D D D D D D ( αc 0 ◦− ) × id   + + + + + + + + + + + + + + + + + + + + + + + + + id × ( −◦ α c 00 0 )   + + + + + + + + + + + + + + + + + + + + + + + + + id × ( ω c 0 0 ◦− )              ( E 1 ( − ) ◦ ω c 0 0 ) × id ) ) id × ( αc 0 0 ◦ G 1 ( − )) u u ( −◦ αc 0 0 ) × id   ' ' ' ' ' ' ' ' ' ' ' ( ω c 0 ◦− ) × id                            E 1 × id # # G G G G G G G G G G G G ◦ , , Y Y Y Y Y ◦   id × G 1 { { w w w w w w w w w w w w ◦ r r e e e e e id × ω c 0 0 ,c 00 0 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? id × α c 0 0 ,c 00 0 1 ] e C C C C C C C C C C C C C C ω c 0 ,c 0 0 1 × id y  { { { { { { { { { { { { { { id × ω c 0 0 ,c 00 0 1 {                After applying the pro duct in terchange (see section 2.6) to 2-morphisms ω c 0 ,c 0 0 1 and α c 0 0 ,c 00 0 1 , the last diagram b ecomes 3.3 n Cat : the hom-c ate gories 45 [ c 0 ,c 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ E c 0 0 ,F c 00 0 ] [ F c 0 ,Gc 0 0 ] × [ c 0 0 ,c 00 0 ] [ E c 0 ,F c 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 0 0 ] × [ F c 0 0 ,Gc 00 0 ] [ E c 0 ,F c 0 0 ] × [ F c 0 0 ,Gc 00 0 ] [ E c 0 ,F c 00 0 ] [ F c 0 ,Gc 00 0 ] [ E c 0 ,Gc 00 0 ] id × ( E 1 ( − ) ◦ ω c 00 0 ) z z id × ( ω c 0 ◦ F 1 ( − )) p p ( F 1 ( − ) ◦ αc 0 0 ) × id . . ( αc 0 ◦ G 1 ( − )) × id $ $ ( E 1 ( − ) ◦ ω c 0 0 ) × id   ( ω c 0 ◦ F 1 ( − )) × id ~ ~ id × ( F 1 ( − ) ◦ αc 00 0 ) id × ( αc 0 0 ◦ G 1 ( − ))   E 1 ( − ) ◦−   −◦ G 1 ( − )   −◦ F 1 ( − )                       id × ( F 1 ( − ) ◦ αc 00 0 ) ? ? ? ? ? ? ? ? ? ?   ? ? ? ? ? ? ? ? ? ? ( ω c 0 ◦ F 1 ( − )) × id                       F 1 ( − ) ◦− ? ? ? ? ? ? ? ? ? ?   ? ? ? ? ? ? ? ? ? ? ◦   −◦ αc 00 0 * * U U U U U U U U U U U U U U U U U U U U U U U U U U U U U ω c 0 ◦− t t i i i i i i i i i i i i i i i i i i i i i i i i i i i i i id × ω c 0 0 ,c 00 0 1 P X ) ) ) ) ) ) ) ) ) ) ) ) α c 0 ,c 0 0 1 × id               ω c 0 ,c 0 0 1 × id c k O O O O O O O O O O O O O O id × α c 0 0 ,c 00 0 1 s { o o o o o o o o o o o o o o By functorialit y of 2-morphisms in ( n − 1) Cat the t wo sides of the diagram get 3.3 n Cat : the hom-c ate gories 46 [ c 0 ,c 0 0 ] × [ c 0 0 ,c 00 0 ] [ c 0 ,c 00 0 ] [ c 0 ,c 00 0 ] [ E c 0 ,F c 00 0 ] [ F c 0 ,Gc 00 0 ] [ E c 0 ,Gc 00 0 ] E 1 ( − ) ◦ ω c 00 0   ω c 0 ◦ F 1 ( − ) u u F 1 ( − ) ◦ αc 00 0 ) ) αc 0 ◦ G 1 ( − )   ◦ z z t t t t t t ◦ $ $ J J J J J J −◦ αc 00 0 K K K K K K K K K K % % K K K K K K K K K K ω c 0 ◦− s s s s s s s s s s s y y s s s s s s s s s s s ω c 0 ,c 00 0 1 ^ f D D D D D D D D D D D D D D D D α c 0 ,c 00 0 1 y  z z z z z z z z z z z z z z z z that is exactly [ ω • 1 α ] c 0 ,c 00 0 1 , and this concludes the pro of. 3.3.2 Units Giv en a morphism of n -categories F : C F / / D , it is p ossible to deﬁne the unit 2-c el l of F , This is denoted id F , with [ id F ] 0 ( c 0 ) = id F c 0 : F c 0 / / F c 0 and [ id F ] c 0 ,c 0 0 1 = id F c 0 ,c 0 0 1 since in the diagram C 1 ( c 0 , c 0 0 ) F c 0 ,c 0 0 1 v v m m m m m m m m m m m m F c 0 ,c 0 0 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q D 1 ( F c 0 , F c 0 0 ) id F c 0 ◦− ( ( Q Q Q Q Q Q Q Q Q Q Q Q D 1 ( F c 0 , F c 0 0 ) −◦ id F c 0 0 v v m m m m m m m m m m m m D 1 ( F c 0 , F c 0 0 ) id F c 0 ,c 0 0 1 k s id F c 0 ◦ − and − ◦ id F c 0 0 are b oth equal to the identit y n − 1-functor ov er D 1 ( F c 0 , F c 0 0 ). It is straigh tforw ard to see that these giv e a 2-morphism, according to our deﬁnition. Prop osition 3.4. L et us ﬁx n -c ate gories C and D . Morphisms b etwe en them and 2-morphisms b etwe en those form a c ate gory, with c omp osition and units given ab ove. 3.3 n Cat : the hom-c ate gories 47 Pr o of. W e will sometimes denote 1-comp ositions of 2-morphisms just b y juxtap osition. W e must prov e that comp osition is asso ciativ e and units are neutral. T o this end, we start considering a diagram: C E   F & & G 8 8 H F F D ω   α   β   W e w ant to prov e ( ω α ) β = ω ( αβ ). F or ev ery c 0 in C 0 , b y asso ciative comp osition of maps [( ω α ) β ] 0 ( c 0 ) = ( ω 0 c 0 ◦ α 0 c 0 ) ◦ β 0 c 0 = ω 0 c 0 ◦ ( α 0 c 0 ◦ β 0 c 0 ) = [ ω ( αβ )] 0 ( c 0 ) . F urthermore for every pair c 0 , c 0 0 in C 0 , asso ciative vertical composition of 2-morphisms of ( n − 1)-categories gives the follo wing diagram for b oth [( ω α ) β ] c 0 c 0 0 1 and [ ω ( αβ )] c 0 c 0 0 1 [ c 0 , c 0 0 ] E 1   F 1 ~ ~ } } } } } } } } } } } } } } } } G 1 A A A A A A A A A A A A A A A A H 1   [ E c 0 , E c 0 0 ] −◦ ω c 0 0   [ F c 0 , F c 0 0 ] ω c 0 ,c 0 0 1 k s ω c 0 ◦− x x p p p p p p p p p p −◦ αc 0 0 & & N N N N N N N N N N N [ Gc 0 , Gc 0 0 ] αc 0 ◦− x x p p p p p p p p p p p −◦ β c 0 0 & & N N N N N N N N N N N [ H c 0 , H c 0 0 ] β c 0 ,c 0 0 1 k s β c 0 ◦−   [ E c 0 , F c 0 0 ] −◦ αc 0 0 & & N N N N N N N N N N [ F c 0 , Gc 0 0 ] ω c 0 ◦− x x p p p p p p p p p p p −◦ β c 0 0 & & N N N N N N N N N N N [ Gc 0 , H c 0 0 ] αc 0 ◦− x x p p p p p p p p p p p [ E c 0 , Gc 0 0 ] −◦ β c 0 0 & & N N N N N N N N N N N [ F c 0 , H c 0 0 ] ω c 0 ◦− x x p p p p p p p p p p p [ E c 0 , H c 0 0 ] α c 0 ,c 0 0 1 k s Finally , w e will show that α id G = α ( id F α = α is prov ed similarly). F or ev ery c 0 in C 0 , b y neutral identities of maps [ α id G ] 0 ( c 0 ) = α 0 c 0 id Gc 0 = α 0 c 0 ; 3.4 n Cat : the sesqui-c ate goric al structur e 48 furthermore, for ev ery pair c 0 , c 0 0 in C 0 , neutral iden tities for v ertical comp o- sition of 2-morphisms of ( n − 1)-categories give: [ c 0 , c 0 0 ] F 1 x x p p p p p p p p p p G 1   G 1 & & N N N N N N N N N N [ F c 0 , F c 0 0 ] −◦ αc 0 0   [ Gc 0 , Gc 0 0 ] id [ Gc 0 , Gc 0 0 ] αc 0 ◦− x x p p p p p p p p p p id N N N N N N N N N N N N N N N N N N N N [ F c 0 , Gc 0 0 ] id N N N N N N N N N N N N N N N N N N N N ≡ [ Gc 0 , Gc 0 0 ] αc 0 ◦− x x p p p p p p p p p p [ F c 0 , Gc 0 0 ] id c c c c c c c c c c c c c c c c c c α c 0 ,c 0 0 1 i q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = α c 0 ,c 0 0 1 3.4 n Cat: the sesqui-categorical structure In the next sections w e will in tro duce reduced left/right comp ositions of morphisms and 2-morphisms of n-categories, in order to show that n Cat has a canonical sesqui-categorical structure. Notice that 0 Cat = Set has a trivial sesqui-categorical structure (all 2-cells are identities), while 1 Cat = Cat has a canonical 2-categorical structure, that inherits a sesqui-categorical structure, forgetting horizontal comp osition of 2-cells. Hence w e may well supp ose n > 1. 3.4.1 Deﬁning reduced left-comp osition Giv en the situation B N / / C F ! ! G = = D α   one deﬁnes reduced horizon tal comp osition N • 0 α : N F ⇒ N G : B → D (or 0-comp osition) in the follo wing wa y: • for ev ery ob ject b 0 in B , [ N • 0 α ] 0 = α 0 ( N ( b 0 )) : F ( N ( b 0 )) → G ( N ( b 0 )) 3.4 n Cat : the sesqui-c ate goric al structur e 49 • for ev ery pair of ob jects b 0 , b 0 0 of B , the diagram b elo w describ es [ N • 0 α ] b 0 ,b 0 0 1 b y means of reduced left comp osition in ( n − 1) Cat : [ N • 0 α ] b 0 ,b 0 0 1 = N b 0 ,b 0 0 1 • 0 α N b 0 ,N b 0 0 1 B 1 ( b 0 , b 0 0 ) N 1   [ N F ] 1   [ N G ] 1   C 1 ( N b 0 , N b 0 0 ) F 1 x x p p p p p p p p p p p G 1 & & N N N N N N N N N N N D 1 ( N F ( b 0 ) , N F ( b 0 0 )) −◦ α N b 0 0 & & N N N N N N N N N N N D 1 ( N G ( b 0 ) , N G ( b 0 0 )) α N b 0 ◦− x x p p p p p p p p p p p D 1 ( N F ( b 0 ) , N G ( b 0 0 )) α N b 0 ,N b 0 0 1 k s T o pro v e that these data give indeed a 2-morphism, unit and comp osition axioms equations (3.6) (3.5) m ust hold. T o this purp ose, let us chose ﬁrst an ob ject b 0 of B and consider the follo wing c hain of diagrams equalities: I u ( b 0 )   [ b 0 , b 0 ] N 1   [ N F ] 1                   [ N G ] 1   6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 [ N b 0 , N b 0 ] F 1 z z u u u u u u u u u G 1 $ $ I I I I I I I I I [ N F ( b 0 ) , N F ( b 0 )] −◦ α N b 0 $ $ I I I I I I I I I [ N G ( b 0 ) , N G ( b 0 )] α N b 0 ◦− z z u u u u u u u u u [ N F ( b 0 ) , N G ( b 0 )] α N b 0 ,N b 0 1 k s ( i ) = I u ( N b 0 )   [ N b 0 , N b 0 ] F 1 z z u u u u u u u u u G 1 $ $ I I I I I I I I I [ N F ( b 0 ) , N F ( b 0 )] −◦ α N b 0 $ $ I I I I I I I I I [ N G ( b 0 ) , N G ( b 0 )] α N b 0 ◦− z z u u u u u u u u u [ N F ( b 0 ) , N G ( b 0 )] α N b 0 ,N b 0 1 k s ( ii ) = I [ α N b 0 ]   [ α N b 0 ]   [ N F ( b 0 ) , N G ( b 0 )] id k s ( i ) holds b y functorialit y w.r.t. units (3.4), and ( ii ) is simply functoriality w.r.t units of α (3.6) for the ob ject N b 0 of C . Concerning comp osition axiom, take three ob jects b 0 , b 0 0 and b 00 0 in B , and consider the follo wing diagram: 3.4 n Cat : the sesqui-c ate goric al structur e 50 [ b 0 , b 0 0 ] × [ b 0 0 , b 00 0 ] id × N 1 z z u u u u u u u u u N 1 × id $ $ I I I I I I I I I [ b 0 , b 0 0 ] × [ N b 0 0 , N b 00 0 ] id × F 1 u u l l l l l l l l l l l l l l id × G 1   [ N b 0 , N b 0 0 ] × [ b 0 0 , b 00 0 ] G 1 × id ) ) R R R R R R R R R R R R R R F 1 × id   [ b 0 , b 0 0 ] × [ N F ( b 0 0 ) , N F ( b 00 0 )] id × ( −◦ α N b 00 0 )   [ N G ( b 0 ) , N G ( b 0 0 )] × [ b 0 0 , b 00 0 ] ( α N b 0 ) × id   [ b 0 , b 0 0 ] × [ N G ( b 0 0 ) , N G ( b 00 0 )] id × ( α N b 0 0 ◦− ) u u l l l l l l l l l l l l l l [ N F ( b 0 ) , N F ( b 0 0 )] × [ b 0 0 , b 00 0 ] ( −◦ α N b 0 0 ) × id ) ) R R R R R R R R R R R R R R [ b 0 , b 0 0 ] × [ N F ( b 0 0 ) , N G ( b 00 0 )] N 1 × id   [ N F ( b 0 ) , N G ( b 0 0 )] × [ b 0 0 , b 00 0 ] id × N 1   [ N b 0 , N b 0 0 ] × [ N F ( b 0 0 ) , N G ( b 00 0 )] F 1 × id   [ N F ( b 0 ) , N G ( b 0 0 )] × [ N b 0 0 , N b 00 0 ] id × G 1   [ N F ( b 0 ) , N F ( b 0 0 )] × [ N F ( b 0 0 ) , N G ( b 00 0 )] ◦ + + W W W W W W W W W W W W W W W W W W W W [ N F ( b 0 ) , N G ( b 0 0 )] × [ N G ( b 0 0 ) , N G ( b 00 0 )] ◦ s s g g g g g g g g g g g g g g g g g g g g [ N F ( b 0 ) , N G ( b 00 0 )] α N b 0 ,N b 0 0 1 × id q y l l l l l l l l l l l l l l l l l l id × α N b 0 0 ,N b 00 0 1 e m R R R R R R R R R R R R R R R R R R By pro duct in terchange rules (see L emma 2.11 , when one of the comp onents is an identit y) 2-cells id × α 1 and α 1 × id can slide along N 1 × id and id × N 1 resp ectiv ely , in order to give: 3.4 n Cat : the sesqui-c ate goric al structur e 51 [ b 0 , b 0 0 ] × [ b 0 0 , b 00 0 ] N 1 × N 1 x x r r r r r r r r r r N 1 × N 1 & & L L L L L L L L L L [ N b 0 , N b 0 0 ] × [ N b 0 0 , N b 00 0 ] id × F 1 u u k k k k k k k k k k k k k k k id × G 1   [ N b 0 , N b 0 0 ] × [ N b 0 0 , N b 00 0 ] G 1 × id ) ) S S S S S S S S S S S S S S S F 1 × id   [ N b 0 , N b 0 0 ] × [ N F ( b 0 0 ) , N F ( b 00 0 )] id × ( −◦ α N b 00 0 )   [ N G ( b 0 ) , N G ( b 0 0 )] × [ N b 0 0 , N b 00 0 ] ( α N b 0 ) × id   [ N b 0 , N b 0 0 ] × [ N G ( b 0 0 ) , N G ( b 00 0 )] id × ( α N b 0 0 ◦− ) u u k k k k k k k k k k k k k k k [ N F ( b 0 ) , N F ( b 0 0 )] × [ N b 0 0 , N b 00 0 ] ( −◦ α N b 0 0 ) × id ) ) S S S S S S S S S S S S S S S [ N b 0 , N b 0 0 ] × [ N F ( b 0 0 ) , N G ( b 00 0 )] F 1 × id   [ N F ( b 0 ) , N G ( b 0 0 )] × [ N b 0 0 , N b 00 0 ] id × G 1   [ N F ( b 0 ) , N F ( b 0 0 )] × [ N F ( b 0 0 ) , N G ( b 00 0 )] ◦ + + X X X X X X X X X X X X X X X X X X X X X X [ N F ( b 0 ) , N G ( b 0 0 )] × [ N G ( b 0 0 ) , N G ( b 00 0 )] ◦ s s f f f f f f f f f f f f f f f f f f f f f f [ N F ( b 0 ) , N G ( b 00 0 )] α N b 0 ,N b 0 0 1 × id q y k k k k k k k k k k k k k k k k k k id × α N b 0 0 ,N b 00 0 1 e m S S S S S S S S S S S S S S S S S S no w, just apply comp osition functoriality for α (3.5) and get: [ b 0 , b 0 0 ] × [ b 0 0 , b 00 0 ] N 1 × N 1   [ N b 0 , N b 0 0 ] × [ N b 0 0 , N b 00 0 ] ◦   [ N b 0 , N b 00 0 ] F 1 x x q q q q q q q q q q G 1 & & M M M M M M M M M M [ N F ( b 0 ) , N F ( b 00 0 )] −◦ α N b 00 0 & & M M M M M M M M M M [ N G ( b 0 ) , N G ( b 00 0 )] α N b 0 ◦− x x q q q q q q q q q q [ N F ( b 0 ) , N G ( b 00 0 )] α N b 0 ,N b 00 0 1 k s = [ b 0 , b 0 0 ] × [ b 0 0 , b 00 0 ] ◦   [ N b 0 , N b 00 0 ] N 1   [ N b 0 , N b 00 0 ] F 1 z z t t t t t t t t t G 1 $ $ J J J J J J J J J [ N F ( b 0 ) , N F ( b 00 0 )] −◦ α N b 00 0 $ $ J J J J J J J J J [ N G ( b 0 ) , N G ( b 00 0 )] α N b 0 ◦− z z t t t t t t t t t [ N F ( b 0 ) , N G ( b 00 0 )] α N b 0 ,N b 00 0 1 k s where the last equalit y is functorialit y of N w.r.t. comp ositions (3.3). And this completes the pro of that left horizon tal comp osition is well deﬁned. 3.4 n Cat : the sesqui-c ate goric al structur e 52 3.4.2 Left-comp osition axioms Giv en the situation A M / / B N / / C F ! ! G / / H = = D α   β   in n Cat , left-comp osition deﬁned ab o v e satisﬁes axioms (L1) to (L4) of Pr op osition 2.2 . (L1) I d C • 0 α = α Pr o of. Let ob jects c 0 , c 0 0 of C b e giv en. It is cle ar that [ I d C • 0 α ] c 0 ( def ) = α I d C ( c 0 ) = α c 0 and also that [ I d C • 0 α ] c 0 ,c 0 0 1 ( def ) = [ I d C ] c 0 ,c 0 0 1 • 0 α c 0 ,c 0 0 1 (1) = I d C 1 ( c 0 ,c 0 0 ) • 0 α c 0 ,c 0 0 1 (2) = α c 0 ,c 0 0 1 where (1) comes from the deﬁnition of identity functors , and (2) is axiom (L1) for ( n − 1) Cat . (L2) M N • 0 α = M • 0 ( N • 0 α ) Pr o of. Let ob jects a 0 , a 0 0 of A b e giv en. Then [ M N • 0 α ] a 0 ( def ) = α M N ( a 0 ) = α N ( M a 0 ) ( def ) = [ N • 0 α ] M a 0 ( def ) = [ M • 0 ( N • 0 α )] a 0 F urthermore, [ M N • 0 α ] a 0 ,a 0 0 1 ( def ) = [ M N ] a 0 ,a 0 0 1 • 0 α M N ( a 0 ) ,M N ( a 0 0 ) 1 = M a 0 ,a 0 0 1 N M a 0 ,M a 0 0 1 • 0 α M N ( a 0 ) ,M N ( a 0 0 ) 1 (1) = M a 0 ,a 0 0 1 • 0 ( N M a 0 ,M a 0 0 1 • 0 α N ( M a 0 ) ,N ( M a 0 0 ) 1 ) ( def ) = M a 0 ,a 0 0 1 • 0 [ N • 0 α ] M a 0 ,M a 0 0 1 ( def ) = [ M • 0 ( N • 0 α )] a 0 ,a 0 0 1 where (1) is axiom (L2) for ( n − 1) Cat . 3.4 n Cat : the sesqui-c ate goric al structur e 53 (L3) N • 0 id F = id N F Pr o of. Let ob jects b 0 , b 0 0 of B b e giv en. T rivially , [ N • 0 id F ] b 0 ( def ) = [ id F ] N b 0 = [ id N F ] b 0 and [ N • 0 id F ] b 0 ,b 0 0 1 ( def ) = N b 0 ,b 0 0 1 • 0 [ id F ] N b 0 ,N b 0 0 1 (1) = N b 0 ,b 0 0 1 • 0 id F N b 0 ,N b 0 0 1 = (2) = id N b 0 ,b 0 0 1 F N b 0 ,N b 0 0 1 = id [ N F ] b 0 ,b 0 0 1 ( def ) = [ id N F ] b 0 ,b 0 0 1 where (1) comes from the deﬁnition of identit y transformation and (2) is axiom (L3) in ( n − 1) Cat . (L4) N • 0 ( α • 1 β ) = ( N • 0 α ) • 1 ( N • 0 β ) Pr o of. Let ob jects b 0 , b 0 0 of B b e giv en. On ob jects: [ N • 0 ( α • 1 β )] b 0 ( def ) = [ α • 1 β ] N b 0 = α N b 0 ◦ β N b 0 ( def ) = [ N • 0 α ] b 0 ◦ [ N • 0 β ] b 0 = [( N • 0 α ) • 1 ( N • 0 β )] b 0 On homs: [ N • 0 ( α • 1 β )] b 0 ,b 0 0 1 ( def ) = N b 0 ,b 0 0 1 • 0 [ α • 1 β ] N b 0 ,N b 0 0 1 (1) = N b 0 ,b 0 0 1 • 0   β N b 0 ,N b 0 0 1 • 0 ( α N b 0 ◦ − )  • 1  α N b 0 ,N b 0 0 1 • 0 ( − ◦ β N b 0 ,N b 0 0 1 )   (2) =  N b 0 ,b 0 0 1 • 0 β N b 0 ,N b 0 0 1 • 0 ( α N b 0 ◦ − )  • 1  N b 0 ,b 0 0 1 • 0 α N b 0 ,N b 0 0 1 • 0 ( − ◦ β N b 0 ,N b 0 0 1 )  ( def ) =  [ N • 0 β ] b 0 ,b 0 0 1 • 0 ([ N ◦ α ] b 0 ◦ − )  • 1  [ N • 0 α ] b 0 ,b 0 0 1 • 0 ( − ◦ [ N • 0 β ] b 0 0 )  (3) = [( N • 0 α ) • 1 ( N • 0 β )] b 0 ,b 0 0 1 where (1) and (3) hold by deﬁnition of v ertical comp osites of 2-morphisms, (2) b y axiom (L4) in ( n − 1) Cat . 3.4.3 Deﬁning reduced righ t-comp osition Giv en the situation C F ! ! G = = D L / / E α   3.4 n Cat : the sesqui-c ate goric al structur e 54 one deﬁnes reduced horizon tal comp osition α • 0 L : F L ⇒ GL : C → E (or 0-comp osition) in the follo wing wa y: • for ev ery ob ject c 0 in C , [ α • 0 L ] 0 = L ( α 0 ( c 0 )) : L ( F ( c 0 )) → L ( G ( c 0 )) • for every pair of ob jects c 0 , c 0 0 of B , the diagram b elow describes [ α • 0 L ] c 0 ,c 0 0 1 b y means of reduced right comp osition in ( n − 1) Cat : [ α • 0 L ] c 0 ,c 0 0 1 = α c 0 ,c 0 0 1 • 0 L F c 0 ,Gc 0 0 1 C 1 ( c 0 , c 0 0 ) F 1 y y t t t t t t t t t G 1 % % J J J J J J J J J D 1 ( F ( c 0 ) , F ( c 0 0 )) −◦ α c 0 0 J J J % % J J J L 1 y y D 1 ( G ( b 0 ) , G ( b 0 0 )) α c 0 ◦− t t t t y y t t t t L 1 % % E 1 ( F L ( c 0 ) , F L ( c 0 0 )) −◦ L ( α c 0 0 ) * * D 1 ( F ( c 0 ) , G ( c 0 0 )) L 1   E 1 ( GL ( c 0 ) , GL ( c 0 0 )) L ( α c 0 ) ◦− t t α c 0 ,c 0 0 1 k s E 1 ( F L ( c 0 ) , GL ( c 0 0 )) T o pro v e that these data give indeed a 2-morphism, unit and comp osition axioms equations (3.6) (3.5) m ust hold. T o this end, chose ﬁrst an ob ject c 0 of C and consider the following c hain of diagrams equalities: I u ( c 0 )   [ c 0 , c 0 ] F 1          G 1   = = = = = = = [ F c 0 , F c 0 ] −◦ α c 0   = = = = = = = [ Gc 0 , Gc 0 ] α c 0 ◦−          [ F c 0 , Gc 0 ] L 1   [ F L ( c 0 ) , GL ( c 0 )] α c 0 ,c 0 1 k s ( i ) = I [ α c 0 ]   [ α c 0 ]   [ F c 0 , Gc 0 ] L 1   [ F c 0 , Gc 0 ] id k s ( ii ) = I [ L ( α c 0 )]   [ L ( α c 0 )]   [ F c 0 , Gc 0 ] id k s where ( i ) holds by unit functoriality of α (3.6), ( ii ) b y functorialit y w.r.t. units (3.4) and by axiom (R3) of reduced right composition in the sesqui- category ( n − 1) Cat . 3.4 n Cat : the sesqui-c ate goric al structur e 55 Concerning composition axiom, let us tak e three ob jects c 0 , c 0 0 and c 00 0 in C , and consider the follo wing diagram: [ c 0 , c 0 0 ] × [ c 0 0 , c 00 0 ] id × ( α c 0 0 ◦ G ( − )) o o id × ( F ( − ) ◦ α c 00 0 ) y y (( α c 0 ◦ G ( − )) × id ) % % ( F ( − ) ◦ α c 0 0 ) × id / / [ c 0 , c 0 0 ] × [ F c 0 0 , Gc 00 0 ] id × L 1   F 1 × id ) ) [ F c 0 , Gc 0 0 ] × [ c 0 0 , c 00 0 ] id × G 1 u u L 1 × id   [ F c 0 , F c 0 0 ] × [ F c 0 0 , Gc 00 0 ] ◦ # # [ F c 0 , Gc 0 0 ] × [ Gc 0 0 , Gc 00 0 ] ◦ { { [ c 0 , c 0 0 ] × [ F L ( c 0 0 ) , GL ( c 00 0 )] [ F L ] 1 × id   [ F c 0 , Gc 00 0 ] L 1   [ F L ( c 0 ) , GL ( c 0 0 )] × [ c 0 0 , c 00 0 ] id × [ GL ] 1   [ F L ( c 0 ) , F L ( c 0 0 )] × [ F L ( c 0 0 ) , GL ( c 00 0 )] ◦ + + W W W W W W W W W W W W W W W W W W W W W [ F L ( c 0 ) , GL ( c 0 0 )] × [ GL ( c 0 0 ) , GL ( c 00 0 )] ◦ s s g g g g g g g g g g g g g g g g g g g g g [ F L ( c 0 ) , GL ( c 00 0 )] id × α c 0 0 ,c 00 0 1 O W ' ' ' ' ' ' ' ' ' ' ' ' α c 0 ,c 0 0 1 × id               Here, internal dotted construction comm utes with external (by pro duct prop erties), hence it can take its place and suggests to apply comp osition functorialit y (3.5) for α , in order to give [ c 0 , c 0 0 ] × [ c 0 0 , c 00 0 ] ◦   [ c 0 , c 00 0 ] F 1 v v m m m m m m m m m m m m m G 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q [ F c 0 , F c 0 ] −◦ α c 00 0 ( ( Q Q Q Q Q Q Q Q Q Q Q Q [ Gc 0 , Gc 0 ] α c 0 ◦− v v m m m m m m m m m m m m [ F c 0 , Gc 00 0 ] L 1   [ F L ( c 0 ) , GL ( c 00 0 )] α c 0 ,c 00 0 1 k s and this complete the pro of horizon tal right comp osition is well deﬁned. 3.4 n Cat : the sesqui-c ate goric al structur e 56 3.4.4 Righ t-comp osition axioms Giv en the diagram C F ! ! G / / H = = D L / / E M / / F α   β   in n Cat , righ t-comp osition deﬁned ab ov e satisﬁes axioms (R1) to (R4) of Pr op osition 2.2 . (R1) α ◦ I d D = α Pr o of. Let ob jects c 0 , c 0 0 of C b e giv en. It is cle ar that [ α • 0 I d D ] c 0 ( def ) = I d D ( α c 0 ) = α c 0 and also that [ α • 0 I d D ] c 0 ,c 0 0 1 ( def ) = α c 0 ,c 0 0 1 • 0 [ I d D ] F c 0 ,Gc 0 0 1 (1) = α c 0 ,c 0 0 1 • 0 I d D 1 ( F c 0 ,Gc 0 0 ) (2) = α c 0 ,c 0 0 1 where (1) comes from the deﬁnition of identity functors , and (2) is axiom (R1) for ( n − 1) Cat . (R2) α • 0 LM = ( α • 0 L ) • 0 M Pr o of. Let ob jects c 0 , c 0 0 of C b e giv en. Then [ α • 0 LM ] c 0 ( def ) = LM ( α c 0 ) = M ( L ( α c 0 )) ( def ) = M ([ α • 0 L ] c 0 ) ( def ) = [( α • 0 L ) • 0 M ] c 0 F urthermore [ α • 0 LM ] c 0 ,c 0 0 1 ( def ) = α c 0 ,c 0 0 1 • 0 [ LM ] F c 0 ,Gc 0 0 1 = α c 0 ,c 0 0 1 • 0 ( L F c 0 ,Gc 0 0 1 M L ( F c 0 ) ,L ( Gc 0 0 ) 1 ) (1) = ( α c 0 ,c 0 0 1 • 0 L F c 0 ,Gc 0 0 1 ) • 0 M L ( F c 0 ) ,L ( Gc 0 0 ) 1 ( def ) = [ α • 0 L ] c 0 ,c 0 0 1 • 0 M L ( F c 0 ) ,L ( Gc 0 0 ) 1 ( def ) = [( α • 0 L ) • 0 M ] c 0 ,c 0 0 1 where (1) is axiom (R2) for ( n − 1) Cat . 3.4 n Cat : the sesqui-c ate goric al structur e 57 (L3) id F • 0 L = id F L Pr o of. Let ob jects c 0 , c 0 0 of C b e giv en. T rivially , [ id F • 0 L ] c 0 ( def ) = = L ([ id F ] c 0 ) (1) = L ( id F c 0 ) (2) = id F L ( c 0 ) where (1) holds by deﬁnition of identit y transformations and (2) from func- torialit y of L . F urthermore, [ id F • 0 L ] c 0 ,c 0 0 1 ( def ) = [ id F ] c 0 ,c 0 0 1 • 0 L F c 0 ,F c 0 0 1 (1) = id F c 0 ,c 0 0 1 • 0 L F c 0 ,F c 0 0 1 = (2) = id F c 0 ,c 0 0 1 L F c 0 ,F c 0 0 1 = id [ F L ] c 0 ,c 0 0 1 ( def ) = [ id F L ] c 0 ,c 0 0 1 where (1) comes from the deﬁnition of identit y transformation and (2) is axiom (R3) in ( n − 1) Cat . (L4) ( α • 1 β ) • 0 L = ( α • 0 L ) • 1 ( β • 0 L ) Pr o of. Let ob jects c 0 , c 0 0 of C b e giv en. On ob jects: [( α • 1 β ) • 0 L ] c 0 ( def ) = L ([ α • 1 β ] c 0 ) = L ( α c 0 ◦ β c 0 ) = L ( α c 0 ) ◦ L ( β c 0 ) = [( α • 0 L ) • 1 ( β • 0 L )] c 0 On homs: [( α • 1 β ) • 0 L ] c 0 ,c 0 0 1 = ( def ) = [ α • 1 β ] c 0 ,c 0 0 1 • 0 L F c 0 ,H c 0 0 1 (1) =   β c 0 ,c 0 0 1 • 0 ( α c 0 ◦ − )  • 1  α c 0 ,c 0 0 1 • 0 ( − ◦ β c 0 0 )   • 0 L F c 0 ,H c 0 0 1 (2) =   β c 0 ,c 0 0 1 • 0 ( α c 0 ◦ − )  • 0 L F c 0 ,H c 0 0 1  • 1   α c 0 ,c 0 0 1 • 0 ( − ◦ β c 0 0 )  • 0 L F c 0 ,H c 0 0 1  (3) =  β c 0 ,c 0 0 1 • 0  L Gc 0 ,H c 0 0 1 ( L ( α c 0 ) ◦ − )   • 1  α c 0 ,c 0 0 1 • 0  L F c 0 ,Gc 0 0 1 ( − ◦ L ( β c 0 0 ))   (4) =   β c 0 ,c 0 0 1 • 0 L Gc 0 ,H c 0 0 1  • 0 ( L ( α c 0 ) ◦ − )  • 1   α c 0 ,c 0 0 1 • 0 L F c 0 ,Gc 0 0 1  • 0 ( − ◦ L ( β c 0 0 ))  ( def ) =  [ β • 0 L ] c 0 ,c 0 0 1 • 0 ([ α • 0 L ] c 0 ◦ − )  • 1  [ α • 0 L ] c 0 ,c 0 0 1 • 0 ( − ◦ [ β • 0 L ] c 0 0 )  (5) = [( α • 0 L ) • 1 ( β • 0 L )] c 0 ,c 0 0 1 where (1) and (5) hold b y deﬁnition of v ertical comp osites of 2-morphisms, (2) b y axiom (R4) in ( n − 1) Cat , (3) by functoriality of L w.r.t. 0-comp osition, and (4) b y axiom (R2) in ( n − 1) Cat . 3.5 Pr o ducts in n Cat 58 3.4.5 Whisk ering axiom Giv en the situation B N / / C F ! ! G = = D L / / E α   a whisk ering op eration may b e deﬁned if the following equation holds: (LR5) ( N ◦ α ) ◦ L = N ◦ ( α ◦ L ) Pr o of. Let ob jects b 0 , b 0 0 of B b e giv en. Then the following follows immediately from deﬁnitions [( N • 0 α ) • 0 L ] b 0 = L ([ N • 0 α ] b 0 ) = L ( α N b 0 ) = [ α • 0 L ] N b 0 = [ N ◦ ( α ◦ L )] b 0 Analogously , consider: [( N • 0 α ) • 0 L ] b 0 ,b 0 0 1 = [ N • 0 α ] b 0 ,b 0 0 1 • 0 L F ( N b 0 ) ,G ( N b 0 0 ) 1 =  N b 0 ,b 0 0 1 • 0 α N b 0 ,N b 0 0 1  • 0 L F ( N b 0 ) ,G ( N b 0 0 ) 1 (1) = N b 0 ,b 0 0 1 • 0  α N b 0 ,N b 0 0 1 • 0 L F ( N b 0 ) ,G ( N b 0 0 ) 1  = N b 0 ,b 0 0 1 • 0 [ α • 0 L ] N b 0 ,N b 0 0 1 = [ N • 0 ( α • 0 L )] b 0 ,b 0 0 1 where everything comes directly from deﬁnitions, but (1) that is exactly the whisk ering in ( n − 1) Cat . 3.5 Pro ducts in n Cat In order to close the induction on the deﬁnition of n Cat , all we need is to sho w that it admits ﬁnite pro ducts, according to the 2-dimensional Universal Pr op erty 2.9 , i.e. to show it admits binary pro ducts and terminal ob jects. 3.5.1 2-univ ersalit y of categorical pro ducts Let t wo n-categories C and D b e giv en. W e know from Prop osition 3.3 that the underlying category b n Cat c admits a (standard) pro duct of C and D : C × D Π C | | x x x x x x x x x Π D " " F F F F F F F F F C D 3.5 Pr o ducts in n Cat 59 No w supp ose we are given tw o 2-morphisms α : A ⇒ A 0 : X → C × D , β : B ⇒ B 0 : X → C × D According to Universal Pr op erty 2.9 , what we w ant to prov e is that there exists a unique 2-morphism θ : T ⇒ T 0 : X → C × D suc h that θ • 0 Π C = α, θ • 0 Π D = β , (3.11) First let us say that T and T 0 are determined by 1-dimensional univ ersal prop ert y: T is such that (and univocally determined by)  T • 0 Π C = A T • 0 Π D = B , T 0 is suc h that (and univocally determined by)  T 0 • 0 Π C = A 0 T 0 • 0 Π D = B 0 . More explicitly , for ev ery pair of ob jects x 0 , x 0 0 in X , T 0 ( x 0 ) = ( A 0 ( x 0 ) , B 0 ( x 0 )) and X 1 ( x 0 , x 0 0 ) T x 0 ,x 0 0 1 / / h A x 0 ,x 0 0 1 , B x 0 ,x 0 0 1 i + + X X X X X X X X X X X X X X X X X X X X X X X X X X X [ C × D ] x 0 ,x 0 0 1 (( Ax 0 , B x 0 ) , ( Ax 0 0 , B x 0 0 )) def C 1 ( Ax 0 , ax 0 0 ) × D ( B x 0 , B x 0 0 ) Similarly for T 0 : T 0 0 ( x 0 ) = ( A 0 0 ( x 0 ) , B 0 0 ( x 0 )) and T 0 x 0 ,x 0 0 1 = h A 0 1 x 0 ,x 0 0 , B 0 1 x 0 ,x 0 0 i Then, θ = h θ 0 , θ 1 i is giv en by: θ 0 ( x 0 ) = ( α 0 ( x 0 ) , β 0 ( x 0 )) and θ x 0 ,x 0 0 1 is giv en by the universal prop erty of pro ducts in ( n − 1) Cat . In fact, a suitable θ 1 w ould ﬁt in the following diagram: X 1 ( x 0 , x 0 0 ) T x 0 ,x 0 0 1 u u l l l l l l l l l l l l l l T 0 1 x 0 ,x 0 0 ) ) S S S S S S S S S S S S S S [ C × D ] 1 (( Ax 0 , B x 0 ) , ( Ax 0 0 , B x 0 0 )) −◦ θ x 0 0 ) ) S S S S S S S S S S S S S S [ C × D ] 1 (( A 0 x 0 , B 0 x 0 ) , ( A 0 x 0 0 , B 0 x 0 0 )) θ x 0 ◦− u u k k k k k k k k k k k k k k [ C × D ] 1 (( Ax 0 , B x 0 ) , ( A 0 x 0 0 , B 0 x 0 0 )) ? k s 3.5 Pr o ducts in n Cat 60 Sp elling out the deﬁnitions, this ma y b e written: X 1 ( x 0 , x 0 0 ) h A x 0 ,x 0 0 1 ,B x 0 ,x 0 0 1 i v v n n n n n n n n n n n n h A 0 1 x 0 ,x 0 0 ,B 0 1 x 0 ,x 0 0 i ( ( R R R R R R R R R R R R R C 1 ( Ax 0 , Ax 0 0 ) × D 1 ( B x 0 , B x 0 0 ) ( −◦ α x 0 0 ) × ( −◦ β x 0 0 ) ( ( P P P P P P P P P P P P C 1 ( A 0 x 0 , A 0 x 0 0 ) × D 1 ( B 0 x 0 , B 0 x 0 0 ) ( α x 0 ◦− ) × ( β x 0 ◦− ) v v l l l l l l l l l l l l l C 1 ( Ax 0 , A 0 x 0 0 ) × D 1 ( B x 0 , B 0 x 0 0 ) θ x 0 ,x 0 0 1 k s hence w e are allow ed to deﬁne θ x 0 ,x 0 0 1 = h α x 0 ,x 0 0 1 , β x 0 ,x 0 0 1 i , and this c hoice would satisfy the universal prop erty of pro ducts.. Concerning the ﬁrst of the (3.11), for ev ery pair of ob jects x 0 , x 0 0 in X [ θ • 0 Π C ] 0 ( x 0 ) = [Π C ] 0 ( θ x 0 ) = π C 0 × D 0 C 0 ( α x 0 , β x 0 ) = α x 0 and also [ θ • 0 Π C ] x 0 ,x 0 0 1 = θ x 0 ,x 0 0 1 • 0 [Π C ] ( Ax 0 ,B x 0 ) , ( A 0 x 0 0 ,B 0 x 0 0 ) 1 = = h α x 0 ,x 0 0 1 , β x 0 ,x 0 0 1 i • 0 Π C 1 ( Ax 0 ,A 0 0 x 0 0 ) = α x 0 ,x 0 0 1 where everything comes directly from deﬁnitions, but the last equality which is given by the universal prop erty deﬁning θ x 0 ,x 0 0 1 in ( n − 1) Cat . The second of the (3.11) can b e pro ved the same wa y . Moreo ver, such a θ is unique. F or, if another η : T ⇒ T 0 : X → C × D is suc h that η • 0 Π C = α, η • 0 Π D = β , then equations ab ov e determine η 0 on ob jects (since a map to a pro duct, C 0 × D 0 , is determined by its pro jections), hence it is equal to θ 0 . On the other side, for ob jects x 0 , x 0 0 in X , and η x 0 ,x 0 0 1 is given considering its comp osition with pro jections by universalit y of the pro duct in ( n − 1) Cat , hence it is equal to θ x 0 ,x 0 0 1 . T o conclude this section, we will see that the just deﬁned pair [ θ 0 , θ − , − 1 ] is indeed a 2-morphism, that is, it ob eys units and composition axioms for n-transformations. T o this purp ose, let us chose a triple of ob jects x 0 , x 0 0 and x 00 0 of X , and let us consider the follo wing diagram: 3.5 Pr o ducts in n Cat 61 [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] id × L x 0 0 ,x 00 0 } } id × L 0 x 0 0 ,x 00 0 u u L x 0 0 ,x 00 0 × id ) ) L 0 x 0 0 ,x 00 0 × id " " [ x 0 , x 0 0 ] × [ Ax 0 0 , A 0 x 00 0 ] × [ B x 0 0 , B 0 x 00 0 ] h A x 0 ,x 0 0 1 ,B x 0 ,x 0 0 1 i × id   [ Ax 0 , A 0 x 0 0 ] × [ B x 0 , B 0 x 0 0 ] × [ x 0 0 , x 00 0 ] id × h A 0 1 x 0 0 ,x 00 0 ,B 0 1 x 0 0 ,x 00 0 i   [ Ax 0 , Ax 0 0 ] × [ B x 0 , B x 0 0 ] × [ Ax 0 0 , A 0 x 00 0 ] × [ B x 0 0 , B 0 x 00 0 ] τ   [ Ax 0 , A 0 x 0 0 ] × [ B x 0 , B 0 x 0 0 ] × [ A 0 x 0 0 , A 0 x 00 0 ] × [ B 0 x 0 0 , B 0 x 00 0 ] τ   [ Ax 0 , Ax 0 0 ] × [ Ax 0 0 , A 0 x 00 0 ] × [ B x 0 , B x 0 0 ] × [ B x 0 0 , B 0 x 00 0 ] C ◦× D ◦ * * U U U U U U U U U U U U U U U U U [ Ax 0 , A 0 x 0 0 ] × [ A 0 x 0 0 , A 0 x 00 0 ] × [ B x 0 , B 0 x 0 0 ] × [ B 0 x 0 0 , B 0 x 00 0 ] C ◦× D ◦ t t h h h h h h h h h h h h h h h h h [ Ax 0 , A 0 x 00 0 ] × [ B x 0 , B 0 x 00 0 ] id × θ x 0 0 ,x 00 0 1 P X ) ) ) ) ) ) ) ) ) ) ) ) θ x 0 ,x 0 0 1 × id               where L x,y = h A x,y 1 , B x,y 1 i  ( − ◦ α y ) × ( − ◦ β y )  L 0 x,y = h A 0 1 x,y , B 0 1 x,y i  ( α x ◦ − ) × ( β x ◦ − )  No w, considering only the righ t branc h of the diagram, where ∆’s express diagonal morphisms, the following equations hold by pro duct interc hange prop erties:  θ x 0 ,x 0 0 1 × id  • 0  id × h A 0 1 x 0 0 ,x 00 0 , B 0 1 x 0 0 ,x 00 0 i  τ  C ◦ × D ◦  = =  h α x 0 ,x 0 0 1 , β x 0 ,x 0 0 1 i × id  • 0  id × h A 0 1 x 0 0 ,x 00 0 , B 0 1 x 0 0 ,x 00 0 i  τ  C ◦ × D ◦  = (∆ × id ) • 0  α x 0 ,x 0 0 1 × β x 0 ,x 0 0 1 × id  • 0 ( id × ∆)( id × A 0 1 x 0 0 ,x 00 0 × B 0 1 x 0 0 ,x 00 0 ) τ  C ◦ × D ◦  = (∆ × ∆) τ • 0  α x 0 ,x 0 0 1 × id × β x 0 ,x 0 0 1 × id  • 0 ( id × A 0 1 x 0 0 ,x 00 0 × id × B 0 1 x 0 0 ,x 00 0 )  C ◦ × D ◦  = (∆ × ∆) τ • 0   α x 0 ,x 0 0 1 × A 0 1 x 0 0 ,x 00 0 • 0 ( C ◦ )  ×  β x 0 ,x 0 0 1 × B 0 1 x 0 0 ,x 00 0 • 0 ( D ◦ )   Similarly , the left branc h gives:  id × θ x 0 0 ,x 00 0 1  • 0  h A 1 x 0 0 ,x 0 0 , B 1 x 0 ,x 0 0 × id i  τ  C ◦ × D ◦  = = (∆ × ∆) τ • 0   A 1 x 0 ,x 0 0 × α x 0 0 ,x 00 0 1 • 0 ( C ◦ )  ×  B 1 x 0 ,x 0 0 × β x 0 0 ,x 00 0 1 • 0 ( D ◦ )   Hence the diagram ab ov e, b eing the vertical comp osite of the tw o branc hes, ma y b e rewritten applying rule (L4): 3.5 Pr o ducts in n Cat 62 = (∆ × ∆) τ • 0 h  α x 0 ,x 0 0 1 × A 0 1 x 0 0 ,x 00 0 • 0 ( C ◦ )  • 1  A 1 x 0 ,x 0 0 × α x 0 0 ,x 00 0 1 • 0 ( C ◦ )   × ×   β x 0 ,x 0 0 1 × B 0 1 x 0 0 ,x 00 0 • 0 ( D ◦ )  • 1  B 1 x 0 ,x 0 0 × β x 0 0 ,x 00 0 1 • 0 ( D ◦ )  i b y comp osition axiom of α and β , and pro duct prop erties, w e get the result: = (∆ × ∆) τ • 0  (( X ◦ ) • 0 α x 0 ,x 00 0 1 ) × (( X ◦ ) • 0 β x 0 ,x 00 0 1 )  = ( X ◦ ) ∆ • 0 ( α x 0 ,x 00 0 1 × β x 0 ,x 00 0 1 ) = ( X ◦ ) • 0 h α x 0 ,x 00 0 1 , β x 0 ,x 00 0 1 i = ( X ◦ ) • 0 θ x 0 ,x 00 0 1 F urthermore, for an ob ject x 0 , consider the follo wing diagram: I u ( x 0 )   [ x 0 , x 0 ] L 0 x 0 ,x 0 ~ ~ L x 0 ,x 0 [ Ax 0 , A 0 x 0 ] × [ B x 0 , B 0 x 0 ] θ x 0 ,x 0 1 k s then, as for the comp osition axiom, u ( x 0 ) • 0 θ x 0 ,x 0 1 = u ( x 0 ) ◦ h α x 0 ,x 0 1 , β x 0 ,x 0 1 i = u ( x 0 )∆ • 0 ( α x 0 ,x 0 1 × β x 0 ,x 0 1 ) = ∆ u ( x 0 ) • 0 ( α x 0 ,x 0 1 × β x 0 ,x 0 1 ) = ∆ • 0  ( u ( x 0 ) • 0 α x 0 ,x 0 1 ) × ( u ( x 0 ) • 0 β x 0 ,x 0 1 )  ( ∗ ) = ∆ • 0  I d [ α x 0 ] × I d [ β x 0 ]  = I d h [ α x 0 ] , [ β x 0 ] i = I d [ θ x 0 ] where ( ∗ ) is giv en by unit axiom of α and β . 3.6 The standar d h -pul lb ack in n Cat 63 3.6 The standard h -pullbac k in n Cat In this section w e give a construction that will b e of fundamental imp ortance for the dev elopment of the theory . The idea is to generalize a classical homotopical construction [ Mat76b ] to n -categories, or b etter to n -group oids, where homotopical asp ects are more than a mere suggestion. W e start considering the follo wing h -pullbacks reference diagram. P Q / / P   C G   A F / / B ε ; C         F or n=0, classical pullback in Set is an instance of h -pullbac k, with 2- morphism ε b eing an iden tity . In fact the category of sets and maps is (seen as) the 2-trivial sesqui-category 0 Cat , and indeed, only condition (1) and (2) survive. Hence, in the next sections, w e will supp ose integer n > 0 b een given. W e exhibit a recursive construction of the standard h -pullbac k satisfying Universal Pr op erty 2.12 . W e will giv e P in the form( P 0 , P − , − 1 ). The set P 0 is the following limit in Set (that yields indeed, also the ob ject- comp onen ts of F , G and ε ): P 0 P 0 v v ε 0   Q 0 ( ( A 0 F 0 A A A A A A A A B 1 s ~ ~ } } } } } } } } t A A A A A A A A C 0 G 0 ~ ~ } } } } } } } } B 0 B 0 Here B 1 is the disjoin t union a b 0 ,b 0 0 ∈ B 0 [ B 1 ( b 0 , b 0 0 )] 0 , and s, t are sour c e and tar get maps of 1-cells. More explicitly , P 0 = { ( a 0 , b 1 , c 0 ) s.t. a 0 ∈ A 0 , c 0 ∈ C 0 , b 1 : F a 0 → Gc 0 ∈ B 1 } P 0 (( a 0 , b 1 , c 0 )) = a 0 , Q 0 (( a 0 , b 1 , c 0 )) = c 0 , ε 0 (( a 0 , b 1 , c 0 )) = b 1 3.6 The standar d h -pul lb ack in n Cat 64 Let us ﬁx t wo element of P 0 : p 0 = ( a 0 , b 1 , c 0 ) , p 0 0 = ( a 0 0 , b 0 1 , c 0 0 ) . The hom-(n-1)category P 1 ( p 0 , p 0 0 ) is granted b y the following h-pullback in ( n − 1) Cat : P 1 ( p 0 , p 0 0 ) Q p 0 ,p 0 0 1 / / P p 0 ,p 0 0 1   C 1 ( c 0 , c 0 0 ) G c 0 ,c 0 0 1   B 1 ( Gc 0 , Gc 0 0 ) b 1 ◦−   A 1 ( a 0 , a 0 0 ) F a 0 ,a 0 0 1 / / B 1 ( F a 0 , F a 0 0 ) −◦ b 0 1 / / B 1 ( F a 0 , Gc 0 0 ) ε p 0 ,p 0 0 1 q y Prop osition 3.5. The p air ( P 0 , P − , − 1 ) yields a n-c ate gory. In order to prov e the statement, we need to sho w constructions for comp osi- tion and units, and to pro ve that they satisﬁes n -category axioms. 3.6.1 Comp osition Supp ose w e are given three elements of P 0 p 0 = ( a 0 , b 1 , c 0 ) , p 0 0 = ( a 0 0 , b 0 1 , c 0 0 ) , p 0 0 = ( a 00 0 , b 00 1 , c 00 0 ) . One deﬁnes ◦ 0 : P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) → P 1 ( p 0 , p 00 0 ) b y means of the universal prop erty of pullback in ( n − 1) Cat . In fact, as P 1 ( p 0 , p 00 0 ) is a pullbac k, we can consider the four-tuple  P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 , p 0 0 ) , P p 0 ,p 0 0 1 × P p 0 0 ,p 00 0 1 , Q p 0 ,p 0 0 1 × Q p 0 0 ,p 00 0 1 , ε p 0 ,p 0 0 ,p 00 0  where the (n-1)-natural transformation ε p 0 ,p 0 0 ,p 00 0 is the comp osite shown b elo w: 3.6 The standar d h -pul lb ack in n Cat 65 [ p 0 , p 0 0 ] × [ a 0 0 , a 00 0 ] id × F 1   P 1 × id { { [ p 0 , p 0 0 ] × [ p 0 0 , p 00 0 ] id × P 1 o o id × Q 1 y y s s s s s s s s s P 1 × id % % K K K K K K K K K Q 1 × id / / [ c 0 , c 0 0 ] × [ p 0 0 , p 00 0 ] G 1 × id   id × Q 1 # # [ a 0 , 0 0 ] × [ a 0 0 , a 00 0 ] ◦   [ p 0 , p 0 0 ] × [ c 0 0 , c 00 0 ] id × G 1   [ a 0 , a 0 0 ] × [ p 0 0 , p 00 0 ] F 1 × id   [ c 0 , c 0 0 ] × [ c 0 0 , c 00 0 ] ◦   [ p 0 , p 0 0 ] × [ F a 0 0 , F a 00 0 ] id × ( −◦ b 00 1 )   [ Gc 0 , Gc 0 0 ] × [ p 0 0 , p 00 0 ] ( b 1 ◦− ) × id   [ p 0 , p 0 0 ] × [ Gc 0 0 , Gc 00 0 ] id × ( b 0 1 ◦− ) x x q q q q q q q q q q [ F a 0 , F a 0 0 ] × [ p 0 0 , p 00 0 ] ( −◦ b 0 1 ) × id & & M M M M M M M M M M [ p 0 , p 0 0 ] × [ F a 0 0 , Gc 00 0 ] P 1 × id   [ F a 0 , Gc 0 0 ] × [ p 0 0 , p 00 0 ] id × Q 1   [ a 0 , a 0 0 ] × [ F a 0 0 , Gc 00 0 ] F 1 × id & & M M M M M M M M M M [ F a 0 , Gc 0 0 ] × [ c 0 0 , c 00 0 ] id × G 1 x x q q q q q q q q q q [ a 0 , a 00 0 ] F 1 # # [ F a 0 , F a 0 0 ] × [ F a 0 0 , Gc 00 0 ] ◦ % % K K K K K K K K K [ F a 0 , Gc 0 0 ] × [ Gc 0 0 , Gc 00 0 ] ◦ y y s s s s s s s s s [ c 0 , c 00 0 ] G 1 { { [ F a 0 , F a 00 0 ] −◦ b 00 1 / / [ F a 0 , Gc 00 0 ] [ Gc 0 , Gc 00 0 ] b 1 ◦− o o id × ε p 0 0 ,p 00 0 1 i q [ [ [ [ [ [ [ [ [ [ [ [ ε p 0 ,p 0 0 1 × id m u c c c c c c c c c c c c (3.12) Dotted outer b order is clearly equal to con tinuous inner b order, hence, by univ ersal prop erty in ( n − 1) Cat , there exists a unique P ◦ 0 : P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) / / P 1 ( p 0 , p 00 0 ) (3.13) suc h that tw o squares b elow commute A 1 ( a 0 , a 0 0 ) × A 1 ( a 0 0 , a 00 0 ) A ◦   P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) P 1 × P 1 o o P ◦   Q 1 × Q 1 / / C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 , c 0 0 ) C ◦   A 1 ( a 0 , a 00 0 ) P 1 ( p 0 , p 00 0 ) P 1 o o Q 1 / / C 1 ( c 0 , c 00 0 ) (3.14) 3.6 The standar d h -pul lb ack in n Cat 66 and P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) P ◦   P 1 ( p 0 , p 00 0 )     ε p 0 ,p 00 0 1 k s B 1 ( F a 0 , Gc 00 0 ) = P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 )     ε p 0 ,p 0 0 ,p 00 0 k s B 1 ( F a 0 , Gc 00 0 ) (3.15) Lemma 3.6. Comp osition P ◦ 0 deﬁne d ab ove is asso ciative, i.e. the diagr am b elow c ommutes in ( n − 1) Cat , for every four-tuple ( p 0 , p 0 0 , p 00 0 , p 000 0 ) of elements of P 0 : P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) × P 1 ( p 00 0 , p 000 0 ) id × P ◦ / / P ◦× id   P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 000 0 ) P ◦   P 1 ( p 0 , p 00 0 ) × P 1 ( p 00 0 , p 000 0 ) P ◦ / / P 1 ( p 0 , p 000 0 ) (3.16) Pr o of. Let us consider the diagrams [ p 0 , p 0 0 ] × [ p 0 0 , p 00 0 ] × [ p 00 0 , p 000 0 ] K * * Q 1 × Q 1 × Q 1 / / P 1 × P 1 × P 1   [ c 0 , c 0 0 ] × [ c 0 0 , c 00 0 ] × [ c 00 0 , c 000 0 ] Ξ( C )   [ p 0 , p 000 0 ] Q 1 / / P 1   [ c 0 , c 000 0 ]   [ a 0 , a 0 0 ] × [ a 0 0 , a 00 0 ] × [ a 00 0 , a 000 0 ] Ξ( A ) / / [ a 0 , a 000 0 ] / / [ F a 0 , Gc 000 0 ] ε p 0 ,p 000 0 1 p x i i i i i i i i i i i i i i i i i i i i i i where morphism Ξ( X ) can b e either the comp osite ( X ◦ × id ) X ◦ or ( id × X ◦ ) X ◦ , with X b eing A or C . The idea of the pro of is to use again univ ersal prop ert y in ( n − 1) Cat to get unique K : P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) × P 1 ( p 00 0 , p 000 0 ) → P 1 ( p 0 , p 000 0 ) that coincides with b oth comp osites of diagram 3.16. T o this end, it suﬃces to sho w that the four-tuples     P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) × P 1 ( p 00 0 , p 000 0 ) ( P 1 × P 1 × P 1 )( A ◦ × id ) A ◦ ( Q 1 × Q 1 × Q 1 )( C ◦ × id ) C ◦ ( P ◦ × id ) P ◦ ε p 0 ,p 000 0 1     3.6 The standar d h -pul lb ack in n Cat 67 and     P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 00 0 ) × P 1 ( p 00 0 , p 000 0 ) ( P 1 × P 1 × P 1 )( id × A ◦ ) A ◦ ( Q 1 × Q 1 × Q 1 )( id × C ◦ ) C ◦ ( id × P ◦ ) P ◦ ε p 0 ,p 000 0 1     are equal. First comp onen ts are identical. Equalit y of second comp onents amoun ts to the asso ciativity axiom for 0- comp osition in A . Equalit y of third comp onents amounts to the as sociativity axiom for 0- comp osition in C . What remains to pro ve is equalit y of fourth comp onents. The starting p oin t is the 2-morphism ( P ◦ × id ) • 0 P ◦ • 0 ε p 0 ,p 000 0 1 , where ε p 0 ,p 000 0 1 : b 1 ◦ [ QG ] 1 ( − ) ⇒ [ P F ] 1 ( − ) ◦ b 000 1 This ma y b e visualized as a diagram: [ p 0 , p 0 0 ] × [ p 0 0 , p 00 0 ] × [ p 00 0 , p 000 0 ] ◦ × id   [ p 0 , p 0 0 ] × [ p 0 0 , p 000 0 ] ◦   [ p 0 , p 000 0 ] P 1 u u l l l l l l l l l l l l l l Q 1 ) ) R R R R R R R R R R R R R R [ a 0 , a 000 0 ] F 1   [ c 0 , c 000 0 ] G 1   [ F a 0 , F a 000 0 ] −◦ b 000 1 ) ) R R R R R R R R R R R R R R [ Gc 0 , Gc 000 0 ] b 1 ◦− u u l l l l l l l l l l l l l [ F c 0 , Gc 000 0 ] ε p 0 ,p 000 0 1 k s that, applying (3.12) and (3.15) to ε p 0 ,p 000 0 1 , equals to 3.6 The standar d h -pul lb ack in n Cat 68 [ p 0 , p 0 0 ] × [ p 0 0 , p 00 0 ] × [ p 00 0 , p 000 0 ] ◦ × id   [ p 0 , p 00 0 ] × [ a 00 0 , a 000 0 ] id × F 1   [ p 0 , p 00 0 ] × [ p 00 0 , p 000 0 ] id × P 1 o o id × Q 1 v v n n n n n n n n n n n n P 1 × id ' ' P P P P P P P P P P P P Q 1 × id / / [ c 0 , c 00 0 ] × [ p 0 0 , p 000 0 ] G 1 × id   [ p 0 , p 00 0 ] × [ c 00 0 , c 000 0 ] id × G 1   [ a 0 , a 00 0 ] × [ p 00 0 , p 000 0 ] F 1 × id   [ p 0 , p 00 0 ] × [ F a 00 0 , F a 000 0 ] id × ( −◦ b 000 1 )   [ Gc 0 , Gc 00 0 ] × [ p 00 0 , p 000 0 ] ( b 1 ◦− ) × id   [ p 0 , p 00 0 ] × [ Gc 00 0 , Gc 000 0 ] id × ( b 00 1 ◦− ) x x p p p p p p p p p p p [ F a 0 , F a 00 0 ] × [ p 00 0 , p 000 0 ] ( −◦ b 00 1 ) × id & & N N N N N N N N N N N [ p 0 , p 00 0 ] × [ F a 00 0 , Gc 000 0 ] P 1 × id   [ F a 0 , Gc 00 0 ] × [ p 00 0 , p 000 0 ] id × Q 1   [ a 0 , a 00 0 ] × [ F a 00 0 , Gc 000 0 ] F 1 × id & & N N N N N N N N N N N [ F a 0 , Gc 00 0 ] × [ c 00 0 , c 000 0 ] id × G 1 x x p p p p p p p p p p p [ F a 0 , F a 00 0 ] × [ F a 00 0 , Gc 000 0 ] ◦ ( ( P P P P P P P P P P P P [ F a 0 , Gc 00 0 ] × [ Gc 00 0 , Gc 000 0 ] ◦ w w n n n n n n n n n n n n [ F a 0 , Gc 000 0 ] id × ε p 00 0 ,p 000 0 1 i q \ \ \ \ \ \ \ \ \ \ \ \ ε p 0 ,p 00 0 1 × id m u b b b b b b b b b b b b No w we can apply (3.12) again, and (3.15) to the right-hand side of the diagram, to express ε p 0 ,p 00 0 1 in terms of ε p 0 ,p 0 0 1 and ε p 0 0 ,p 00 0 1 . 3.6 The standar d h -pul lb ack in n Cat 69 [ p 0 ,p 0 0 ] × [ p 0 0 ,p 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 00 0 ] × [ p 00 0 ,p 000 0 ] [ F a 0 ,Gc 0 0 ] × [ p 0 0 ,p 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 0 0 ] × [ F a 0 0 ,Gc 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 00 0 ] × [ F a 00 0 ,Gc 000 0 ] [ F a 0 ,Gc 00 0 ] × [ p 00 0 ,p 000 0 ] [ F a 0 ,Gc 000 0 ] ◦ × id z z * *     id × ε p 0 0 ,p 00 0 1 × id       ( PF 1 ( − ) ◦− ) × id , , ( −◦ Q G 1 ( − )) × id   PF 1 ( − ) ◦− + + −◦ Q G 1 ( − ) s s ε p 0 ,p 0 0 1 × id × id           k s id × ε p 00 0 ,p 000 0 1 h p X X X X X X X X X X X X X X Moreo ver, in order to shift the 2-morphism ε p 00 0 ,p 000 0 1 up, we apply pr o duct inter change rules to the left-hand side. What we get is the diagram: 3.6 The standar d h -pul lb ack in n Cat 70 [ p 0 ,p 0 0 ] × [ p 0 0 ,p 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 0 0 ] × [ p 0 0 ,p 00 0 ] × [ F a 00 0 ,Gc 000 0 ] [ F a 0 ,Gc 0 0 ] × [ p 0 0 ,p 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 0 0 ] × [ F a 0 0 ,Gc 00 0 ] × [ p 00 0 ,p 000 0 ] [ p 0 ,p 0 0 ] × [ F a 0 0 ,Gc 000 0 ] [ F a 0 ,Gc 00 0 ] × [ p 00 0 ,p 000 0 ] [ F a 0 ,Gc 000 0 ] id × ( PF 1 ( − ) ◦− )   * *     id × ε p 0 0 ,p 00 0 1 × id     t t ( PF 1 ( − ) ◦− ) × id , , id × ( −◦ Q G 1 ( − )) q q ( −◦ Q G 1 ( − )) × id   PF 1 ( − ) ◦− + + −◦ Q G 1 ( − ) s s ε p 0 ,p 0 0 1 × id × id           k s id × id × ε p 00 0 ,p 000 0 1 3 3 3 3 U ] 3 3 3 3 The dotted arrow ﬁts the diagram prop erly , making the tw o regions commute. Hence the whole diagram is perfectly symmetric, and calculations ma y b e carried on doing the steps in rev erse order, and gain the result. 3.6.2 Units As for comp osition, we get unit morphisms by universal prop ert y in ( n − 1) Cat . Supp ose an elemen t p 0 of P 0 is ﬁxed. I u ( c 0 ) / / u ( a 0 )   C 1 ( c 0 , c 0 ) b 1 ◦ G ( − )   A 1 ( a 0 , a 0 ) F ( − ) ◦ b 1 / / B 1 ( F a 0 , Gc 0 ) The square ab ov e commutes, hence it is an iden tity 2-morphism, o ver the same base deﬁning ε p 0 ,p 0 1 , and this implies the existence of a unique P u ( p 0 ) : I / / P 1 ( p 0 , p 0 ) . 3.6 The standar d h -pul lb ack in n Cat 71 Lemma 3.7. Units deﬁne d ab ove ar e neutr al w.r.t. 0-c omp osition, i.e. , for every p air p 0 , p 0 0 , in P 0 , ( L ) and ( R ) c ommute. P 1 ( p 0 , p 0 0 × I ) ( R ) id × u ( p 0 0 ) / / P 1 ( p 0 , p 0 0 ) × P 1 ( p 0 0 , p 0 0 ) ◦   P 1 ( p 0 , p 0 0 ) ( L ) id / / λ ∼ =   ρ ∼ = O O P 1 ( p 0 , p 0 0 ) I × P 1 ( p 0 , p 0 0 ) u ( p 0 ) × id / / P 1 ( p 0 , p 0 ) × P 1 ( p 0 , p 0 0 ) ◦ O O Pr o of. W e show only the comm utativity of ( L ), the other b eing similar. Hence let us consider the comp osition [ p 0 , p 0 0 ] λ ∼ =   P 1   Q 1   I × [ p 0 , p 0 0 ] u ( p 0 ) × id   [ p 0 , p 0 ] × [ p 0 , p 0 0 ] ◦   [ p 0 , p 0 0 ] P 1 w w o o o o o o o o o o o Q 1 ' ' O O O O O O O O O O O [ a 0 , a 0 0 ] F ( − ) ◦ b 0 1 ' ' O O O O O O O O O O O [ c 0 , c 0 0 ] b 1 ◦ G ( − ) w w o o o o o o o o o o o [ F a 0 , Gc 0 0 ] ε p 0 ,p 0 0 1 k s (3.17) It is easy to see that b oth sides comm ute with dotted arrows. In fact λ ( P u ( p 0 ) × id )( P ◦ ) Q 1 ( i ) = λ ( P u ( p 0 ) × id )( Q 1 × Q 1 )( C ◦ ) ( ii ) = λ ( C u ( c 0 ) × Q 1 )( C ◦ ) ( iii ) = Q 1 where ( i ) holds b y (3.14), ( ii ) b y deﬁnition of P u , ( iii ) b y the unit axiom in C . Similarly for the left-hand side. If one shows that comp osition ab o v e is equal to ε p 0 ,p 0 0 1 , the univ ersal prop erty of pullbac ks implies ( L ). Now we can reformulate it with the help of (3.12) 3.6 The standar d h -pul lb ack in n Cat 72 and (3.15): [ p 0 , p 0 0 ] λ ∼ =   I × [ p 0 , p 0 0 ] u ( p 0 ) × id   [ p 0 , p 0 ] × [ p 0 , p 0 0 ] v v   id × ε p 0 ,p 0 0 1 W _ 7 7 7 7 7 7 7 7 [ b 1 ] × id # # G G G G G G G G G G G G G G G G [ p 0 , p 0 ] × [ F a 0 , Gc 0 0 ] P F 1 ( − ) ◦− # # G G G G G G G G G G G G G G G G [ F a 0 , Gc 0 ] × [ p 0 , p 0 0 ] −◦ QG 1 ( − ) { { w w w w w w w w w w w w w w w w [ F a 0 , Gc 0 0 ] where we hav e somehow abusively replaced the identit y 2-morphism on the right hand side, with its source (= target) 1-morphism. Hence all the righ t-hand side, being an identit y , ma y be cancelled. Finally , b y pr o duct inter change [ p 0 , p 0 0 ] λ ∼ =   I × [ p 0 , p 0 0 ]     id × ε p 0 ,p 0 0 1 k s I × [ F a 0 , Gc 0 0 ] u ( F a 0 ) ◦−   [ F a 0 , Gc 0 0 ] (1) = [ p 0 , p 0 0 ]     ε p 0 ,p 0 0 1 k s [ F a 0 , Gc 0 0 ] λ ∼ =   I × [ F a 0 , Gc 0 0 ] u ( F a 0 ) ◦−   [ F a 0 , Gc 0 0 ] (2) = ε p 0 ,p 0 0 1 where (1) holds by naturality of λ ( − ) : ( − ) ⇒ I × ( − ), and (2) by neutral iden tities in B . 3.6 The standar d h -pul lb ack in n Cat 73 3.6.3 Pro jections and ε So far w e prov ed that the pair P = ( P 0 , P − , − 1 ) is indeed a n -category . In order to sho w it is a part of a pullback four-tuple, we should prov e that P = ( P 0 , P − , − 1 ) , Q = ( Q 0 , Q − , − 1 ) , ε = ( ε 0 , ε − , − 1 ) pro duced in stating the deﬁnitions ab ov e, constitute resp ectiv ely tw o n- categorie morphisms and one 2-morphism. But this has b een already prov ed throughout the last sections. In fact, universal deﬁnition of 0-comp osition ab o ve reveals that this is just the one that makes P , Q and ε functorial. Similarly , univ ersal deﬁnition of 0-units reveals that these are just the ones that mak e P , Q and ε functorial. 3.6.4 Univ ersal property The ﬁnal step in proving that n Cat admits h -pullbac ks, is to show that the four-tuple ( P , P , Q, ε ) satisﬁes univ ersal prop ert y of h -Pullbacks ( UP 2.12). T o this aim, let us supp ose a n-category X b een giv en, together with mor- phisms and 2-morphisms M : X → A , N : X → C , ω : M F ⇒ N G On ob jects, as P 0 is a limit in Set , it suﬃces to consider the cone ov er the same diagram deﬁning the latter, whose commutativit y is a consequence of the v ery deﬁnition of ω : X 0 M 0 v v n n n n n n n n n n n n n n n ω 0   N 0 ( ( P P P P P P P P P P P P P P P A 0 F 0 A A A A A A A A B 1 s ~ ~ } } } } } } } } t A A A A A A A A C 0 G 0 ~ ~ } } } } } } } } B 0 B 0 This yields a unique map L 0 : X 0 → P 0 , suc h that: L 0 P 0 = M 0 , L 0 Q 0 = N 0 , L 0 ε 0 = ω 0 (3.18) On homs, let us ﬁx ob jects x 0 and x 0 0 in P . By the universal prop erty in dimension n − 1, the four-tuple ( X 1 ( x 0 , x 0 0 ) , M x 0 ,x 0 0 1 , N x 0 ,x 0 0 1 , ε x 0 ,x 0 0 1 ) giv es a unique morphism L x 0 ,x 0 0 1 : X 1 ( x 0 , x 0 0 ) → P 1 ( Lx 0 , Lx 0 0 ) suc h that L x 0 ,x 0 0 1 • 0 P Lx 0 ,Lx 0 0 1 = M x 0 ,x 0 0 1 L x 0 ,x 0 0 1 • 0 Q Lx 0 ,Lx 0 0 1 = N x 0 ,x 0 0 1 (3.19) L x 0 ,x 0 0 1 • 0 ε Lx 0 ,Lx 0 0 1 = ω x 0 ,x 0 0 1 3.6 The standar d h -pul lb ack in n Cat 74 Claim : the pair ( L 0 , L − , − 1 ) constitutes a n -functor L : X → P . Pr o of. The pro of is divided in tw o parts. 1. F unctoriality w.r.t. c omp ositions. Let us ﬁx a triple x 0 , x 0 0 , x 00 0 of ob jects of X . What w e w ant to pro v e is the diagram b elo w commutes: X 1 ( x 0 , x 0 0 ) × X 1 ( x 0 0 , x 00 0 ) X ◦ / / L 1 × L 1   X 1 ( x 0 , x 00 0 ) L 1   P 1 ( Lx 0 , Lx 0 0 ) × P 1 ( Lx 0 0 , Lx 00 0 ) P ◦ / / P 1 ( Lx 0 , Lx 00 0 ) Then, let us consider the h -pullbac k deﬁning P 1 ( Lx 0 , Lx 00 0 ). If we can sho w that the horizonal comp osition of b oth comp osites ab o v e with ε p 0 ,p 00 0 1 coincide, uniqueness forces ( L 1 × L 1 ) • 0 P ◦ = X ◦ • 0 L 1 . Hence, let us follow the chain of equalities b elo w: [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] X ◦   [ x 0 , x 00 0 ] L 1   [ Lx 0 , Lx 00 0 ] | | " " ε Lx 0 ,Lx 00 0 1 k s [ M F x 0 , N Gx 00 0 ] (1) = [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] X ◦   [ x 0 , x 00 0 ]     ω x 0 ,x 00 0 1 k s [ M F x 0 , N Gx 00 0 ] (2) = [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] y y   id × ω x 0 0 ,x 00 0 1 > > > > > > [ c > > > > > >   % % ω x 0 ,x 0 0 1 × id       |        [ x 0 , x 0 0 ] × [ M F x 0 0 , N Gx 00 0 ] M F 1 ( − ) ◦− & & M M M M M M M M M M [ M F x 0 , N Gx 0 0 ] × [ x 0 0 , x 00 0 ] −◦ N G 1 ( − ) x x q q q q q q q q q q [ M F x 0 , N Gx 00 0 ] (1) holds by the third equation of (3.19), (2) by functorialit y w.r.t. comp osi- 3.6 The standar d h -pul lb ack in n Cat 75 tion of ω , (3) = [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] id × L 1 y y r r r r r r r r r r L 1 × id % % L L L L L L L L L L [ x 0 , x 0 0 ] × [ Lx 0 0 , Lx 00 0 ] | | " " id × ε Lx 0 0 ,Lx 00 0 1 k s [ Lx 0 , Lx 0 0 ] × [ x 0 0 , x 00 0 ] | | " " ε Lx 0 ,Lx 0 0 1 × id k s [ x 0 , x 0 0 ] × [ LP F x 0 0 , LQGx 00 0 ] LP F 1 ( − ) ◦− % % L L L L L L L L L L [ LP F x 0 , LQGx 0 0 ] × [ x 0 0 , x 00 0 ] −◦ LQG 1 ( − ) y y r r r r r r r r r r [ M F x 0 , N Gx 00 0 ] here, (3) is a full consequence of equations (3.19) and pro duct in terchange, (4) = [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] L 1 × L 1   [ Lx 0 , Lx 0 0 ] × [ Lx 0 0 , Lx 00 0 ] x x   id × ε Lx 0 0 ,Lx 00 0 1 : : : : : : : : Y a : : : : : :   & & ε Lx 0 ,Lx 0 0 1 × id         ~        [ x 0 , x 0 0 ] × [ P F x 0 0 , QGx 00 0 ] P F 1 ( − ) ◦− ' ' P P P P P P P P P P P P [ P F x 0 , QGx 0 0 ] × [ x 0 0 , x 00 0 ] −◦ QG 1 ( − ) w w n n n n n n n n n n n n [ P F x 0 , QGx 00 0 ] (5) = [ x 0 , x 0 0 ] × [ x 0 0 , x 00 0 ] L 1 × L 1   [ Lx 0 , Lx 0 0 ] × [ Lx 0 0 , Lx 00 0 ] P ◦   [ Lx 0 , Lx 00 0 ]     ε Lx 0 ,Lx 00 0 1 k s [ M F x 0 , N Gx 00 0 ] (4) is obtained sliding the L 1 ’s up, and functoriality w.r.t. comp osition of ε giv es (5). 2. F unctoriality w.r.t. units. Let us ﬁx an ob ject x 0 in X . What we w ant to pro ve is the diagram below comm utes: I u ( x 0 ) / / u ( Lx 0 ) ) ) T T T T T T T T T T T T T T T T T T T X 1 ( x 0 , x 0 ) L 1   P 1 ( Lx 0 , Lx 0 ) W e pro ceed in a similar w ay . Let us consider the pullback deﬁning P 1 ( Lx 0 , Lx 00 0 ). If w e show that the horizontal comp osition of b oth comp osites ab o v e with 3.6 The standar d h -pul lb ack in n Cat 76 ε Lx 0 ,Lx 0 1 coincide, uniqueness forces u ( x 0 ) • 0 L 1 = u ( Lx 0 ). Hence, let us follo w the chain of equalities b elow: I u ( x 0 )   [ x 0 , x 0 ] L 1   [ Lx 0 , Lx 0 ]     ε Lx 0 ,Lx 0 1 k s [ M F x 0 , N Gx 0 ] (1) = I u ( x 0 )   [ x 0 , x 0 ]     ω Lx 0 ,Lx 0 1 k s [ M F x 0 , N Gx 0 ] (2) = I     I d [ ω x 0 ] k s [ M F x 0 , N Gx 0 ] (3) = I u ( Lx 0 )   [ Lx 0 , Lx 0 ]     ε Lx 0 ,Lx 0 1 k s [ M F x 0 , N Gx 0 ] (1) holds by the third of the (3.19), (2) is just ω , the functoriality of units, and since b y (3.18), ω x 0 = ε Lx 0 , (3) is obtained b y ε unit functoriality . Once we ha v e v eriﬁed that L is a n-functor, equations (3.18) and (3.19) taken together are exactly conditions 1., 2. and 3. of Universal Pr op erty 2.12 . What is still missing is uniqueness, but this is implied by the proofs. In fact let us supp ose there is another ˆ L X → P satisfying universal prop erty . Conditions 1., 2. and 3. of (2.12) imply: ˆ L 0 P 0 = M 0 , ˆ L 0 Q 0 = N 0 , ˆ L 0 ε 0 = ω 0 and ˆ L x 0 ,x 0 0 1 • 0 P ˆ Lx 0 , ˆ Lx 0 0 1 = M x 0 ,x 0 0 1 ˆ L x 0 ,x 0 0 1 • 0 Q ˆ Lx 0 , ˆ Lx 0 0 1 = N x 0 ,x 0 0 1 ˆ L x 0 ,x 0 0 1 • 0 ε ˆ Lx 0 , ˆ Lx 0 0 1 = ω x 0 ,x 0 0 1 Since L 0 and the L 1 ’s where determined univ o cally by Universal Prop erties of limits in Set and of h -pullbac ks in ( n − 1) Cat , uniqueness of those forces ˆ L 0 = L 0 and ˆ L − , − 1 = L − , − 1 Hence w e prov ed the Theorem 3.8. The sesqui-c ate gory n Cat admits h -pul lb acks. 3.6 The standar d h -pul lb ack in n Cat 77 3.6.5 Pullbac ks and h -pullbac ks It is p ossible to recov er the usual notion of pullback by means of a similar inductiv e construction: a pullbac k is a univ ersal triple < Q , P , Q > suc h that P F = QG : Q Q / / P   C G   A F / / B ( pb ) where Q 0 is the pullbac k in Set Q 0 Q 0 / / P 0   C 0 G 0   A 0 F 0 / / B 0 ( pb ) and for every pair of ob jects ( a 0 , c 0 ) and ( a 0 0 , c 0 0 ) of Q 0 , the following pullbac k in ( n − 1) Cat Q 1  ( a 0 , c 0 ) , ( a 0 0 , c 0 0 )  Q ( a 0 ,c 0 ) , ( a 0 0 ,c 0 0 ) 1 / / P ( a 0 ,c 0 ) , ( a 0 0 ,c 0 0 ) 1   C 1 ( c 0 , c 0 0 ) G c 0 ,c 0 0 1   A 1 ( a 0 , a 0 0 ) F a 0 ,a 0 0 1 / / B 1 ( F a 0 = Gc 0 , F a 0 0 = Gc 0 0 ) ( pb ) In fact this giv es an h -pullbac k in the trivial (= 2-discrete) sesqui-category o ver the category b n Cat c : the triple < Q , P , Q > suc h that P F = QG ma y b e seen as a four-tuple < Q , P , Q, id : P F ⇒ QG > , and so on. . . Chapter 4 n -Group oids and exact sequences The sesqui-category n Cat of strict and small n-categories, deﬁned so far, has a naturally arising notion of equiv alence that ma y b e deﬁned recursively . This giv es a notion of n-group oid, equiv alent to that of Kaprano v and V o evodsky in [ KV91 ], as a we akly invertible strict n-c ate gory (see App endix A for a comparison). 4.1 n -Equiv alences Deﬁnition 4.1. L et n-c ate gory morphism F : C → D b e given. F is c al le d equiv alence of n-categories if it satisﬁes the fol lowing pr op erties: n = 0 F is an isomorphism in Set . n > 0 1. F is essential ly surje ctive on obje cts, i.e. for every obje ct d 0 of D , ther e exists an obje ct c 0 of C and a 1-c el l d 1 : F c 0 → d 0 such that for every d 0 0 in C , the morphisms d 1 ◦ − : D 1 ( d 0 , d 0 0 ) → D 1 ( F c 0 , d 0 0 ) − ◦ d 1 : D 1 ( d 0 0 , F c 0 ) → D 1 ( d 0 0 , d 0 ) ar e e quivalenc es of (n-1)c ate gories. 2. for every p air c 0 , c 0 0 in C , F c 0 ,c 0 0 1 : C 1 ( c 0 , c 0 0 ) → D 1 ( F c 0 , F c 0 0 ) 4.1 n -Equivalenc es 79 is an e quivalenc e of (n-1)c ate gories. F rom deﬁnition ab o ve, one gets the following Deﬁnition 4.2. A 1-c el l c 1 : c 0 → c 0 0 of a n-c ate gory C is said to b e w eakly inv ertible , or simply an equiv alence , if, for every obje ct ¯ c 0 of C , the morphisms c 1 ◦ − : C 1 ( c 0 0 , ¯ c 0 ) → C 1 ( c 0 , ¯ c 0 ) − ◦ c 1 : C 1 (¯ c 0 , c 0 ) → C 1 (¯ c 0 , c 0 0 ) ar e (natur al) e quivalenc es of ( n − 1) c ate gories. 4.1.1 In v erses When a 1-cell is w eakly in vertible, then it has indeed left and right (quasi) in verses. In fact for c 1 : c 0 → c 0 0 , c 1 ◦ − : C 1 ( c 0 0 , c 0 ) → C 1 ( c 0 , c 0 ) to b e an equiv alence implies that for the 1-cell 1 c 0 : c 0 → c 0 there exists a pair ( c ∗ 1 , c 2 : c 1 ◦ c ∗ 1 ∼ + 3 1 c 1 ) , similarly for − ◦ c 1 : C 1 ( c 0 0 , c 0 ) → C 1 ( c 0 0 , c 0 0 ) implies there exists a pair ( c † 1 , c 0 2 : c † 1 ◦ c 1 ∼ + 3 1 c 0 1 ) . 4.1.2 Prop erties Lemma 4.3. L et n-functors C F / / D G / / E b e given. Then if F and G ar e e quivalenc es, so is F • 0 G . Pr o of. The comp osite of isomorphisms in Set is trivially an isomorphism. Hence w e may supp ose n > 0. 1. for ev ery pair c 0 , c 0 0 , [ F • 0 G ] c 0 ,c 0 0 1 is an equiv alence. In fact [ F • 0 G ] c 0 ,c 0 0 1 = F c 0 ,c 0 0 1 • 0 G F c 0 ,F c 0 0 1 t wo comp onent on the right-hand side are indeed equiv alences by h yp othesis, and so is their comp osites by induction. 4.1 n -Equivalenc es 80 2. F or an y ob ject e 0 of E there exists a pair ( d 0 , e 1 : Gd 0 → e 0 ). Similarly , for any ob ject d 0 in D there is a pair ( c 0 , d 1 : F c 0 → d 0 ). Hence, for giv en e 0 , those pro duce a pair ( c 0 , G ( F c 0 ) Gd 1 / / Gd 0 e 1 / / e 0 ) That left and righ t 0-comp ositions (in D ) with Gd 1 ◦ e 1 are equiv alences, is a statemen t in volving a composition of equiv alences of (n-1)categories, hence giv en by induction. In fact, b y deﬁnition − ◦ ( Gd 1 ◦ e 1 ) = ( − ◦ Gd 1 ) • 0 ( − ◦ e 1 ) and ( Gd 1 ◦ e 1 ) ◦ − = ( e 1 ◦ − ) • 0 ( Gd 1 ◦ − ) Deﬁnition 4.4. A 2-morphism of n-c ate gories α : F ⇒ G : C → D is an e quivalenc e 2-morphism, or n-natur al e quivalenc e, if n = 1 α is a natur al isomorphism. n > 1 1. for any obje ct c 0 in C , the 1-c el l α c 0 is an e quivalenc e 2. for any p air of obje cts c 0 , c 0 0 in C , the (n-1)tr ansformation α c 0 ,c 0 0 1 is an e quivalenc e 2-morphism It is not precisely in the aims of this work, nev ertheless it is worth men tioning the follo wing Prop osition 4.5. n-c ate gories, n-functors and n-natur al e quivalenc es form a sesqui-c ate gory, denote d n Cat eq , Notice that in n Cat eq , and a fortiori in n Gp d (t.b.d.) equiv alences hav e more nice prop erties, lik e they are h -pullbac k stable, hav e the 2of3 prop erty and so on. Deﬁnition 4.6. A morphism of n -c ate gories F : C → D is c al le d h -surje ctive if n = 0 4.1 n -Equivalenc es 81 F is a surje ctive map. n > 0 1. F is essential ly surje ctive on obje cts, i.e. for every obje ct d 0 of D , ther e exists a p air ( c 0 , d 1 : Lc 0 ˜ → d 0 ) , with d 1 an e quivalenc e, 2. for every p air of obje cts c 0 , c 0 0 of C , the morphism F c 0 ,c 0 0 1 : C 1 ( c 0 , c 0 0 ) → D 1 ( F c 0 , F c 0 0 ) is h -surje ctive. Deﬁnition 4.7. A morphism of n -c ate gories F : C → D is c al le d faithful if n = 0 F is a inje ctive map. n > 0 for every p air c 0 , c 0 0 , the (n-1)functor F c 0 ,c 0 0 1 is faithful. The notion of h -surjectiv e is weak er than (implied by) that of equiv alence. In fact, more is true: Prop osition 4.8. A morphism of n-c ate gories F : C → D is an e quivalenc e pr e cisely when it is faithful and h -surje ctive. Pr o of. When n = 0 this is the characterization of bijective maps as injective plus surjectiv e. Hence supp ose n > 0. Let F b e an equiv alence. Then F is essentially surjectiv e b y deﬁnition. More, for ev ery c 0 , c 0 0 , F c 0 ,c 0 0 1 is an equiv alence in ( n − 1) Cat , therefore h -surjectiv e b y induction. Finally , the last is also faithful by induction, and this concludes the ﬁrst implication. Con versely , let F b e faithful and h -surjectiv e. Then it is essen tially surjective b y deﬁnition. More, for every c 0 , c 0 0 , F c 0 ,c 0 0 1 ’s are faithful and h -surjectiv e, and inductiv e hypothesis implies they are equiv alences. Notice that faithfulness can b e reform ulated saying that the n-functor is surje ctive on e quations . In fact, from the globular p oin t of view, this is equiv alen t to saying that equal n -cells in the image of F come from equal n -cells of C . Under this p ersp ectiv e, to b e h -surjectiv e amounts to b eing (w eakly) surjective in any dimension, up to n − 1, and last prop osition sa ys precisely that an equiv alence is (w eakly) surjectiv e on k -cell, with 0 ≤ k ≤ n . 4.2 n -Gr oup oids 82 R emark 4.9 . The notions of h -surjectiv e morphisms and of equiv alences reduce to w ell kno wn ones, when considered in low dimension. In fact, in dimension one those are stated explicitly in the deﬁnitions as the ﬁrst step of the inductiv e pro cess. F or n = 1 an h -surjectiv e morphism is a functor which is full and essen tially surjective on ob ject. Hence the notion of equiv alence is the usual one. Finally we state a useful Lemma, whose pro of is part of the pro of of L emma 4.3 : Lemma 4.10. L et n -functors C F / / D G / / E b e given. Then if F and G ar e h -surje ctive, so is F • 0 G . 4.2 n -Group oids Deﬁnition 4.11. The deﬁnition is inductive on n . n = 0 A 0 -gr oup oid is a 0 -c ate gory, i.e. a set. n > 0 A n-gr oup oid is a n -c ate gory C such that: 1. every 1-c el l of C is an e quivalenc e; 2. for every p air of obje cts c 0 , c 0 0 of C the ( n − 1) c ate gory C 1 ( c 0 , c 0 0 ) is a ( n − 1) gr oup oid. W e denote by n Gp d the sub-sesqui-category of n Cat generated by n - group oids. Prop osition 4.12. F or every given natur al numb er n , the fol lowing is a diagr am of inclusions: nGp d   / / o  ( ∗ )   ? ? ? ? ? ? ? ? ? ? ? ? ? ? n Cat n Cat eq  / @ @               Pr o of. The case = 1 is well known, hence supp ose n > 1. The only inclusion to b e pro ved is the one marked ( ∗ ). T o this end, it suﬃce to show that 2-morphisms of n-group oids are equiv alences. But, condition 4.3 The sesqui-functor π 0 83 1. of L emma 4.4 ab ov e is automatically satisﬁed, as n-group oid’s 1-cells are alw ays equiv alences, condition 2. is given by induction. Prop osition 4.13. n Gp d is close d under h-pul lb acks. Pr o of. F or n = 0 the result holds trivially . Hence let us supp ose n > 0. F or the h -pullbac k P Q / / P   C G   A F / / B ε ; C         a 1-cell ( a 0 a 1   , F a 0 F a 1   b 1 / / Gc 0 Gc 1   b 2 u } s s s s s s s s s s s s , c 0 ) c 1   ( a 0 0 , F a 0 0 b 0 1 / / Gc 0 0 , c 0 0 ) induces equiv alences because a 1 , c 1 and b 2 do. T urning to homs, the statement is relativ e to ( n − 1) group oids and holds by induction. 4.3 The sesqui-functor π 0 Purp ose of this section is to in tro duce the family of sesqui-functors { π ( n ) 0 } n ∈ N ∗ that extends the iso-classes functor Gp d → Set . Deﬁnition/Prop osition 4.14. F or any inte ger n > 0 , ther e exists a clas- sifying sesqui-functor π ( n ) 0 : nGp d → ( n − 1 ) Gp d ac c or ding to the fol lowing inductive deﬁnition. Mor e over, it c ommutes with ﬁnite pr o ducts and it pr eserves e quivalenc es. n = 1 π (1) 0 : n Gp d → ( n − 1) Gp d is the functor (= trivial sesqui-functor) Gp d → Set that assigns to a group oid C the set | C | of isomorphism classes of ob jects of C . It commutes with ﬁnite pro ducts: in fact the terminal set {∗} is exactly the classiﬁed terminal group oid I , and in a pro duct of group oids C × D , an isomorphism ( c 1 , d 1 ) is a pair of isomorphisms c 1 in C and d 1 in D . 4.3 The sesqui-functor π 0 84 Finally , it sends equiv alences of category in isomorphism-maps. n > 1 4.3.1 π 0 on ob jects Let a n -group oid C b e giv en. Then, π ( n ) 0 C = ([ π ( n ) 0 C ] 0 , [ π ( n ) 0 C ] 1 ( − , − )), where • [ π ( n ) 0 C ] 0 = C 0 • for ev ery pair c 0 , c 0 0 in [ π ( n ) 0 C ] 0 , [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) = π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 )) F or a triple of ob jects c 0 , c 0 0 , c 00 0 , comp osition is the dotted arro w b elow: [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) × [ π ( n ) 0 C ] 1 ( c 0 0 , c 00 0 ) π ( n ) 0 C ◦ / / def [ π ( n ) 0 C ] 1 ( c 0 , c 00 0 ) def π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 )) × π ( n − 1) 0 ( C 1 ( c 0 0 , c 00 0 )) (1) π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 , c 0 0 )) π ( n − 1) ( C ◦ ) / / π ( n − 1) 0 ( C 1 ( c 0 , c 00 0 )) F or an ob ject c 0 , unit morphism is the dotted arro w b elow: I ( n ) u ( c 0 ) / / (2) [ π ( n ) 0 C ] 1 ( c 0 , c 0 ) def π ( n − 1) 0 ( I ( n − 1) ) π ( n − 1) 0 ( u ( c 0 )) / / π ( n − 1) 0 ( C 1 ( c 0 , c 0 )) Equalities (1) and (2) hold b ecause π ( n − 1) 0 comm utes with (ﬁnite) pro ducts b y induction hypothesis. Claim 4.15. The p air ([ π ( n ) 0 C ] 0 , [ π ( n ) 0 C ] − , − 1 ) , with c omp osition and units as deﬁne d ab ove, satisﬁes axioms for a (n-1) c ate gory. Mor e over, π ( n ) 0 C is a (n-1) gr oup oid. 4.3 The sesqui-functor π 0 85 Pr o of. W e hav e to prov e asso ciativit y and unit axioms. Concerning asso ciativit y , let ob jects c 0 , c 0 0 , c 00 0 , c 000 0 b e giv en. Let us consider the follo wing equalities of morphisms [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) × [ π ( n ) 0 C ] 1 ( c 0 0 , c 00 0 ) × [ π ( n ) 0 C ] ( c 00 0 , c 000 0 )1 → [ π ( n ) 0 C ] 1 ( c 0 , c 000 0 ) , ( π 0 C ◦ c 0 ,c 0 0 ,c 00 0 × id ) • 0 π 0 C ◦ c 0 ,c 00 0 ,c 000 0 ( def ) = π ( n − 1) 0 ( C ◦ c 0 ,c 0 0 ,c 00 0 × id ) • 0 π ( n − 1) 0 ( C ◦ c 0 ,c 00 0 ,c 000 0 ) (1) = π ( n − 1) 0 (( C ◦ c 0 ,c 0 0 ,c 00 0 × id ) • 0 C ◦ c 0 ,c 00 0 ,c 000 0 ) (2) = π ( n − 1) 0 (( id × C ◦ c 0 0 ,c 00 0 ,c 000 0 ) • 0 C ◦ c 0 ,c 0 0 ,c 000 0 ) (3) = π ( n − 1) 0 ( id × C ◦ c 0 0 ,c 00 0 ,c 000 0 ) • 0 π ( n − 1) 0 ( C ◦ c 0 ,c 0 0 ,c 000 0 ) ( def ) = ( id × π 0 C ◦ c 0 0 ,c 00 0 ,c 000 0 ) • 0 π 0 C ◦ c 0 ,c 0 0 ,c 000 0 where, (1) and (3) hold by functoriality of π ( n − 1) 0 , (2) by asso ciativity of 0-comp osition in C . T urning to left-units, for every pair c 0 , c 0 0 , one has the following equalities of morphisms: [ π ( n ) 0 C ] 1 ( c 0 0 , c 0 ) → [ π ( n ) 0 C ] 1 ( c 0 0 , c 0 ) λ • 0 ( id × u ( c 0 )) • 0 ◦ c 0 0 ,c 0 ,c 0 ( def ) = π ( n − 1) 0 ( λ ) • 0 π ( n − 1) 0 ( id × u ( c 0 )) • 0 π ( n − 1) 0 ( ◦ c 0 0 ,c 0 ,c 0 ) (1) = π ( n − 1) 0 ( λ • 0 ( id × u ( c 0 )) • 0 ◦ c 0 0 ,c 0 ,c 0 ) (2) = π ( n − 1) 0 ( id C 1 ( c 0 0 ,c 0 ) ) (3) = id π ( n − 1) 0 C 1 ( c 0 0 ,c 0 ) ( def ) = id [ π ( n ) 0 C ] 1 ( c 0 0 ,c 0 ) where 1 and 3 hold b y functoriality of π ( n − 1) 0 , (2) by neutrality of 0-iden tities in C . Righ t units are dealt the same wa y . Finally , in order to sho w that π ( n ) 0 C is a ( n − 1)group oid, tw o facts hav e to b e pro ved: 1. all 1-cells of π ( n ) 0 C are equiv alences. 2. all homs [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) are ( n − 2)group oids. The ﬁrst fact is an easy consequence of the v ery deﬁnition of π 0 on comp osi- tions. In fact, let ob ject ¯ c 0 and 1-cell ˜ x : c 0 → c 0 0 in π ( n ) C b e given. Then b y deﬁnition, − ◦ ˜ x = π ( n − 1) 0 ( − ◦ x ), where x = ˜ x if n > 1, or { x } ∼ = ˜ x if n = 1. The result follows, for π ( n − 1) 0 preserv es equiv alences. 4.3 The sesqui-functor π 0 86 T o prov e the second statement, let us consider the ( n − 2)category [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ). This is deﬁned to b e π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 )), where C 1 ( c 0 , c 0 0 ) is a ( n − 1)group oid. F or π ( n − 1) 0 the result follo ws by induction. 4.3.2 π 0 on morphisms Let a n-functor F : C → D b e giv en. Then, π ( n ) 0 F = ([ π ( n ) 0 F ] 0 , [ π ( n ) 0 F ] − , − 1 ), where • [ π ( n ) 0 F ] 0 = F 0 • for ev ery pair c 0 , c 0 0 in [ π ( n ) 0 C ] 0 [ π ( n ) 0 F ] c 0 ,c 0 0 1 = π ( n − 1) 0 ( F c 0 ,c 0 0 1 ) Claim 4.16. The p air ([ π ( n ) 0 F ] 0 , [ π ( n ) 0 F ] − , − 1 ) satisﬁes axioms for ( n − 1) functors. Pr o of. W e hav e to prov e functorialit y w.r.t. c omposition and units. Concerning comp osition, let ob jects c 0 , c 0 0 , c 00 0 b e given. Let us consider the follo wing equalities of morphisms [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) × [ π ( n ) 0 C ] 1 ( c 0 0 , c 00 0 ) → [ π ( n ) 0 D ] 1 ( F c 0 , F c 00 0 ) π ( n ) 0 C ◦ • 0 [ π ( n ) 0 F ] c 0 ,c 00 0 1 ( def ) = π ( n − 1) 0 ( C ◦ ) • 0 π n − 1 0 ( F c 0 ,c 00 0 1 ) (1) = π ( n − 1) 0 ( C ◦ • 0 F c 0 ,c 00 0 1 ) (2) = π ( n − 1) 0 (( F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1 ) • 0 D ◦ ) (3) = π ( n − 1) 0 ( F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1 ) • 0 π ( n − 1) 0 ( D ◦ ) ( def ) = ([ π ( n ) 0 F ] c 0 ,c 0 0 1 × [ π ( n ) 0 F ] c 0 0 ,c 00 0 1 ) • 0 π ( n ) 0 D ◦ (1) and (3) are justiﬁed by the sesqui-functor π ( n − 1) 0 preserving comp osition, (2) is functorialit y w.r.t. comp osition of F . T urning to identities, for every c 0 one has the following equalities of mor- phisms I ( n ) → [ π ( n ) 0 ] 1 ( F c 0 , F c 0 ) u ( c 0 ) • 0 [ π ( n ) 0 D ] c 0 ,c 0 1 ( def ) = π ( n − 1) 0 ( u ( c 0 )) • 0 π ( n − 1) 0 ( F c 0 ,c 0 1 ) (1) = π ( n − 1) 0 ( u ( c 0 ) • 0 F c 0 ,c 0 1 ) (2) = π ( n − 1) 0 ( u ( F c 0 ) ( def ) = u ( F c 0 ) where (1) holds for π ( n − 1) 0 preserving comp osition, (2) by functorialit y w.r.t. units of F . 4.3 The sesqui-functor π 0 87 4.3.3 The underlying functor In order to b e a sesqui-functor, π 0 m ust restrict to a functor b et ween the underlying categories: b π ( n ) 0 c : b nGp d c → b ( n − 1 ) Gp d c Claim 4.17. Given the situation C F / / D G / / E in n Gp d , the assignments given ab ove satisfy 1. π ( n ) 0 ( F • 0 G ) = π ( n ) 0 F • 0 π ( n ) 0 G 2. π ( n ) 0 ( id C ) = id π ( n ) 0 C Pr o of. Let us chec k ﬁrst num b er 1. [ π ( n ) 0 ( F • 0 G )] 0 ( def ) = [ F • 0 G ] 0 ((1)) = F 0 G 0 ( def ) = [ π ( n ) 0 F ] 0 • 0 [ π ( n ) 0 G ] 0 where (1) holds b y comp osition of n-functors. Moreo ver, for ob jects c 0 , c 0 0 , [ π ( n ) 0 ( F • 0 G )] c 0 ,c 0 0 1 ( def ) = π ( n − 1) 0 ([ F • 0 G ] c 0 ,c 0 0 1 ) (1) = π ( n − 1) 0 ( F c 0 ,c 0 0 1 • 0 G F c 0 ,F c 0 0 1 ) (2) = π ( n − 1) 0 ( F c 0 ,c 0 0 1 ) • 0 π ( n − 1) 0 ( G F c 0 ,F c 0 0 1 ) ( def ) = π ( n ) 0 ( F c 0 ,c 0 0 1 ) • 0 π ( n ) 0 ( G F c 0 ,F c 0 0 1 ) where (1) holds again by comp osition of n-functors, and (2) by functorialit y of π ( n − 1) 0 . No w, let us chec k num b er 2. [ π ( n ) 0 ( id C )] 0 = [ id C ] 0 = id C 0 = id [ π ( n ) 0 C ] 0 = h id π ( n ) 0 C i 0 follo ws straight from deﬁnitions. 4.3 The sesqui-functor π 0 88 F or ob jects c 0 , c 0 0 [ π ( n ) ( id C )] c 0 ,c 0 0 1 = π ( n − 1) 0 ([ id C ] c 0 ,c 0 0 1 ) = π ( n − 1) 0 ( id C 1 ( c 0 ,c 0 0 ) ) (1) = id π ( n − 1) 0 ( C 1 ( c 0 ,c 0 0 )) = id [ π ( n ) 0 C ] 1 ( c 0 ,c 0 0 ) = h id π ( n ) 0 C i c 0 ,c 0 0 1 where every equality ab ov e comes from deﬁnitions, but (1) that holds by functorialit y of π ( n − 1) 0 . 4.3.4 π 0 on 2-morphisms The action of π ( n ) 0 on 2-morphisms is more sensible to deﬁne. Indeed, on ob jects and morphisms w e had the obje ct-p art of deﬁnitions that were mere equalities, while the homs w ere given by induction. No w the situation is diﬀerent, since the obje ct-p art of a 2-cell is a map inv olving also the 1-cells of co domain n -group oid. That is the reason why w e must analyze carefully what happ ens in lo w dimension, in order to start induction properly . n = 2 Let us consider π 00 0 : 2 Gp d → Gp d : C F   G A A D α   7→ π 00 0 C π 00 0 F $ $ π 00 0 G : : π 00 0 D π 00 0 α   (in order to simplify notation, for lo w dimensions we use primes). No w it is clear that [ π 00 0 D ] 1 = D 1 / ∼ , where ∼ is the equiv alence relation on the set of 1-cells of D giv en b y iso-2-cells. Call p : D 1 → D 1 / ∼ the canonic pro jection on to the quotient. Then w e let [ π 00 0 α ] 0 = α 0 · p This is well deﬁned, since equiv alence classes in D 1 resp ect 1-cell’s sources and targets. Moreov er they are compatible with 0-comp osition, i.e. p ( d 1 ◦ d 0 1 ) = p ( d 1 ) ◦ p ( d 0 1 ). In fact for every ob ject c 0 of C , one has [ π 00 0 α ] 0 ( c 0 ) = { α 0 ( c 0 ) } ∼ : F c 0 = [ π 00 0 F ]( c 0 ) → [ π 00 0 G ]( c 0 ) = Gc 0 4.3 The sesqui-functor π 0 89 Hence if w e choose a 1-cell ˜ c 1 : c 0 → c 0 0 , sa y ˜ c 1 = { c 1 } ∼ , then π 00 0 sends the 2-isomorphism α c 0 ,c 0 0 1 ( c 1 ) in the equalit y [ π 00 0 α ] c 0 ◦ { Gc 1 } ∼ = { F c 1 } ∼ ◦ [ π 00 0 α ] c 0 0 This prov es that π 00 0 α is a natural isomorphism of group oids, i.e. a 2- morphisms in Gp d . n > 2 More generally , supp ose w e are given a 2-morphism α : F ⇒ G : C → D in n Gp d . Then we deﬁne π ( n ) 0 α as the pair ([ π ( n ) 0 α ] 0 , [ π ( n ) 0 α ] − , − 1 ), where • [ π ( n ) 0 α ] 0 = α 0 , • for ev ery pair of ob jects c 0 , c 0 0 of C , [ π ( n ) 0 α ] c 0 ,c 0 0 1 = π ( n − 1) 0 ( α c 0 ,c 0 0 1 ) . Claim 4.18. The p air ([ π ( n ) 0 α ] 0 , [ π ( n ) 0 α ] − , − 1 ) satisﬁes axioms for ( n − 1) tr ansformations. Pr o of. It is w ell-deﬁned on ob jects, since n > 2. Moreov er, it is w ell-deﬁned also on homs. In fact, given ob jects c 0 and c 0 0 , consider the diagram: [ π ( n ) 0 C ] 1 ( c 0 , c 0 0 ) D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D [ π ( n ) 0 G ] c 0 ,c 0 0 1 / / [ π ( n ) 0 F ] c 0 ,c 0 0 1   [ π ( n ) 0 D ] 1 ( Gc 0 , Gc 0 0 ) z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z [ π ( n ) 0 α ] c 0 ◦−   π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 )) / / π ( n − 1) 0 F c 0 ,c 0 0 1   π ( n − 1) 0 ( D 1 ( Gc 0 , Gc 0 0 )) π ( n − 1) 0 ( α c 0 ◦− )   π ( n − 1) 0 ( D 1 ( F c 0 , F c 0 0 )) / / π ( n − 1) 0 ( D 1 ( F c 0 , Gc 0 0 )) [ π ( n ) 0 D ] 1 ( F c 0 , F c 0 0 ) z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z −◦ [ π ( n ) 0 α ] c 0 0 / / [ π ( n ) 0 D ] 1 ( Gc 0 , Gc 0 0 ) D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D π ( n − 1) 0 α c 0 ,c 0 0 1 r r r r r r r r r r t | r r r r r r r r r r π ( n − 1) 0 G c 0 ,c 0 0 1 π ( n − 1) 0 ( −◦ α c 0 0 ) 4.3 The sesqui-functor π 0 90 Up and left squares are the deﬁnition of π ( n ) 0 on n-functors (w.r.t. hom- comp onen ts), do wn and right commute by deﬁnition of comp osition in π ( n ) 0 ( D ), and of [ π ( n ) 0 α ] 0 ab o ve. And that is all. In fact, coherence w.r.t. comp osition and units are satisﬁed b e- cause their diagrams-equations are π n − 1 0 of corresponding diagrams-equations that hold for α . F or the same reason π ( n ) 0 1. is functorial w.r.t. vertical comp osition and units of 2-morphisms 2. preserv es reduced horizontal comp osition i.e. it is a sesqui-functor. Pr o of. 1. Supp ose w e are given ω : E ⇒ F : C → D α : F ⇒ G : C → D in n Gp d . When n > 2, for every ob ject c 0 in π ( n ) 0 C one has [ π ( n ) 0 ( ω α )] 0 ( c 0 ) (1) = ( ω α ) 0 ( c 0 ) (2) = ω 0 c 0 ◦ α 0 c 0 (3) = [ π ( n ) 0 ( ω )] 0 ( c 0 ) ◦ [ π ( n ) 0 ( α )] 0 ( c 0 ) where (1) and (3) hold b y deﬁnition of π ( n ) 0 , and (2) by deﬁnition of v ertical comp osition. When n = 2 , one has [ π ( n ) 0 ( ω α )] 0 ( c 0 ) (1) = p  ( ω α ) 0 ( c 0 )  (2) = p  ω 0 c 0 ◦ α 0 c 0  (3) = p ( ω 0 c 0 ) ◦ p ( α 0 c 0 ) (4) = [ π ( n ) 0 ( ω )] 0 ( c 0 ) ◦ [ π ( n ) 0 ( α )] 0 ( c 0 ) where (1) and (4) hold by deﬁnition of π ( n ) 0 , (2) b y deﬁnition of v ertical comp osition and (3) for p is compatible with 0-comp osition. No w supp ose we are given ob jects c 0 , c 0 0 of π ( n ) 0 C . Then one has: [ π ( n ) 0 ( ω α )] c 0 ,c 0 0 1 (1) = π ( n − 1) 0 ([ ω α ] c 0 ,c 0 0 1 ) (2) = π ( n − 1) 0  ( α c 0 ,c 0 0 1 • 0 R [ ω c 0 ◦ − ]) • 1 ( ω c 0 ,c 0 0 1 • 0 R [ − ◦ α c 0 0 ])  (3) = ( π ( n − 1) 0 α c 0 ,c 0 0 1 • 0 R π ( n − 1) 0 [ ω c 0 ◦ − ]) • 1 ( π ( n − 1) 0 ω c 0 ,c 0 0 1 • 0 R π ( n − 1) 0 [ − ◦ α c 0 0 ]) (4) = ([ π ( n ) 0 α ] c 0 ,c 0 0 1 • 0 R π ( n − 1) 0 [ ω c 0 ◦ − ]) • 1 ([ π ( n ) 0 ω ] c 0 ,c 0 0 1 • 0 R π ( n − 1) 0 [ − ◦ α c 0 0 ]) (5) = ([ π ( n ) 0 α ] c 0 ,c 0 0 1 • 0 R [[ π ( n ) 0 ω ] c 0 ◦ − ]) • 1 ([ π ( n ) 0 ω ] c 0 ,c 0 0 1 • 0 R [ − ◦ [ π ( n ) 0 α ] c 0 0 ]) 4.3 The sesqui-functor π 0 91 where (1) and (4) hold b y the deﬁnition of π ( n ) 0 on 2-morphisms w.r.t. homs, (2) by deﬁnition of vertical composition, (3) b ecause π ( n ) 0 is a sesqui-functor (induction), (5) b y deﬁnition of π ( n ) 0 D 0-comp osition. Concerning units, let us consider id F : F ⇒ F : [ π ( n ) 0 id F ] 0 ( c 0 ) = [ id F ] 0 ( c 0 ) = id F c 0 = id [ π ( n ) 0 F ] c 0 and also [ π ( n ) 0 id F ] c 0 ,c 0 0 1 = π ( n − 1) 0 ([ id F ] c 0 ,c 0 0 1 ) = id π ( n − 1) 0 ( F c 0 ,c 0 0 1 ) = id [ π ( n ) 0 F ] c 0 ,c 0 0 1 . 2. W e prov e the statement for reduced left-comp osition. Supp ose 2- morphism α as ab o ve, and morphism N : B → C b e given. When n > 2, for an y ob jects b 0 of B , one has [ π ( n ) 0 ( N • 0 L α )] 0 ( b 0 ) (1) = [ N • 0 L α ] 0 ( b 0 ) (2) = α 0 ( N ( b 0 )) (3) = [ π ( n ) 0 α ] 0 ( N ( b 0 )) (4) = [ π ( n ) 0 α ] 0 ([ π ( n ) 0 N ]( b 0 )) (5) = [ π ( n ) 0 N • 0 L π ( n ) 0 α ] 0 ( b 0 ) where (1), (3) and (4) hold b y deﬁnition of π ( n ) 0 , (2) and (5) b y deﬁnition of reduced left-comp osition, Moreo ver, let us choose tw o ob jects b 0 and b 0 0 in B . Then one has [ π ( n ) 0 ( N • 0 L α )] b 0 ,b 0 0 1 (1) = π ( n − 1) 0 ([ N • 0 L α ] b 0 ,b 0 0 1 ) (2) = π ( n − 1) 0 ( N b 0 ,b 0 0 1 • 0 L α N b 0 ,N b 0 0 1 ) (3) = π ( n − 1) 0 ( N b 0 ,b 0 0 1 ) • 0 L π ( n − 1) 0 ( α N b 0 ,N b 0 0 1 ) (4) = [ π ( n ) 0 N ] b 0 ,b 0 0 1 • 0 L [ π ( n ) 0 α ] π ( n ) 0 N ( b 0 ) ,π ( n ) 0 N ( b 0 0 ) 1 where (1) and (4) hold b y deﬁnition of π ( n ) 0 , (2) by deﬁnition of reduced left-comp osition and (3) b y induction hypothesis. When n = 2 the calculation can b e carried on similarly , as w e did for v ertical comp osites ab ov e. 4.3 The sesqui-functor π 0 92 Finally , concerning reduced right-composition, the pro of is similar, as the deﬁnition. 4.3.5 π 0 comm utes with (ﬁnite) pro ducts W e will sho w that π ( n ) 0 preserv es binary pro ducts and the terminal ob ject. Prop osition 4.19. L et C and D b e n-gr oup oids. Then 1. π ( n ) 0 ( C × D ) = π ( n ) 0 C × π ( n ) 0 D 2. π ( n ) 0  I ( n )  = I ( n − 1) Pr o of. 1. Consider the follo wing equalities: [ π ( n ) 0 ( C × D )] 0 (1) = π ( n − 1) 0 ([ C × D ] 0 ) (2) = π ( n − 1) 0 ( C 0 × D 0 ) (3) = π ( n − 1) 0 C 0 × π ( n − 1) 0 D 0 (4) = [ π ( n ) 0 C ] 0 × [ π ( n ) 0 D ] 0 Moreo ver, let ob jects ( c 0 , d 0 ) and ( c 0 0 , d 0 0 ) of C × D b e given. Then [ π ( n ) 0 ( C × D )] ( c 0 ,d 0 ) , ( c 0 0 ,d 0 0 ) 1 (1) = π ( n − 1) 0  [ C × D ] ( c 0 ,d 0 ) , ( c 0 0 ,d 0 0 ) 1  (2) = π ( n − 1) 0  C c 0 ,c 0 0 1 × D d 0 ,d 0 0 1  (3) = π ( n − 1) 0 ( C c 0 ,c 0 0 1 ) × π ( n − 1) 0 ( D d 0 ,d 0 0 1 ) (4) = [ π ( n ) 0 C ] c 0 ,c 0 0 1 × [ π ( n ) 0 D ] d 0 ,d 0 0 1 In b oth cases, (1) and (4) follow from the deﬁnition of π ( n ) 0 , (2) holds b y deﬁnition of pro ducts and (3) by induction. 2. Consider the follo wing equalities: h π ( n ) 0  I ( n ) i 0 (1) = h I ( n ) i 0 (2) = {∗} (3) = h I ( n − 1) i 0 (4) = h π ( n ) 0  I ( n − 1) i 0 4.3 The sesqui-functor π 0 93 where (1) and (4) hold by deﬁnition of π ( n ) 0 , (2) and (3) hold by deﬁnition of terminal n-category . h π ( n ) 0  I ( n ) i ∗ , ∗ 1 (1) = π ( n − 1) 0 h I ( n ) i ∗ , ∗ 1  (2) = π ( n − 1) 0  I ( n − 1)  (3) = I ( n − 2) (4) = h I ( n − 1) i ∗ , ∗ 1 where (1) and (4) follow from the deﬁnition of π ( n ) 0 , (2) holds by deﬁnition of terminal n-category and (3) b y induction. 4.3.6 π 0 preserv es equiv alences Prop osition 4.20. L et F : C → D b e an e quivalenc e of n-gr oup oids. Then π ( n ) 0 F : π ( n ) 0 C → π ( n ) 0 D is an e quivalenc e of (n-1)-gr oup oids. Pr o of. If n = 1, then F is an equiv alence of categories, then π 0 0 F is clearly an isomorphism. Hence we may well supp ose n > 1. 1. Let ob jects c 0 , c 0 0 of C b e giv en. Then, b y deﬁnition, [ π ( n ) 0 F ] c 0 ,c 0 0 1 = π ( n − 1) 0 ( F c 0 ,c 0 0 1 ). This is an equiv alences of (n-2)group oids, since F c 0 ,c 0 0 1 is an equiv alence of (n-1)group oids and π ( n − 1) 0 preserv es equiv alences b y induction. 2. Let an ob ject d 0 of π ( n ) 0 D b e given. This is indeed an ob ject of D , hence there exists a pair ( c 0 , d 1 : F c 0 → d 0 ) with d 1 b eing an equiv alence in D . No w, c 0 is also an ob ject of π ( n ) 0 C , and d 1 (ev entually { d 1 } ∼ if n = 2) is a 1-cell (hence an equiv alence) of the (n-1) group oid π ( n ) 0 D . 4.3.7 A remark on π 0 By the discussion in the previous section, π ( n ) 0 is clearly extendible to a functor b n Cat c → b ( n − 1) Cat c on underlying categories. Diﬃculties arise when w e try to extend it further to a sesqui-functor. In fact, even for n = 1 our deﬁnition fails, since a 4.3 The sesqui-functor π 0 94 natural transformation α : F ⇒ G : C → D do es not imply that the maps π 0 F ⇒ π 0 G : π 0 C → π 0 D are equal. This is the case when α is a natural isomorphism, i.e. for every c 0 in C , α c 0 is an isomorphism. This case may b e generalized in order to get a sesqui-functor on n -categories that remo ve the obstruction b y c onsidering only n -natural equiv alences: π ( n ) 0 : n Cat eq → ( n − 1) Cat eq This giv es a chance to develop the theory in a more general case. On the other hand, if one w ants to keep into account al l n-transformations, one direction can b e to consider a generalization of c onne cte d-c omp onents functor, rather then iso-classes functor. A t the momen t, w e ha ve not explored this p ersp ective since it seems to give rise to a completely diﬀeren t theory , not consisten t with low-dimensional problems we aim to generalize. 4.4 The discr etizer 95 4.4 The discretizer Purp ose of this section is to in tro duce the family of sesqui-functors { D ( n ) } n ∈ N that extends the discr ete-gr oup oid functor Set → Gp d . Deﬁnition/Prop osition 4.21. F or any inte ger n > 0 , ther e exists a n - discr ete sesqui-functor D ( n ) : ( n − 1) Gp d → n Gp d ac c or ding to the fol lowing r e cursive deﬁnition. Mor e over it c ommutes with ﬁnite pr o ducts and it pr eserves e quivalenc es n = 1 D (1) : Set → Gp d is the functor (= trivial sesqui-functor) that assigns to a set C the discrete group oid D ( C ), i.e. with ob jects the elements of C and only iden tity arrows. n > 1 4.4.1 D ( n ) on ob jects and morphisms Let a ( n − 1)-group oid C b e giv en. Then D ( n ) C = ([ D ( n ) C ] 0 , [ D ( n ) C ] 1 ( − , − )), where • [ D ( n ) C ] 0 = C 0 • for ev ery pair c 0 , c 0 0 in [ D ( n ) C ] 0 , [ D ( n ) C ] 1 ( c 0 , c 0 0 ) = D ( n − 1) ( C 1 ( c 0 , c 0 0 )) F or a triple of ob jects c 0 , c 0 0 , c 00 0 , comp osition is the dotted arro w: [ D ( n ) C ] 1 ( c 0 , c 0 0 ) × [ D ( n ) C ] 1 ( c 0 0 , c 00 0 ) D ( n ) C ◦ / / def [ D ( n ) C ] 1 ( c 0 , c 00 0 ) def D ( n − 1) ( C 1 ( c 0 , c 0 0 )) × D ( n − 1) ( C 1 ( c 0 0 , c 00 0 )) (1) D ( n − 1) ( C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 , c 0 0 )) D ( n − 1) ( C ◦ ) / / D ( n − 1) 0 ( C 1 ( c 0 , c 00 0 )) 4.4 The discr etizer 96 F or an ob ject c 0 , unit morphism is the dotted arro w: I ( n ) u ( c 0 ) / / (2) [ D ( n ) 0 C ] 1 ( c 0 , c 0 ) def D ( n − 1) 0 ( I ( n − 1) ) D ( n − 1) 0 ( u ( c 0 )) / / D ( n − 1) 0 ( C 1 ( c 0 , c 0 )) Equalities (1) and (2) hold b ecause D ( n − 1) comm utes with (ﬁnite) pro ducts b y inductive hypothesis. Let a ( n − 1)functor F : C → D b e giv en. Then, D ( n ) F = ([ D ( n ) F ] 0 , [ D ( n ) F ] − , − 1 ), where • [ D ( n ) F ] 0 = F 0 • for ev ery pair c 0 , c 0 0 in [ D ( n ) C ] 0 [ D ( n ) F ] c 0 ,c 0 0 1 = D ( n − 1) ( F c 0 ,c 0 0 1 ) Notice that, since deﬁnitions of π ( n ) 0 and of D ( n ) are formally identical, pro ving that all ab ov e is consistent is a matter of a syntactical substitution of the ﬁrst with the second in the corresp onding pro ofs concerning π ( n ) 0 . Hence w e hav e deﬁned the following functor b etw een underlying categories: b D ( n ) c : b ( n − 1 ) Gp d c → b nGp d c 4.4.2 D ( n ) on 2-morphisms Unlik e that of π ( n ) 0 , the deﬁnition of D ( n ) is straightforw ard since the b egin- ning of induction. Let α : F ⇒ G : C → D in n Gp d b e giv en. As usual, D ( n ) ( α ) = ([ D ( n ) α ] 0 , [ D ( n ) α ] − , − 1 ) where • [ D ( n ) α ] 0 = α 0 • for ev ery pair of ob jects c 0 , c 0 0 in [ D C ] 0 [ D ( n ) α ] c 0 ,c 0 0 1 = D ( n − 1) ( α c 0 ,c 0 0 1 ) Mo dulo the syn tactical conv ersion mentioned ab ov e, we can prov e that D ( n ) is a sesqui-functor that commutes with pro ducts and preserv es equiv alences. Hence it is w ell deﬁned on n -group oids. 4.5 The adjunction π ( n ) 0 a D ( n ) 97 4.4.3 A remark on D ( n ) Diﬀeren tly from π ( n ) 0 , the deﬁnition of D ( n ) extends with no changes to n -categories. In fact it lives more naturally in an n -categorical setting, and our deﬁnition is just its restriction to n -group oids 4.5 The adjunction π ( n ) 0 a D ( n ) 4.5.1 In lo w dimension The follo wing adjunction has b een extensiv ely studied by category-theorist: Gp d π 0 * * ⊥ Set D k k Let us describ e it brieﬂy , as it will b e the ﬁrst step of an inductiv e deﬁnition for general n-group oids. co-unit Giv en a set S , it is clear that π 0 ( D ( S )) = S , hence co-unit  is the iden tity . unit F or a group oid C , the unit η = η C : C → D ( π 0 ( C )) is the pr oje ction given by η C ( c 0 ) = [ c 0 ] ∼ , and for c 1 : c 0 → c 0 0 , η C ( c 1 ) = id [ c 0 ] ∼ = id [ c 0 0 ] ∼ 4.5.2 The general setting Here and in the follo wing, let an integer n > 1 b e given. F or an (n-1)group oid S , the co-unit of the adjunction is still the identit y . In fact π ( n ) 0 ( D ( n ) ( S )) = S . Pr o of. [ π ( n ) 0 ( D ( n ) ( S ))] 0 = [ D ( n ) ( S ))] 0 = S 0 b y deﬁnition, as n > 1. On the other side, for ob jects s 0 , s 0 0 one has [ π ( n ) 0 ( D ( n ) ( S ))] 1 ( s 0 , s 0 0 ) = π ( n − 1) 0 ([ D ( n ) ( S )] 1 ( s 0 , s 0 0 )) = π ( n − 1) 0 ( D ( n − 1) ( S 1 ( s 0 , s 0 0 ))) = S 1 ( s 0 , s 0 0 ) where the last equalit y is precisely the induction hypothesis. Same argumen t holds for morphisms. Concerning the unit of the adjunction, w e state the following 4.5 The adjunction π ( n ) 0 a D ( n ) 98 Deﬁnition 4.22. L et us ﬁx n-gr oup oid C . Then η ( n ) C : C → D ( n )  π ( n ) 0 ( C )  c onsists of the fol lowing data: • h η ( n ) C i 0 = id C 0 : C 0 → h D ( n )  π ( n ) 0 ( C ) i 0 = C 0 • for any p air c 0 , c 0 0 of obje cts of C , h η ( n ) C i c 0 ,c 0 0 1 is the dotte d arr ow b elow: C 1 ( c 0 , c 0 0 ) / / η ( n − 1) C 1 ( c 0 ,c 0 0 ) ( ( [ D ( n ) ( π ( n ) 0 ( C ))] 1 ( c 0 , c 0 0 ) D ( n − 1) ([ π ( n ) 0 ( C )] 1 ( c 0 , c 0 0 )) D ( n − 1) ( π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 ))) Claim 4.23. The p air < id C 0 , η ( n − 1) C 1 ( c 0 ,c 0 0 ) > is a n-functor. Pr o of. The diagrams mark ed ( i ) and ( ii ) express functorialit y w.r.t. comp o- sition and units resp ectiv ely , for any triple c 0 , c 0 0 , c 00 0 of ob jects of C : C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) C ◦ / / η ( n − 1) C 1 ( c 0 ,c 0 0 ) × η ( n − 1) C 1 ( c 0 0 ,c 00 0 )   C 1 ( c 0 , c 00 0 ) η ( n − 1) C 1 ( c 0 ,c 00 0 )   D ( n − 1)  π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 ))  × D ( n − 1)  π ( n − 1) 0 ( C 1 ( c 0 0 , c 00 0 ))  D ( n ) π ( n ) 0 C ◦ W W W W W W W W W W W W W W W + + W W W W W W W W W W W W W W W ( i ) D ( n − 1)  π ( n − 1) 0 ( C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ))  D ( n − 1) π ( n − 1) 0 ( C ◦ ) / / D ( n − 1)  π ( n − 1) 0 ( C 1 ( c 0 , c 00 0 ))  I ( n − 1) C u ( c 0 ) / / D ( n ) π ( n ) 0 C u ( c 0 ) U U U U U U U U U U U U U U U U * * U U U U U U U U U U U C 1 ( c 0 , c 0 ) η ( n − 1) C 1 ( c 0 ,c 0 )   D ( n − 1)  π ( n − 1) 0  I ( n − 1)  D ( n − 1) π ( n − 1) 0 ( C u ( c 0 ) ) / / D ( n − 1)  π ( n − 1) 0 ( C 1 ( c 0 , c 0 ))  ( ii ) 4.5 The adjunction π ( n ) 0 a D ( n ) 99 Lo wer triangles commute b y deﬁnition. Commutativit y of external diagrams will b e prov ed by ﬁnite induction in the follo wing lemmas. This will imply that ( i ) and ( ii ) comm ute. First w e need the following conv entional Notation 4.24. Given the n -c ate gory C , for 0 ≤ s < m ≤ n we write c m : c s + 3 _ _ _ _ c 0 s me aning c m : c m − 1 → c 0 m − 1 : · · · : c s +1 → c 0 s +1 : c s → c 0 s ar e m -c el l, ( m − 1) -c el ls, . . . , ( s + 1) -c el ls, s -c el ls of C . F urthermor e we inductively deﬁne C m ( c m − 1 , c 0 m − 1 ) := [ C m − 1 ( c m − 2 , c 0 m − 2 )] 1 ( c m − 1 , c 0 m − 1 ) b eing C 1 ( c 0 , c 0 0 ) given by the deﬁnition of n -c ate gory. Lemma 4.25. Given c n − j − 1 , k n − j − 1 : c 0 + 3 _ _ _ _ c 0 0 , c 0 n − j − 1 , k 0 n − j − 1 : c 0 0 + 3 _ _ _ _ c 00 0 the fol lowing diagr am c ommutes in j - Gp d : C n − j ( c n − j − 1 , k n − j − 1 ) × C n − j ( c 0 n − j − 1 , k 0 n − j − 1 ) ◦ 0 / / η ( j ) C n − j (  ,  ) × η ( j ) C n − j (  ,  )   C n − j ( c n − j − 1 ◦ 0 c 0 n − j − 1 , k n − j − 1 ◦ 0 k 0 n − j − 1 ) η ( j ) C n − j (  ,  )   D ( j ) π ( j ) 0 ( C n − j ( c n − j − 1 , k n − j − 1 )) × D ( j ) π ( j ) 0  C n − j ( c 0 n − j − 1 , k 0 n − j − 1 )  D ( j ) π ( j ) 0  C n − j ( c n − j − 1 ◦ 0 c 0 n − j − 1 , k n − j − 1 ◦ 0 k 0 n − j − 1 )  D ( j ) π ( j ) 0  C n − j ( c n − j − 1 , k n − j − 1 ) × C n − j ( c 0 n − j − 1 , k 0 n − j − 1 )  D ( j ) π ( j ) 0 ( ◦ 0 ) 4 4 h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h (4.1) wher e we write (  ,  ) when substitutes ar e cle ar fr om the c ontext. Pr o of. By ﬁnite induction o ver j . Diagram ab ov e for j = 1 is the follo wing square, of group oids and functors 4.5 The adjunction π ( n ) 0 a D ( n ) 100 C n − 1 ( c n − 2 , k n − 2 ) × C n − 1 ( c 0 n − 2 , k 0 n − 2 ) ◦ 0 / / η (1) C n − 1 (  ,  ) × η (1) C n − 1 (  ,  )   C n − 1 ( c n − 2 ◦ 0 c 0 n − 2 , k n − 2 ◦ 0 k 0 n − 2 ) η (1) C n − 1 (  ,  )   D (1) π (1) 0 ( C n − 1 ( c n − 2 , k n − 2 )) × D (1) π (1) 0  C n − 1 ( c 0 n − 2 , k 0 n − 2 )  D (1) π (1) 0  C n − 1 ( c n − 2 , k n − 2 ) × C n − 1 ( c 0 n − 2 , k 0 n − 2 )  D (1) π (1) 0 ( ◦ 0 ) / / D (1) π (1) 0  C n − 1 ( c n − 2 ◦ 0 c 0 n − 2 , k n − 2 ◦ 0 k 0 n − 2 )  F or this to commute, it must commute on ob jects and on arrows. In order to pro ve this, let us consider c n : c n − 1 → k n − 1 : c n − 2 → k n − 2 and c 0 n : c 0 n − 1 → k 0 n − 1 : c 0 n − 2 → k 0 n − 2 Moreo ver, let us recall that η (1) X := < [ I d X 0 ] ∼ , I d [ dom ( − )] ∼ > and apply: for the pair ( c n − 1 , c 0 n − 1 ), on the lo wer-left one has ( c n − 1 , c 0 n − 1 )  [ η (1) ] 0 × [ η (1) ] 0 / / ([ c n − 1 ] ∼ , [ c 0 n − 1 ] ∼ ) = [( c n − 1 , c 0 n − 1 )] ∼  [ ◦ 0 ] ∼ / / [ c n − 1 ◦ 0 c 0 n − 1 ] ∼ , where on the upp er-righ t ( c n − 1 , c 0 n − 1 )  ◦ 0 / / c n − 1 ◦ 0 c 0 n − 1 = c n − 1 ◦ 0 c 0 n − 1  [ η (1) ] 0 / / [ c n − 1 ◦ 0 c 0 n − 1 ] ∼ Similarly for arro ws, on the low er-left one has ( c n , c 0 n )  [ η (1) ] 1 × [ η (1) ] 1 / / ( I d [ c n − 1 ] ∼ , I d [ c 0 n − 1 ] ∼ ) = I d [( c n − 1 ,c 0 n − 1 )] ∼  [ ◦ 0 ] ∼ / / I d [ c n − 1 ◦ 0 c 0 n − 1 ] ∼ , 4.5 The adjunction π ( n ) 0 a D ( n ) 101 while on the upp er-righ t ( c n , c 0 n )  ◦ 0 / / c n ◦ 0 c 0 n = c n ◦ 0 c 0 n  [ η (1) ] 0 / / I d [ c n − 1 ◦ 0 c 0 n − 1 ] ∼ , and this completes the case j = 1. No w let us assume, as induction hypothesis, that Lemma holds for j − 1. W e will pro ve it holds for j . T o this end, supp ose w e are given c n − j +1 : c n − j → k n − j : c n − j − 1 → k n − j − 1 and c 0 n − j +1 : c 0 n − j → k 0 n − j : c 0 n − j − 1 → k 0 n − j − 1 . F or the pair ( c n − j , c 0 n − j ), on the lo wer-left one has ( c n − j , c 0 n − j )  [ η ( j ) ] 0 × [ η ( j ) ] 0 / / ( c n − j , c 0 n − j ) = ( c n − j , c 0 n − j )  ◦ 0 / / c n − j ◦ 0 c 0 n − j , where on the upp er-righ t ( c n − j , c 0 n − j )  ◦ 0 / / c n − j ◦ 0 c 0 n − j = c n − j ◦ 0 c 0 n − j  [ η ( j ) ] 0 / / c n − j ◦ 0 c 0 n − j F or homs, let us sa y that h Diagram (4 . 1) for j i 1  ( c n − j , c 0 n − j ) , ( k n − j , k 0 n − j )  is indeed h Diagram (4 . 1) for ( j − 1) i . Induction completes the pro of. In the same w ay one can prov e the following 4.5 The adjunction π ( n ) 0 a D ( n ) 102 Lemma 4.26. Given c n − j − 1 : c 0 + 3 _ _ _ _ c 0 , the fol lowing diagr am c ommutes in j - Gp d : I ( j ) C u 0 ( c 0 ) / / C n − j ( c n − j − 1 , c n − j − 1 ) η j C n − j ( c n − j − 1 ,c n − j − 1 )   D ( j ) π ( j ) 0  I ( j )  D ( j ) π ( j ) 0 ( C u 0 ( c 0 ) ) / / D ( j ) π ( j ) 0 ( C n − j ( c n − j − 1 , c n − j − 1 )) (4.2) Finally w e can conclude the Pr o of. (of Claim 4.23). The ﬁrst of the diagrams of Claim 4.23 amounts to diagram (4.1) with j = ( n − 1), the second amounts to diagram (4.2) again with j = ( n − 1). Prop osition 4.27. η ( n ) : id n Gp d ⇒ D ( n )  π ( n ) 0 ( − )  is a natur al tr ansformation of sesqui-functors. Pr o of. It suﬃces to show that, for any α : F ⇒ G : C → D , α • 0 η ( n ) D = η ( n ) C • 0 D ( n ) ( π ( n ) 0 ( α )) C η C / / F G ~ ~ D ( π 0 ( C )) D ( π 0 ( F ))   D ( π 0 ( G ))   D η D / / D ( π 0 ( D )) α + 3 D ( π 0 ( α )) + 3 By induction on n . n = 1 The adjunction of underlying categories and functors is w ell kno wn. It ex- tends plainly to sesqui-categories: in fact, in Gp d , α ( c 0 )’s are isomorphisms. Hence, D 0 ( π 0 0 ( α )) is an equalit y of functors. 4.5 The adjunction π ( n ) 0 a D ( n ) 103 n > 1 On ob jects, let us ﬁx c 0 in C 0 . Then [ α • 0 η ( n ) D ] 0 ( c 0 ) = η ( n ) D ( α ( c 0 )) = [ η ( n ) D 1 ( F c 0 ,Gc 0 ) ] 0 ( α ( c 0 )) = α ( c 0 ) (or [ α ( c 0 )] ∼ if n = 2). On the other side, [ η ( n ) C • 0 D ( n ) ( π ( n ) 0 α )] 0 ( c 0 ) = [ D ( n ) ( π ( n ) 0 α )] 0 ([ η ( n ) C ] 0 ( c 0 )) = [ D ( n ) ( π ( n ) 0 α )] 0 ( c 0 ) = [ π ( n ) 0 α ] 0 ( c 0 ) = α ( c 0 ) (or [ α ( c 0 )] ∼ if n = 2). On homs, let us ﬁx one more ob ject c 0 0 . Then [ α • 0 η ( n ) D ] c 0 ,c 0 0 1 = α c 0 ,c 0 0 1 • 0 [ η ( n ) D ] F c 0 ,Gc 0 0 1 = α c 0 ,c 0 0 1 • 0 η ( n − 1) D 1 ( F c 0 ,Gc 0 0 ) on the other side, [ η ( n ) C • 0 D n ( π n 0 α )] c 0 ,c 0 0 1 = [ η ( n ) C ] c 0 ,c 0 0 1 • 0 [ D ( n ) ( π ( n ) 0 α )] c 0 ,c 0 0 1 = η ( n − 1) C 1 ( c 0 ,c 0 0 ) • 0 D ( n − 1) ( π ( n − 1) 0 ( α c 0 ,c 0 0 1 )) Equalit y of the tw o sides α c 0 ,c 0 0 1 • 0 η ( n − 1) D 1 ( F c 0 ,Gc 0 0 ) = η ( n − 1) C 1 ( c 0 ,c 0 0 ) • 0 D ( n − 1) ( π ( n − 1) 0 ( α c 0 ,c 0 0 1 )) is giv en by induction hypothesis. Theorem 4.28. F or every p ositive inte ger n , π ( n ) 0 a D ( n ) Pr o of. After the discussion ab ov e, triangular identities will b e pro ved in the follo wing form: D ( n ) η ( n ) D ( n ) + 3 L L L L L L L L L L L L L L L L L L L L L L D ( n ) π ( n ) 0 D ( n ) D ( n )  ( n ) = D ( n ) ( i ) D ( n ) π ( n ) 0 π ( n ) 0 η ( n ) + 3 J J J J J J J J J J J J J J J J J J J J J J π ( n ) 0 D ( n ) π ( n ) 0  ( n ) π ( n ) 0 = π ( n ) 0 ( ii ) π ( n ) 0 Diagram ( i ) comm utes. In fact, for a (n-1)groupoid S one has η ( n ) D ( n ) S = id D ( n ) S . By induction on n . F or n = 1, S = S is a set and η 0 D 0 S is giv en by [ η 0 D 0 S ] 0 : s 0 7→ [ s 0 ] ∼ = s 0 Since D 0 S is a discrete category , it only has identit y arro ws, hence [ η 0 D 0 S ] 0 = id b y functoriality . 4.6 n -Discr ete h -pul lb acks 104 F or n > 1, b y deﬁnition one has [ η D ( n ) S ] 0 = id [ D ( n ) S ] 0 Moreo ver, for any pair of ob jects s 0 , s 0 0 of S , [ η D ( n ) S ] s 0 ,s 0 0 1 = η ( n − 1) [ D ( n ) S ] s 0 ,s 0 0 1 = η ( n − 1) D ( n − 1) ( S 1 ( s 0 ,s 0 0 )) ( ♣ ) = id D ( n − 1) ( S 1 ( s 0 ,s 0 0 )) = id [ D ( n ) S ] 1 ( s 0 ,s 0 0 ) = [ id D ( n ) S ] 1 ( s 0 , s 0 0 ) where all equalities hold by deﬁnition, but ( ♣ ) that is given b y induction h yp othesis. Diagram ( ii ) commutes. In fact, for a n-group oid C one has π ( n ) 0 ( η ( n ) C ) = id π ( n ) 0 C . By induction on n . F or n = 1, i.e. for a group oid C , π 0 0 ( η 0 C ) = [ π 0 0 ( η 0 C )] 0 , and it is giv en by [ π 0 0 ( η 0 C )] 0 : [ c 0 ] ∼ 7→ [ π 0 0 ] 0 ([ η 0 C ] 0 ([ c 0 ] ∼ )) = π 0 0 ([ c 0 ] ∼ ) = [ c 0 ] ∼ F or n > 1, b y deﬁnition one has [ π ( n ) 0 ( η ( n ) C )] 0 = id [ π ( n ) 0 C ] 0 Moreo ver, for any pair of ob jects c 0 , c 0 0 of C , [ π ( n ) 0 ( η ( n ) C )] c 0 ,c 0 0 1 = π ( n − 1) 0 ([ η ( n ) C ] c 0 ,c 0 0 1 ) = π ( n − 1) 0 ( η ( n ) C 1 ( c 0 ,c 0 0 ) ) ( ♣ ) = id π ( n − 1) 0 ( C 1 ( c 0 ,c 0 0 )) = id [ π ( n ) 0 C ] 1 ( c 0 ,c 0 0 ) where all equalities hold by deﬁnition, but ( ♣ ) that is given b y induction h yp othesis. 4.6 n -Discrete h -pullbac ks An application of the adjunction π ( n ) 0 a D ( n ) is the following useful result, just a sp ecial case of more general h -limits preserv ation prop ert y: 4.6 n -Discr ete h -pul lb acks 105 Lemma 4.29. Sesqui-functor D ( n ) : ( n − 1) Gp d → n Gp d pr eserves h - pul lb acks. Pr o of. W e omit the sup erscripts b eing alw ays ( n ). Let us consider D of the h -pullbac k ( P , P , Q, φ ) in ( n − 1) Gp d P Q / / P   C G   A F / / φ 4 , w e can now apply π 0 Q N / / M   D C DG   D A DF / / ω 5 = s s s s s s s s s s s s D B  π 0 / / π 0 Q π 0 N / / π 0 M   C G   A F / / π 0 ω 4 < q q q q q q q q q q q q B univ ersal prop erty for P gives then a unique L : π 0 Q → P suc h that ( i ) L • 0 P = π 0 M ( ii ) L • 0 Q = π 0 N ( iii ) L • 0 φ = π 0 ω Hence the comp osition η Q • 0 D L witnesses the fact that D ( P ) is an h -pullbac k. In fact ( i ) 0 η Q • 0 D L • 0 D P = η Q • 0 D ( L • 0 P ) ( i ) = η Q • 0 π 0 M ( a ) = M • 0 η D A ( b ) = M ( ii ) 0 η Q • 0 D L • 0 D Q = η Q • 0 D ( L • 0 Q ) ( ii ) = η Q • 0 π 0 N ( a ) = N • 0 η D C ( b ) = N ( iii ) 0 η Q • 0 D L • 0 D φ = η Q • 0 D ( L • 0 φ ) ( iii ) = η Q • 0 π 0 ω ( a ) = ω • 0 η D C ( b ) = ω where ( a ) b y naturality of η and ( b ) by triangular iden tities. Hence we can sa y that the h -pullbac k of a n -discrete diagram is itself n - discrete. 4.7 Exact se quenc es of n-gr oup oids 106 4.7 Exact sequences of n-group oids 4.7.1 P oin tedness and h -ﬁb ers A p oin ted n -category is simply a n -category C with a chosen ob ject ∗ C . A morphism of p oin ted n -categories F : C → D is a n -functor such that F ( ∗ C ) = ∗ D A 2-morphism of p ointed n -functors α : F ⇒ G : C → D is a natural n -transformation suc h that α 0 ( ∗ C ) = id ∗ D : ∗ D → ∗ D . The data described ab o ve form a sesqui-category , sub-sesqui-category of n Cat that w e will denote n Cat ∗ . Similarly one deﬁnes the sesqui-category n Gp d ∗ of p oin ted n-group oids, sub-sesqui-category of n Gp d . Subscripts of the star will b e often omitted. Notice that deﬁnitions of p ointed morphism and of p ointed 2-morphisms imply that n Cat ∗ and nGp d ∗ are closed under ﬁnite pro ducts and h -pullbac ks. Deﬁnition 4.30. Given a morphism of n-gr oup oids F : C → D , and an obje ct d of D , the past h-ﬁb er F ( p ) and the future h-ﬁb er F ( f ) of F over d ar e given by the fol lowing h-pul lb acks r esp. F ( p ) F,d ! / / E   I [ d ]   C F / / D ε z  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ F ( f ) F,d E / / !   C F   I [ d ] / / D ε z  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ wher e [ d ] is the c onstant functor over an obje ct d , as usual. R emark 4.31 . Distinction betw een future and past h -ﬁb ers, makes more sense in n -categorical context. There, in fact, the h -ﬁb er may come in diﬀeren t tastes: the lax-version that uses lax-2-morphisms, the pseudo-v ersion, that uses equiv alence transformations. Nev ertheless, even for n-groupoids, keeping trac k of the direction of 2-morphisms is necessary in order to recognize the problem of coherently choosing in verses of cells, when this problem arises. Still, w e will often omit sup erscripts (and subscripts), as they will b e clear from diagrams. R emark 4.32 . The notion of strict ﬁb er, or simply ﬁb er, is recov ered by the strict pullbac k (see Section 3.6.5). 4.7.2 Equiv alences and h -surjectiv e morphisms of n-group oids The notions of h-surje ctive morphism and of e quivalenc e get simpler in the con text of n-group oid. Concerning b oth cases, it is the notion of essen tial surjectivit y itself to b e inv olved. In fact, for a n-group oids morphism L : A → K , to be essen tially surjectiv e on ob jects amoun ts to the follo wing prop ert y: 4.7 Exact se quenc es of n-gr oup oids 107 Essen tial surjectivit y (1) for any ob ject k 0 of K , there exists a pair ( a 0 , k 1 : La 0 → k 0 ) with a 0 ob ject of A and k 1 1-cell of K . That is: it is no longer necessary to ask for k 1 to b e an equiv alence, since ev ery cell in a n-group oid is indeed an equiv alence. With the notion of h -ﬁb er in mind, we can further reformulate the notion of essen tial surjectivity . Essen tial surjectivit y (2) for any ob ject k 0 of K , the h -ﬁb er F ( p ) L,k 0 is not empt y . Finally , in p oin ted case, ﬁb ers assume a sp ecial meaning, as referred by the follo wing Deﬁnition 4.33. L et G : B → C b e a morphism of n-gr oup oids. Then the ﬁb er K = F ( p ) G, ∗ K K / / 0   B G / / C κ           is c al le d (p ast) the h -kernel of G . W e will call universal prop erty of h -k ernels, the universal property of pullbac ks sp ecialized for these kind of pullbac ks. 4.7.3 Exact sequences Deﬁnition 4.34. L et the fol lowing diagr am in nGp d ∗ b e given: A F / / 0   B G / / C ε   We c al l the triple ( F , ε, G ) exact in B if the c omp arison n-functor L : A → K , given by the universal pr op erty of the h-kernel ( K , K , κ ) , is h -surje ctive. A L   F   ? ? ? ? ? ? ? 0   B G / / C K K ? ?        0 G G ε             κ O W & & & & & & & & & & ≡ 4.7 Exact se quenc es of n-gr oup oids 108 This notion of exactness is a straightforw ard extensions of the notion intro- duced b y Vitale in [Vit02]. Of course it reduces to usual exactness for p ointed sets and group. Moreov er it is preserved by one-p oint susp ension and by discretization, hence an exact sequence of groups ma y be considered as an exact sequence of one-p oint group oids, as w ell as an exact sequence of p ointed discrete group oids (with a group structure). In the categorical group (p ointed group oid) situation, it has sho wn its useful- ness in extending homological algebraic structures in a 1-dimensional con text. 4.7.4 π 0 preserv es exactness In the following paragraphs w e will show that, giv en a three-term exact sequence in n Gp d , the sesqui-functor π ( n ) 0 pro duces a three-term exact sequence in ( n − 1) Gp d . Preliminary Lemmas clarify the relations b etw een preserv ation of exactness and its main ingredients: h -surjectivit y and the notion of h -pullbac k. Lemma 4.35. L et us c onsider the fol lowing h -pul lb ack diagr am: P S / / R   Z H   B ε ; C         G / / C The c omp arison L : π ( n ) 0 P → Q with the h -pul lb ack of π ( n ) 0 ( G ) and π ( n ) 0 ( H ) is h -surje ctive. π ( n ) 0 P L " " E E E E E E E E E E π ( n ) 0 S   π ( n ) 0 R % % Q Q / / P   π ( n ) 0 Z π ( n ) 0 H   π ( n ) 0 B γ 8 @ x x x x x x x x x x x x π ( n ) 0 G / / π ( n ) 0 C π ( n ) 0 ε ; C Pr o of. By induction on n . n = 1 The h -pullbac k P has ob jects ( b 0 , Gb 0 c 1 / / H z 0 , z 0 ) 4.7 Exact se quenc es of n-gr oup oids 109 and arro ws ( b 1 , = , z 1 ) : ( b 0 , c 1 , z 0 ) → ( b 0 0 , c 0 1 , z 0 0 ) where the “=” sta ys for the commutativ e square Gb 0 c 1 / / Gb 1   H z 0 H z 1   Gb 0 0 c 0 1 / / H z 0 0 Hence the set π 0 0 ( P ) has elemen ts the classes [ b 0 , c 1 , z 0 ] ∼ . On the other side, the set Q is a usual pullbac k in Set . It has elements the pairs ([ b 0 ] ∼ , [ z 0 ] ∼ ) such that π 0 0 G ([ b 0 ] ∼ ) = π 0 0 H ([ z 0 ] ∼ ), i.e. [ Gb 0 ] ∼ = [ H z 0 ] ∼ , i.e. such that there exists c 1 : Gb 0 → H z 0 . Then the comparison L = L 0 : [ b 0 , c 1 , z 0 ] ∼ 7→ ([ b 0 ] ∼ , [ z 0 ] ∼ ) is clearly surjectiv e. n = 2 No w the h -pullback P is a 2-group oid with ob jects ( b 0 , Gb 0 c 1 / / H z 0 , z 0 ) . Arro ws are of the form ( b 1 , c 2 , z 1 ) : ( b 0 , c 1 , z 0 ) → ( b 0 0 , c 0 1 , z 0 0 ) i.e.       b 0 b 1   b 0 0 , Gb 0 c 1 / / Gb 1   H z 0 H z 1   Gb 0 0 c 0 1 / / c 2 8 @ y y y y y y y y y y H z 0 0 , z 0 z 1   z 0 0       Finally 2-cells are of the form ( b 2 , = , z 2 ) : ( b 1 , c 2 , z 1 ) ⇒ ( b 0 1 , c 0 2 , z 0 1 ) i.e.       b 1 b 2   b 0 1 , Gb 1 ◦ c 0 1 c 2 + 3 Gb 2 ◦ c 0 1   c 1 ◦ H z 1 c 1 ◦ H z 2   Gb 0 1 ◦ c 0 1 c 0 2 + 3 c 1 ◦ H z 0 1 , z 1 z 2   z 0 1       4.7 Exact se quenc es of n-gr oup oids 110 Therefore the group oid π 00 0 P has ob jects ( b 0 , c 1 , z 0 ) and arro ws [ b 1 , c 2 , z 1 ] ∼ . On the other side, the group oid Q has ob jects ( b 0 , Gb 0 [ c 1 ] ∼ / / H z 0 , z 0 ) and arro ws ([ b 1 ] ∼ , = , [ z 1 ] ∼ ) with [ b 1 ] ∼ : b 0 → b 0 0 in π 00 0 B and [ z 1 ] ∼ : z 0 → z 0 0 in π 00 0 Z suc h that Gb 0 [ c 1 ] ∼ / / [ Gb 1 ] ∼   H z 0 [ H z 1 ] ∼   Gb 0 0 ( ♥ ) [ c 0 1 ] ∼ / / H z 0 0 Hence the comparison L : ( b 0 , c 1 , z 0 ) 7→ ( b 0 , c 1 , z 0 ) [ b 1 , c 2 , z 1 ] ∼ 7→ ([ b 1 ] ∼ , [ z 1 ] ∼ ) is h -surjectiv e. In fact it is an identit y (hence strictly surjective) on ob jects, and full on homs. Let us ﬁx a pair of ob jects ( b 0 , c 1 , z 0 ) and ( b 0 0 , c 0 1 , z 0 0 ) in the domain, and an arrow ([ b 1 ] ∼ , = , [ z 1 ] ∼ ) in Q , where the “=” is the diagram ( ♥ ) ab o ve. Then [ c 1 ◦ H z 1 ] ∼ = [ Gb 1 ◦ c 0 1 ] ∼ if, and only if, there exists c 2 : c 1 ◦ H z 1 → Gb 1 ◦ c 0 1 . In other words we get an arrow [ b 1 , c 2 , z 1 ] ∼ of π 00 0 P that L sends in ([ b 1 ] ∼ , = , [ z 1 ] ∼ ), i.e. L is full. n > 2 On ob jects, L 0 : [ π ( n ) 0 P ] 0 → Q 0 is the iden tity . In fact, for big n [ π ( n ) 0 P ] 0 = P 0 , the last b eing the set-theoretical limit ov er the diagram B 0 G 0 A A A A A A A A C 1 d ~ ~ | | | | | | | | c B B B B B B B B Z 0 H 0 ~ ~ } } } } } } } } C 0 C 0 No w, for n > 2 this diagram coincides with the one deﬁning Q 0 : [ π ( n ) 0 B ] 0 [ π ( n ) 0 G ] 0 ! ! D D D D D D D D [ π ( n ) 0 C ] 1 d } } z z z z z z z z c ! ! D D D D D D D D [ π ( n ) 0 Z ] 0 [ π ( n ) 0 H ] 0 } } z z z z z z z z [ π ( n ) 0 C ] 0 [ π ( n ) 0 C ] 0 4.7 Exact se quenc es of n-gr oup oids 111 Univ ersality of limits gives L 0 = id . On homs, let us ﬁx tw o ob jects p 1 = ( b 0 , c 1 , z 0 ) and p 0 1 = ( b 0 0 , c 0 1 , z 0 0 ) of [ π ( n ) 0 P ] 0 = P 0 and compute L p 1 ,p 0 1 1 as usual: by means of universal prop erty of h -pullbac ks. In fact the diagram [ π ( n ) 0 P ] 1 ( p 0 , p 0 0 ) L p 0 ,p 0 0 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q [ π ( n ) 0 S ] 1 * * [ π ( n ) 0 R ] 1 ' ' Q 1 ( p 0 , p 0 0 ) Q 1 / / P 1   [ π ( n ) 0 Z ] 1 ( z 0 , z 0 0 ) [ π ( n ) 0 H ] 1   [ π ( n ) 0 C ] 1 ( H z 0 , H z 0 0 ) c 1 ◦−   [ π ( n ) 0 B ] 1 ( b 0 , b 0 0 ) [ π ( n ) 0 G ] 1 / / [ π ( n ) 0 C ] 1 ( Gb 0 , Gb 0 0 ) −◦ c 0 1 / / [ π ( n ) 0 C ] 1 ( Gb 0 , H z 0 0 ) σ q y l l l l l l l l l l l l l l l l l l l l [ π ( n ) 0 ε ] 1 q y is the same as (and determined b y) π ( n − 1) 0 ( P 1 ( p 0 , p 0 0 )) L p 0 ,p 0 0 1 ) ) R R R R R R R R R R R R R R R π ( n − 1) 0 S 1 + + π ( n − 1) 0 R 1 ' ' Q 1 ( p 0 , p 0 0 ) Q 1 / / P 1   [ π ( n ) 0 Z ] 1 ( z 0 , z 0 0 ) π ( n − 1) 0 H 1   π ( n − 1) 0 ( C 1 ( H z 0 , H z 0 0 )) c 1 ◦−   π ( n − 1) 0 ( B 1 ( b 0 , b 0 0 )) π ( n − 1) 0 G 1 / / π ( n − 1) 0 ( C 1 ( Gb 0 , Gb 0 0 )) −◦ c 0 1 / / π ( n − 1) 0 ( C 1 ( Gb 0 , H z 0 0 )) σ q y j j j j j j j j j j j j j j j j j j j j j j π ( n − 1) 0 ( ε 1 ) q y This shows that L p 0 ,p 0 0 1 is itself a comparison b etw een π 0 of an h -pullbac k and a h -pullbac k of a π 0 of a diagram (of (n-1)groupoids), hence it is h -surjectiv e b y induction hypothesis. In conclusion w e hav e shown that L = < L 0 , L − , − 1 > is h -surjectiv e. Lemma 4.36. If the n -functor L : A → K is h -surje ctive, then also π ( n ) 0 ( L ) is h -surje ctive. Pr o of. By induction on n . 4.7 Exact se quenc es of n-gr oup oids 112 n = 1 Let L b e a h -surjectiv e functor b etw een groupoids, i.e. L is full and essen tially surjectiv e on ob jects. Therefore for an element [ k 0 ] ∼ ∈ π 0 0 K there exists a pair ( a 0 , k 1 : La 0 → k 0 ) Hence ( π 0 0 L )([ a 0 ] ∼ ) = [ La 0 ] ∼ = [ k 0 ] ∼ in π 0 0 K . n = 2 Let L b e a h -surjectiv e morphism b etw een 2-group oids, i.e. 1. for an y k 0 there exist ( a 0 , k 1 : La 0 → k 0 ). 2. for an y pair a 0 , a 0 0 , L a 0 ,a 0 0 1 : A 1 ( a 0 , a 0 0 ) → K 1 ( La 0 , La 0 0 ) is h -surjectiv e. Since [ π 00 0 L ] 0 = L 0 , for an y k 0 one has [ k 1 ] ∼ : La 0 → k 0 , and this pro ves the ﬁrst condition. Moreo ver, once we ﬁx a pair a 0 , a 0 0 , by deﬁnition one has [ π 00 0 L ] a 0 ,a 0 0 1 = π 0 0 ( L a 0 ,a 0 0 1 ). Hence it is h -surjective by previous case. n > 2 More generally , a morphism L of n -group oids is h -surjectiv e when conditions 1. and 2. ab ov e b oth hold. Since [ π ( n ) 0 L ] 0 = L 0 and [ π ( n ) 0 ( K )] 1 = π ( n − 1) 0 ( K 1 ), condition 1. for L and for π ( n ) 0 L is the same, hence it holds. Moreo ver, whence we ﬁx a pair a 0 , a 0 0 , by deﬁnition one has [ π ( n ) 0 L ] a 0 ,a 0 0 1 = π ( n − 1) 0 ( L a 0 ,a 0 0 1 ). Hence it is h -surjective by induction hypothesis. Finally w e are ready to state and prov e the following imp ortant Theorem 4.37. Given an exact se quenc e in n Gp d ∗ A F / / 0 ! ! B G / / C ε   the se quenc e π ( n ) 0 A π ( n ) 0 F / / 0 π ( n ) 0 B π ( n ) 0 G / / π ( n ) 0 C π ( n ) 0 ε   4.8 The sesqui-functor π 1 113 is exact in ( n − 1) Gp d ∗ Pr o of. Let us consider the diagram Ker  π ( n ) 0 G  ' ' O O O O O O O O O O O O O 0 $ $ π ( n ) 0  Ker G  L 0 O O / / π ( n ) 0 B π ( n ) 0 G / / π ( n ) 0 C π ( n ) 0 A π ( n ) 0 L O O π ( n ) 0 F 7 7 n n n n n n n n n n n n n n n κ         L is the comparison in n Gp d , hence h -surjectiv e b y hypothesis. Therefore π ( n ) 0 L is h -surjectiv e by L emma 4.36 . L 0 is the comparison in (n-1) Gp d , h -surjectiv e by L emma 4.35 . Finally , their comp osition is again h -surjectiv e by L emma 4.10 , and it is the comparison b et ween π ( n ) 0 A and the kernel of π ( n ) 0 G b y uniqueness in universal prop ert y of h -kernels. 4.8 The sesqui-functor π 1 Purp ose of this section is to introduce the family of sesqui-functors { π ( n ) 1 } n ∈ N that extends the isos-of-the-p oint functor Gp d ∗ → Set ∗ that assigns to eac h p oin ted group oid the p oin ted (hom-)set of endo-arrows of the p oint. Deﬁnition/Prop osition 4.38. F or any inte ger n > 0 , ther e exists a sesqui- functor π ( n ) 1 : n Gp d ∗ → ( n − 1) Gp d ∗ c ontr a-variant on 2-morphisms, ac c or ding to the fol lowing r e cursive deﬁni- tion. n = 1 π (1) 1 is the functor (= trivial sesqui-functor) Gp d ∗ → Set ∗ that assigns to a pointed group oid C the pointed set C ( ∗ , ∗ ). It can b e considered contra- v arian t, since 2-morphisms in Set are equalities. n > 1 Let a n -group oid C b e giv en. Then π ( n ) 1 C = C 1 ( ∗ , ∗ ). Let a morphism of n -group oids F : C → D b e given. Then π ( n ) 1 F = F ∗ , ∗ 1 . 4.8 The sesqui-functor π 1 114 Of course these assignmen ts giv e indeed a functor b et ween underlying cate- gories. In fact π ( n ) 1 ( id C ) = [ id C ] ∗ , ∗ 1 = id C 1 ( ∗ , ∗ ) = id π ( n ) 1 C and for ev ery other G : D → E , one has π ( n ) 1 ( F • 0 G ) = [ F • 0 G ] ∗ , ∗ 1 = F ∗ , ∗ 1 • 0 G ∗ , ∗ 1 = π ( n ) 1 ( F ) • 0 π ( n ) 1 ( G ) . Let a 2-morphism α : F ⇒ G : C → D b e given. Then w e deﬁne π ( n ) 1 ( α ) = α ∗ , ∗ 1 : π ( n ) 1 ( G ) ⇒ π ( n ) 1 ( F ) . In fact, since comp ositions with identities gives identit y functors, con tra- v ariance is explained b y the following diagram: C 1 ( ∗ , ∗ ) F ∗ , ∗ 1 y y s s s s s s s s s s G ∗ , ∗ 1 % % K K K K K K K K K K D 1 ( ∗ , ∗ ) −◦ 1 ∗ % % K K K K K K K K K K D 1 ( ∗ , ∗ ) 1 ∗ ◦− y y s s s s s s s s s s D 1 ( ∗ , ∗ ) α ∗ , ∗ 1 k s In order to show that so-deﬁned π ( n ) 1 is indeed a sesqui-functor, tw o facts ha ve to b e prov ed regarding π ( n ) 1 . 1. it is functorial on hom-categories. 2. it preserv es reduced horizontal comp ositions. Pr o of. 1. Supp ose w e are given ω : E ⇒ F : C → D α : F ⇒ G : C → D in n Gp d . Then π ( n ) 1 ( ω • 1 α ) = [ ω • 1 α ] ∗ , ∗ 1 = α ∗ , ∗ 1 • 1 ω ∗ , ∗ 1 = π ( n ) 1 ( α ) • 1 π ( n ) 1 ( ω ) π ( n ) 1 ( id F ) = [ id F ] ∗ , ∗ 1 = id F ∗ , ∗ 1 = id π ( n ) 1 ( F ) 2. W e prov e the statement for reduced left-comp osition. Supp ose 2- morphism α as ab o ve, and morphism N : B → C b e given. Then π ( n ) 1 ( N • 0 L α ) = [ N • 0 L α ] ∗ , ∗ 1 = N ∗ , ∗ 1 • 0 L α ∗ , ∗ 1 = π ( n ) 1 ( N ) • 0 L π ( n ) 1 ( α ) . Finally , concerning reduced right-composition, the pro of is similar, as the deﬁnition 4.8 The sesqui-functor π 1 115 The follo wing prop osition is a direct consequence of the deﬁnitions. Hence it needs no pro of. Prop osition 4.39. Sesqui-functor π ( n ) 1 c ommutes with ﬁnite pr o ducts and pr eserves e quivalenc es. R emark 4.40 . The deﬁnition of π ( n ) 1 giv en here makes it of diﬃcult use w.r.t. the inductive setting developed so far, where ev erything is giv en as a pair, where the ﬁrst comp onent lives in Set , the second in ( n − 1) Cat (or ( n − 1) Gp d ). This motives a further search for a diﬀeren t (but equiv alen t) deﬁnition of π ( n ) 1 , see Cor ol lary 6.15 . Thereafter it will also b e shown that π ( n ) 1 preserv es exactness, as a consequence of universalit y of its deﬁnition, although this could b e pro ved here directly . Chapter 5 3 -Morphisms of n -categories 5.1 What structure for n Cat? So far w e ha ve shown that n-categories organizes naturally into a sesqui- category , ditto for n -group oids. This gives a setting to deal not only with n -categories and n -functors, but also with their 2-morphisms, namely lax-n- transformations. Y et the necessit y of introducing 3-morphisms (lax- n -mo diﬁcations) takes us out of that comfortable setting, in to the unknown territory of sesqui- categorically enric hed structures. F ollo wing this suggestion, we hav e named the new setting sesqui 2 -c ate gory . This notion is closely related with that of Gra y -category [ Gra76 , Gra74 ] (or 3D-T as see [ Cra00 ]) and incorp orates a horizontal dimension raising comp osition of 2-morphisms. In fact the set of axioms which deﬁne the former is a subset of those deﬁning the latter. In order to fully justify the name c hosen to denote suc h a structure, it would b e interesting to inv estigate explicitly the enric hment that generates this notion from that of sesqui-category . What we present here is a treatable inductive approac h, comprehensive of a useful c haracterization given in The or em 5.3 . Deﬁnition 5.1. A (smal l) sesqui 2 -category C c onsists of: • A 3-trunc ate d r eﬂexive globular set C • : C 3 d 2 / / c 2 / / C 2 e 2 o o d 1 / / c 1 / / C 1 e 1 o o d 0 / / c 0 / / C 0 e 0 o o with op er ations • m : C p c m × d m C q → C p + q − m − 1 , m < min( p, q ) 5.1 What structur e for n Cat ? 117 such that the fol lowing axioms hold: ( i ) F or every p air C , D ∈ C 0 , the lo c alization C ( C , D ) is a sesqui-c ate gory, with - obje ct ar e F , G, etc. ∈ C 1 ( C , D ) - for any p air of obje cts F , G , 1-c el ls ar e α, β , etc. ∈ C 2 ( F , G ) - for any p air of 1-c el ls α, γ : F → G , 2-c el ls ar e Λ , Σ , etc. ∈ C 3 ( α, β ) k-c omp ositions ar e r estrictions of • k +1 -c omp ositions: - 0-c omp osition of 1-c el ls of C ( C , D )  0 := • 1 : C 2 c 1 × d 1 C 2 → C 2 - left/right r e duc e d 0-c omp ositions of 1-c el l with a 2-c el l of C ( C , D )  0 L := • 1 : C 2 c 1 × d 1 C 3 → C 3  0 R := • 1 : C 3 c 1 × d 1 C 2 → C 3 - 1-c omp ositions of 2-c el ls of C ( C , D )  1 := • 2 : C 3 c 2 × d 2 C 3 → C 3 ( ii ) F or every morphism F : C → D and obje cts B , E of C − • 0 F : C ( B , C ) → C ( B , D ) F • 0 − : C ( D , E ) → C ( C , E ) ar e sesqui-functors. ( iii ) F or every obje ct C and obje cts B , D of C , if we denote id C = e 0 ( C ) , − • 0 id C : C ( B , C ) → C ( B , C ) id C • 0 − : C ( C , D ) → C ( C , D ) ar e identity sesqui-functors. ( iv ) (naturality axioms) F or every p air of 0-c omp osable 2-morphisms α : F ⇒ G : C → D and β : H ⇒ K : D → E 5.1 What structur e for n Cat ? 118 ( a ) α • 0 β : ( F • 0 β ) • 1 ( α • 0 K ) → ( α • 0 H ) • 1 ( G • 0 β ) F or every 2-morphisms ε : L ⇒ M : B → C and β : H ⇒ K : D → E , and for every 3-morphism Λ : α _ * 4 ω : F ⇒ G : C → D ( b ) ( α • 0 β ) • 2  (Λ • 0 H ) • 1 ( G • 0 β )  =  ( F • 0 β ) • 1 (Λ • 0 K )  • 2 ( ω • 0 β ) and ( c )  ( L • 0 Λ) • 1 ( ε • 0 G )  • 2 ( ε • 0 ω ) = ( ε • 0 α ) • 2  ( ε • 0 F ) • 1 ( M • 0 Λ)  ( v ) (functoriality axioms) F or every 2-morphisms ω : D ⇒ E : B → C and γ : H ⇒ L : D → E and every p air of 1-c omp osable 2-morphisms α : F ⇒ G : C → D and β : G ⇒ H : C → D ( a ) ( α • 1 β ) • 0 γ =  ( α • 0 γ ) • 1 ( β • 0 L )  • 2  ( α • 0 K ) • 1 ( β • 0 γ )  and ( b ) ω • 0 ( α • 1 β ) =  ( ω • 0 α ) • 1 ( E • 0 β )  • 2  ( D • 0 α ) • 1 ( ω • 0 β )  ( v i ) (asso ciativity axiom) F or every 0-c omp osable triple x ∈ [ C ( B , C )] p , y ∈ [ C ( C , D )] q and z ∈ [ C ( D , E )] r , with p + q + r ≤ 2 ( x • 0 y ) • 0 z = x • 0 ( y • 0 z ) ( v ii ) (identit y axioms) 5.1 What structur e for n Cat ? 119 F or morphisms E : B → C and H : D → E , and 2-morphism α : F ⇒ G : C → D , id F • 0 α = id F • 0 α , α • 0 id G = id α • 0 G R emark 5.2 . 1. Axiom ( iv )( c ) is b etter understo o d when visualized as in the follo wing diagram (same notation) F • 0 H ω • 0 H   α • 0 H + 3 Λ • 0 H  J T F • 0 β   G • 0 H G • 0 β   F • 0 K α • 0 β k 0 ; k k k k k k k k k k k k k k k k k k k k k k k k k k k α • 0 K + 3 G • 0 K = F • 0 H ω • 0 H + 3 F • 0 β   G • 0 H G • 0 β   F • 0 K ω • 0 β k 0 ; k k k k k k k k k k k k k k k k k k k k k k k k k k k ω • 0 K + 3 α • 0 K A I G • 0 K Λ • 0 K  J T The same can b e claimed for axiom ( iv )( b ). 2. Axiom ( v )( a ) is b etter understo o d when visualized as in the following diagram (same notation) F • 0 K ( α • 1 β ) • 0 K + 3 F • 0 γ   H • 0 K H • 0 γ   F • 0 L ( α • 1 β ) • 0 L + 3 ( α • 1 β ) • 0 γ k 0 ; k k k k k k k k k k k k k k k k k k k k k k k k k k k H • 0 L = F • 0 K α • 0 K + 3 F • 0 γ   G • 0 K β • 0 K + 3 G • 0 γ   H • 0 K H • 0 γ   F • 0 L α • 0 L + 3 α • 0 γ q 2 @ q q q q q q q q q q q q q q q q q q G • 0 L β • 0 L + 3 β • 0 γ q 2 @ q q q q q q q q q q q q q q q q q q H • 0 L The same can b e claimed for axiom ( v )( b ). Theorem 5.3. L et C • b e a 3-trunc ate d r eﬂexive globular set. Then the fol lowing two statements ar e e quivalent. 1. C is a (smal l) sesqui 2 -category 2. Axioms ( i ) , ( ii ) and ( iii ) of Deﬁnition 5.1 hold, mor e over ( v iii ) The 2-trunc ation C 2 d 1 / / c 1 / / C 1 e 1 o o d 0 / / c 0 / / C 0 e 0 o o of C • is a sesqui-c ate gory. ( ix ) F or every 2-morphism α : F ⇒ G : C → D and obje cts B , E of C − • 0 α : − • 0 F ⇒ − • 0 G : C ( B , C ) → C ( B , D ) α • 0 − : F • 0 − ⇒ G • 0 − : C ( D , E ) → C ( C , E ) ar e lax natur al tr ansformations of sesqui-functors. 5.1 What structur e for n Cat ? 120 ( x ) F or every morphism F : C → D of C − • 0 id F − • 0 F ⇒ − • 0 F id F • 0 − : F • 0 − ⇒ F • 0 − ar e identic al natur al tr ansformations. ( xi ) (reduced asso ciativit y axiom) F or every 0-c omp osable triple x ∈ [ C ( B , C )] p , y ∈ [ C ( C , D )] q and z ∈ [ C ( D , E )] r , with p + q + r = 2 ( x • 0 y ) • 0 z = x • 0 ( y • 0 z ) i.e. for 3-morphism Λ , 2-morphisms α, β and morphisms F , G of C , the fol lowing e quations hold, when c omp osites exist: (Λ • 0 F ) • 0 G = Λ • 0 ( F • 0 G ) ( α • 0 β ) • 0 F = α • 0 ( β • 0 F ) ( F • 0 Λ) • 0 F = F • 0 (Λ • 0 G ) ( α • 0 F ) • 0 β = α • 0 ( F • 0 β ) (Λ • 0 F ) • 0 G = Λ • 0 ( F • 0 G ) ( F • 0 α ) • 0 β = F • 0 ( α • 0 β ) Pr o of. First we prov e that 1 . implies 2 . . Condition ( v iii ) is equiv alent to satisfying prop erties ( L 1) to ( L 4), ( R 1) to ( R 4) and ( LR 5) of Pr op osition 2.2 . No w, ( L 1) and ( R 1) hold by ( iii ), ( L 2) and ( R 2) by ( iv ), ( L 3), ( R 3), ( L 4) and ( R 4) b y ( ii ), ( LR 5) by ( v i ). Condition ( ix ) holds. In fact let us recall Deﬁnition 2.7 . Assignment on ob jects (=1-cells) is given by 0-comp osition, naturality by ( iv ) and functorialit y by ( v ) (comp ositions) and ( v ii ) (units). Condition ( x ) holds to o. In fact this is implied by ( ix ) ab ov e and ( v ii ). Finally ( xi ) is a subset of ( v i ). Con versely we prov e that 2 . implies 1 . . Conditions ( iv ) and ( v ) hold by ( ix ). Condition ( v i ) holds b y ( xi ) for the cases p + q + r = 2. What is still to pro ve is the case p + q + r = 0 and the case p + q + r = 1, that are giv en by ( v iii ). Finally ( v ii ) is a consequence of ( ix ) and ( x ). R emark 5.4 . Notice that the c haracterization given b y The or em 5.3 is some- ho w redundan t. Nev ertheless its usefulness is that it mak es a v ailable practical rules in order to deal with calculations in a sesqui 2 -categorical en vironment. 5.2 L ax n -mo diﬁc ation 121 5.2 Lax n -mo diﬁcation Purp ose of the rest of the c hapter is to give a pro of of the following Theorem 5.5. The sesqui-c ate gory n Cat , endowe d with 3-morphism, their c omp ositions, whiskering and dimension r aising 0-c omp osition of 2-morphisms is a sesqui 2 -c ate gory. This is done b y means of the characterization given in The or em 5.3 . As usual the approac h is genuinely inductive, starting with the well known deﬁnition of a mo diﬁc ation in Cat [Bor94]. Hence supp ose given an integer n > 1. Let us consider the follo wing situation in n Cat : α, β : F ⇒ G : C → D A lax n -mo diﬁc ation Λ : α _ * 4 β C F   G B B D α   β   Λ _ * 4 is a pair (Λ 0 , Λ 1 ), where • Λ 0 : C 0 / / ‘ c 0 ∈ C 0 [ D 2 ( α 0 ( c 0 ) , β 0 ( c 0 ))] 0 is a map such that, for ev ery c 0 in C 0 , Λ 0 ( c 0 ) : α 0 ( c 0 ) + 3 β 0 ( c 0 ) . L et us p oint out that subscript “ 0 ” is sometimes omitte d (as in α ( c 0 ) ), or c 0 is itself subscripte d (as in α c 0 ). • ( n-natur ality ) for every pair of ob jects c 0 , c 0 0 of C , a 3-morphism of 5.2 L ax n -mo diﬁc ation 122 ( n − 1)categories that ﬁl ls the following diagram: C 1 ( c 0 , c 0 0 ) F c 0 ,c 0 0 1                               G c 0 ,c 0 0 1   > > > > > > > > > > > > > > > > > > > > > > > > > > > > D 1 ( F c 0 , F c 0 0 ) −◦ αc 0 0   −◦ β c 0 0 . . D 1 ( Gc 0 , Gc 0 0 ) αc 0 ◦−   β c 0 ◦− p p D 1 ( F c 0 , Gc 0 0 ) −◦ Λ c 0 0             Λ c 0 ◦− 6 6 6 6 6 6   6 6 6 6 α c 0 ,c 0 0 1 k s β c 0 ,c 0 0 1 k s Λ c 0 ,c 0 0 1  J T i.e. G c 0 ,c 0 0 1 • 0 ( − ◦ αc 0 0 ) α c 0 ,c 0 0 1 + 3 id • 0 (Λ c 0 ◦− )   F c 0 ,c 0 0 1 • 0 ( − ◦ αc 0 0 ) id • 0 ( −◦ Λ c 0 )   G c 0 ,c 0 0 1 • 0 ( β c 0 ◦ − ) Λ c 0 ,c 0 0 1 l 0 < l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l β c 0 ,c 0 0 1 + 3 F c 0 ,c 0 0 1 • 0 ( − ◦ β c 0 0 ) These data must ob ey to functoriality axioms described by the following equations of 3-diagrams in (n-1) Cat : • ( functoriality w.r.t. 0 -c omp osition ) for ev ery triple c 0 , c 0 0 , c 00 0 of ob jects of C 5.2 L ax n -mo diﬁc ation 123 C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) id × β c 0 0 ,c 00 0 1 ( ( " " | | id × α c 0 0 ,c 00 0 1 v v C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) β c 0 ,c 0 0 1 × id ( ( " " | | α c 0 ,c 0 1 × id v v C 1 ( c 0 , c 0 0 ) × D 1 ( F c 0 0 , Gc 00 0 ) w  w w w w w w w w w w id × F ( − ) ◦ Λ c 00 0 k s id × Λ c 0 0 ◦ G ( − ) k s w  id × Λ c 0 0 ,c 00 0 1 U $ / U U U U U U U U U U U U U U U U U U F ( − ) ◦− & & L L L L L L L L L L L L L L L L L L L L L L L D 1 ( F c 0 , Gc 0 0 ) × C 1 ( c 0 0 , c 00 0 ) w  w w w w w w w w w w F ( − ) ◦ Λ c 0 0 × id k s Λ c 0 ◦ G ( − ) × id k s w  Λ c 0 ,c 0 0 1 × id U $ / U U U U U U U U U U U U U U U U U U −◦ G ( − ) x x r r r r r r r r r r r r r r r r r r r r r r r D 1 ( F c 0 , Gc 00 0 ) = (5.1) C 1 ( c 0 , c 0 0 ) × C 1 ( c 0 0 , c 00 0 ) ◦   C 1 ( c 0 , c 00 0 ) F ( − ) ◦ Λ c 00 0 ( ( " " | | Λ c 0 ◦ G ( − ) v v D 1 ( F c 0 , Gc 00 0 ) 7 ? w w w w w w w w w w α c 0 ,c 00 0 1 k s β c 0 ,c 00 0 1 k s 7 ? Λ c 0 ,c 00 0 1 d m w d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d namely: (Λ c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1 ) • 2 ( F c 0 ,c 0 0 1 ◦ Λ c 0 0 ,c 00 0 1 ) = ( − ◦ − ) • 0 Λ c 0 ,c 00 0 1 where the 2-dimensional in tersection is the 2-morphism F ( − ) ◦ Λ c 0 0 ◦ G ( − ). • ( functoriality w.r.t. units ) for every ob ject c 0 of C 5.3 n Cat ( C , D ) : the underlying c ate gory 124 I ( n − 1) u ( c 0 )   C 1 ( c 0 , c 0 ) F ( − ) ◦ Λ c 0 ( ( " " | | Λ c 0 ◦ G ( − ) v v D 1 ( F c 0 , Gc 0 ) 7 ? w w w w w w w w w w α c 0 ,c 0 1 k s β c 0 ,c 0 1 k s 7 ? Λ c 0 ,c 0 1 d m w d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d = I ( n − 1) % %     y y D 1 ( F c 0 , Gc 0 ) [Λ c 0 ] < D             [Λ c 0 ] < D I d f n x f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f (5.2) namely: u ( c 0 ) • 0 Λ c 0 ,c 0 1 = I d [Λ c 0 ] W e write [Λ c 0 ] for the constant 2-morphism given by Λ c 0 ; in this case it is b et ween constant morphisms: [Λ c 0 ] : [ αc 0 ] ⇒ [ β c 0 ] : I ( n − 1) → D 1 ( F c 0 , Gc 0 ) Notice that b oth functoriality axioms for 3-morphisms reduce to those for 2-morphisms, when we consider only identit y 3-morphisms (i.e. 2-morphisms c onsider e d as 3-morphisms). In the same wa y functoriality axioms for 2-morphisms reduce to those for 1-morphisms, when we consider only iden tity 3-morphisms (i.e. 2-morphisms c onsider e d as 3-morphisms). 5.3 n Cat ( C , D ) : the underlying category Here and in the following three sections w e consider n -categories C and D b e giv en. W e consider a sesqui-category structure ov er the category n Cat ( C , D ). As we did in deﬁning the sesqui-category n Cat , w e start by showing the underlying category structure. This has b een already detailed in section 3.3, hence it suﬃces to recall that: • ob jects of b n Cat ( C , D ) c are n -functors C → D ; 5.4 n Cat ( C , D ) : the hom-c ate gories 125 • arro ws of b n Cat ( C , D ) c n -lax transformation b etw een them. Comp osition is 2-morphisms 1-comp osition, ob vious units. 5.4 n Cat ( C , D ) : the hom-categories Let us ﬁx n -functors F , G : C → D . W e hav e to deﬁne categories  n Cat ( C , D )  ( F , G ), or more simply n Cat ( F , G ). • Ob jects of n Cat ( F , G ) are 2-morphisms α : F ⇒ G ; • Arro ws α → β are 3-morphisms of n -categories. 5.4.1 Comp osition F or 3-morphisms Λ = (Λ 0 , Λ − , − 1 ) : α → β and Σ = (Σ 0 , Σ − , − 1 ) : β → γ their 2-comp osition Λ • 2 Σ : α → γ is given by the following data: • ( on obje cts ) [Λ • 2 Σ] 0 : c 0 7→ Λ c 0 ◦ 1 Σ c 0 i.r. αc 0 Λ c 0 + 3 β c 0 Σ c 0 + 3 γ c 0 • ( on homs ) F or c hosen ob jects c 0 , c 0 0 one has [Λ • 2 Σ] c 0 ,c 0 0 1 =   G c 0 ,c 0 0 1 • 0 (Λ c 0 ◦ − )  • 1 Σ c 0 ,c 0 0 1  • 2  Λ c 0 ,c 0 0 1 • 1  F c 0 ,c 0 0 1 • 0 ( − ◦ Σ c 0 0 )   C 1 ( c 0 , c 0 0 ) F c 0 ,c 0 0 1                                          G c 0 ,c 0 0 1   : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : D 1 ( F c 0 , F c 0 0 ) −◦ αc 0 0   −◦ β c 0 0 K K K K K K K K K K K K K K % % K K K K K K K K K K K K K K −◦ γ c 0 0 / / D 1 ( Gc 0 , Gc 0 0 ) αc 0 ◦−   β c 0 ◦− s s s s s s s s s s s s s s y y s s s s s s s s s s s s s s γ c 0 ◦− o o D 1 ( F c 0 , Gc 0 0 ) −◦ Λ c 0 0             Λ c 0 ◦−   3 3 3 3 3 3 3 3 3 3 −◦ Σ c 0 0             Σ c 0 ◦−   3 3 3 3 3 3 3 3 3 3 α c 0 ,c 0 0 1 k s β c 0 ,c 0 0 1 k s γ c 0 ,c 0 0 1 k s Λ c 1 ,c 2 1  J T Σ c 1 ,c 2 1  J T 5.4 n Cat ( C , D ) : the hom-c ate gories 126 W e can represen t this also as a 2-dimensional pasting, sometimes useful in pro ofs: G c 0 ,c 0 0 1 • 0 ( αc 0 ◦ − ) id • 0 (Λ c 0 ◦− ) + 3 α c 0 ,c 0 0 1   G c 0 ,c 0 0 1 • 0 ( β c 0 ◦ − ) id • 0 (Σ c 0 ◦− ) + 3 β c 0 ,c 0 0 1   Λ c 0 ,c 0 0 1 n q ~ n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n G c 0 ,c 0 0 1 • 0 ( γ c 0 ◦ − ) γ c 0 ,c 0 0 1   Σ c 0 ,c 0 0 1 n q ~ n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n F c 0 ,c 0 0 1 • 0 ( − ◦ αc 0 0 ) id • 0 ( −◦ Λ c 0 0 ) + 3 F c 0 ,c 0 0 1 • 0 ( − ◦ β c 0 0 ) id • 0 ( −◦ Σ c 0 0 ) + 3 F c 0 ,c 0 0 1 • 0 ( − ◦ γ c 0 0 ) Notice that ( − ◦ Λ c 0 0 ) • 1 ( − ◦ Σ c 0 0 ) = − ◦ (Λ c 0 0 ◦ Σ c 0 0 ) = − ◦ [Λ • 2 Σ] c 0 0 (Λ c 0 ◦ − ) • 1 (Σ c 0 ◦ − ) = (Λ c 0 ◦ Σ c 0 ) ◦ − = [Λ • 2 Σ] c 0 ◦ − These data form indeed a 3-morphism. In fact let us consider the following diagram for ev ery triple of ob jects c 0 , c 0 0 , c 00 0 of C αc 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ G c 0 0 ,c 00 0 1 ( − ) [Λ • 2 Σ] c 0 ◦ id + 3 α c 0 ,c 0 0 1 ◦ id   γ c 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ G c 0 0 ,c 00 0 1 ( − ) γ c 0 ,c 0 0 1 ◦ id   [Λ • 2 Σ] c 0 ,c 0 0 1 ◦ id i o z i i i i i i i i i i i i i i i i i i i i i i i i i i i F c 0 ,c 0 0 1 ( − ) ◦ αc 0 0 ◦ G c 0 0 ,c 00 0 1 ( − ) id ◦ α c 0 0 ,c 00 0 1   id ◦ [Λ • 2 Σ] c 0 0 ◦ id + 3 F c 0 ,c 0 0 1 ( − ) ◦ γ c 0 0 ◦ G c 0 0 ,c 00 0 1 ( − ) id ◦ γ c 0 0 ,c 00 0 1   id ◦ [Λ • 2 Σ] c 0 0 ,c 00 0 1 i o z i i i i i i i i i i i i i i i i i i i i i i i i i i i F c 0 ,c 0 0 1 ( − ) ◦ F c 0 0 ,c 00 0 1 ( − ) ◦ αc 00 0 id ◦ [Λ • 2 Σ] c 00 0 + 3 F c 0 ,c 0 0 1 ( − ) ◦ F c 0 0 ,c 00 0 1 ( − ) ◦ γ c 00 0 b y deﬁnition of [Λ • 2 Σ] − , − 1 one has 5.4 n Cat ( C , D ) : the hom-c ate gories 127 αc 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ G c 0 0 ,c 00 0 1 ( − ) Λ c 0 ◦ id + 3 α c 0 ,c 0 0 1 ◦ id   β c 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ G c 0 0 ,c 00 0 1 ( − ) Σ c 0 ◦ id + 3 β c 0 ,c 0 0 1 ◦ id   Λ c 0 ,c 0 0 1 ◦ id j o { j j j j j j j j j j j j j j j j j j j j j j j j γ c 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ G c 0 0 ,c 00 0 1 ( − ) γ c 0 ,c 0 0 1 ◦ id   Σ c 0 ,c 0 0 1 ◦ id j o { j j j j j j j j j j j j j j j j j j j j j j j j F c 0 ,c 0 0 1 ( − ) ◦ αc 0 0 ◦ G c 0 0 ,c 00 0 1 ( − ) id ◦ α c 0 0 ,c 00 0 1   id ◦ Λ c 0 0 ◦ id + 3 F c 0 ,c 0 0 1 ( − ) ◦ β c 0 0 ◦ G c 0 0 ,c 00 0 1 ( − ) id ◦ β c 0 0 ,c 00 0 1   id ◦ Σ c 0 0 ◦ id + 3 id ◦ Λ c 0 0 ,c 00 0 1 j o { j j j j j j j j j j j j j j j j j j j j j j j j F c 0 ,c 0 0 1 ( − ) ◦ γ c 0 0 ◦ G c 0 0 ,c 00 0 1 ( − ) id ◦ γ c 0 0 ,c 00 0 1   id ◦ Σ c 0 0 ,c 00 0 1 j o { j j j j j j j j j j j j j j j j j j j j j j j j F c 0 ,c 0 0 1 ( − ) ◦ F c 0 0 ,c 00 0 1 ( − ) ◦ αc 00 0 id ◦ Λ c 00 0 + 3 F c 0 ,c 0 0 1 ( − ) ◦ F c 0 0 ,c 00 0 1 ( − ) ◦ β c 00 0 id ◦ Σ c 00 0 + 3 F c 0 ,c 0 0 1 ( − ) ◦ F c 0 0 ,c 00 0 1 ( − ) ◦ γ c 00 0 (5.3) This diagram is unambiguous b ecause interc hange holds on separate comp o- nen ts of pro duct ( pr o duct inter change in dimension n − 1, with intersection the constan t [ β c 0 0 ]). Hence we get αc 0 ◦ G c 0 ,c 00 0 1 ( − ◦ − ) Λ c 0 ◦ id + 3 α c 0 ,c 00 0 1   β c 0 ◦ G c 0 ,c 00 0 1 ( − ◦ − ) Σ c 0 ◦ id + 3 β c 0 ,c 00 0 1   Λ c 0 ,c 00 0 1 l p | l l l l l l l l l l l l l l l l l l l l l γ c 0 ◦ G c 0 ,c 00 0 1 ( − ◦ − ) γ c 0 ,c 00 0 1   Σ c 0 ,c 00 0 1 l p | l l l l l l l l l l l l l l l l l l l l l F c 0 ,c 00 0 1 ( − ◦ − ) ◦ αc 00 0 id ◦ Λ c 00 0 + 3 F c 0 ,c 00 0 1 ( − ◦ − ) ◦ αc 00 0 id ◦ Σ c 00 0 + 3 F c 0 ,c 00 0 1 ( − ◦ − ) ◦ αc 00 0 More simply for an y ob ject c 0 of C one has αc 0 ◦ G c 0 ,c 0 1 ( u ( c 0 )) Λ c 0 ◦ id + 3 α c 0 ,c 0 1   β c 0 ◦ G c 0 ,c 0 1 ( u ( c 0 )) Σ c 0 ◦ id + 3 β c 0 ,c 0 1   Λ c 0 ,c 0 1 n q ~ n n n n n n n n n n n n n n n n n n γ c 0 ◦ G c 0 ,c 0 1 ( u ( c 0 )) γ c 0 ,c 0 1   Σ c 0 ,c 0 1 n q ~ n n n n n n n n n n n n n n n n n n F c 0 ,c 0 1 ( u ( c 0 )) ◦ αc 0 id ◦ Λ c 0 + 3 F c 0 ,c 0 1 ( u ( c 0 )) ◦ αc 0 id ◦ Σ c 0 + 3 F c 0 ,c 0 1 ( u ( c 0 )) ◦ αc 0 = (5.4) [ αc 0 ] [Λ c 0 ] + 3 [ β c 0 ] [Σ c 0 ] + 3 id p r  p p p p p p p p p p p p p p p [ γ c 0 ] id p r  p p p p p p p p p p p p p p p [ αc 0 ] [Λ c 0 ] + 3 [ β c 0 ] [Σ c 0 ] + 3 [ γ c 0 ] where, as usual, square brac kets mean c onstant . 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 128 5.4.2 Units F or any 2-morphism β : F ⇒ G : C → D its identit y 3-morphisms id β is giv en by: • ( on obje cts ) [ id β ] 0 : c 0 7→ id β c 0 • ( on homs ) F or c hosen ob jects c 0 , c 0 0 one has [ id β ] − , − 1 = id β − , − 1 It is immediate to c heck that ab ov e pair is indeed a 3-morphisms. Similarly asso ciativity and neutral units follows from same prop erties for 2-cells and from diagrams (5.3) and (5.4) suitably adapted ( adding one mor e c olumn, for what c onc erns asso ciativity, trivializing one c olumn, for what c onc erns units ). 5.5 n Cat ( C , D ) : the sesqui-categorical structure In the this se ction we will show that hom-categories n Cat ( C , D ) underly a structure of sesqui-categories, with 2-cells pro vided b y 3-morphisms of n-categories. T o this end w e deﬁne reduced left/right 1-comp osition of a 3-morphism with a 2-morphism, according to the following reference diagram. C E * * F $ $ G z z H t t D ω + 3 α + 3 β + 3 σ + 3 Λ    5.5.1 Reduced left-composition The 3-morphism ω • 1 Λ : ω • 1 α _ * 4 ω • 1 β : E + 3 G : C / / D is giv en by the data b elow 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 129 • ( on obje cts ) F or an ob ject c 0 of C [ ω • 1 Λ] 0 : c 0 7→ E c 0 ω c 0   F c 0 αc 0 & & β c 0 x x Λ c 0 + 3 Gc 0 • ( on homs ) F or ob jects c 0 , c 0 0 of C [ ω • 1 Λ] c 0 ,c 0 0 1 =  Λ c 0 ,c 0 0 1 • 0 ( ω c 0 ◦ − )  • 1  ω c 0 ,c 0 0 1 • 0 ( − ◦ β c 0 0 )  = ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 ) ω c 0 ◦ αc 0 ◦ G c 0 ,c 0 0 1 ( − ) [ ω • 1 α ] c 0 ,c 0 0 1 + 3 ω c 0 ◦ α c 0 ,c 0 0 1 N N N N N N N N N N N N N N N N # + N N N N N N N N N N N N N N [ ω • 1 Λ] c 0 ◦ G c 0 ,c 0 0 1 ( − ) = ω c 0 ◦ Λ c 0 ◦ G c 0 ,c 0 0 1 ( − )   E c 0 ,c 0 0 1 ( − ) ◦ ω c 0 0 ◦ αc 0 0 E c 0 ,c 0 0 1 ( − ) ◦ [ ω • 1 Λ] c 0 0 = E c 0 ,c 0 0 1 ( − ) ◦ ω c 0 0 ◦ Λ c 0 0   ω c 0 ◦ F c 0 ,c 0 0 1 ( − ) ◦ αc 0 0 ω c 0 ,c 0 0 1 ◦ αc 0 0 3 ; ω c 0 ◦ F c 0 ,c 0 0 1 ( − ) ◦ Λ c 0 0   ω c 0 ◦ Λ c 0 ,c 0 0 1 p 2 ? p p p p p p p p p p p p p p p p p p p p p p p p p p p ω c 0 ◦ F c 0 ,c 0 0 1 ( − ) ◦ β c 0 0 ω c 0 ,c 0 0 1 ◦ β c 0 0 N N N N N N N N N N N N N N N N # + N N N N N N N N N N N N N N ω c 0 ◦ β c 0 ◦ G c 0 ,c 0 0 1 ( − ) ω c 0 ◦ β c 0 ,c 0 0 1 p p p p p p p p p p p p p p p p 3 ; p p p p p p p p p p p p p p [ ω • 1 β ] c 0 ,c 0 0 1 + 3 E c 0 ,c 0 0 1 ( − ) ◦ ω c 0 0 ◦ β c 0 0 The pair < [ ω • 1 Λ] 0 , [ ω • 1 Λ] − , − 1 > forms indeed a 3-morphism of n-categories. Pr o of. W e hav e to show that it satisﬁes comp osition and unit axioms. Let us b egin with comp osition, and ﬁx a triple c 0 , c 0 0 , c 00 0 of C . Notice that, in order to keep diagrams in the page we denote ◦ 0 -comp osition by juxtap osition, and subscripts for transformations on ob jects are used. 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 130 ω c 0 α c 0 G c 0 ,c 0 0 1 G c 0 0 ,c 00 0 1 ω c 0 α c 0 ,c 0 0 1 id F F F F F F F F F F F F  ' F F F F F F F F F F ω c 0 Λ c 0 G c 0 ,c 0 0 1 id   E c 0 ,c 0 0 1 ω c 0 0 α c 0 0 G c 0 0 ,c 00 0 1 E c 0 ,c 0 0 1 ω c 0 0 Λ c 0 0 G c 0 0 ,c 00 0 1   id ω c 0 0 α c 0 0 ,c 00 0 1 G G G G G G G G G G G G  ' G G G G G G G G G G ω c 0 F c 0 ,c 0 0 1 α c 0 0 G c 0 0 ,c 00 0 1 ω c 0 ,c 0 0 1 α c 0 0 id w w w w w w w w w w w w 7 ? w w w w w w w w w w ω c 0 F c 0 ,c 0 0 1 Λ c 0 0 id   E c 0 ,c 0 0 1 ω c 0 0 F c 0 0 ,c 00 0 1 α c 00 0 id ω c 0 0 F c 0 0 ,c 00 0 1 Λ c 00 0   ω c 0 Λ c 0 ,c 0 0 1 id x 5 D x x x x x x x x x x x x x x x x x x x x x ω c 0 F c 0 ,c 0 0 1 β c 0 0 G c 0 0 ,c 00 0 1 ω c 0 ,c 0 0 1 β c 0 0 id G G G G G G G G G G G G  ' G G G G G G G G G G id ω c 0 0 Λ c 0 0 ,c 00 0 1 w 5 C w w w w w w w w w w w w w w w w w w w w w E c 0 ,c 0 0 1 ω c 0 0 F c 0 0 ,c 00 0 1 β c 00 0 id ω c 0 0 ,c 00 0 1 β c 00 0  ' G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G ω c 0 β c 0 G c 0 ,c 0 0 1 G c 0 0 ,c 00 0 1 ω c 0 β c 0 ,c 0 0 1 id x x x x x x x x x x x x 7 ? x x x x x x x x x x E c 0 ,c 0 0 1 ω c 0 0 β c 0 0 G c 0 0 ,c 00 0 id 1 id ω c 0 0 β c 0 0 ,c 00 0 1 w w w w w w w w w w w w 7 ? w w w w w w w w w w E c 0 ,c 0 0 1 E c 0 0 ,c 00 0 1 ω c 00 0 β c 00 0 By pro duct in terchange it is clear that  ω c 0 ,c 0 0 1 ◦ α c 0 0 ◦ id G c 0 0 ,c 00 0 1  • 1  id E c 0 ,c 0 0 1 ◦ ω c 0 0 ◦ Λ c 0 0 ,c 00 0 1  =  ω c 0 ◦ F c 0 ,c 0 0 1 ◦ Λ c 0 0 ,c 00 0 1  • 1  ω c 0 ,c 0 0 1 ◦ id F c 0 0 ,c 00 0 1 ◦ β c 00 0  and the diagram can b e re-dra wn ω c 0 α c 0 G c 0 ,c 0 0 1 G c 0 0 ,c 00 0 1 ω c 0 α c 0 ,c 0 0 1 id F F F F F F F F F F F F  ' F F F F F F F F F F ω c 0 Λ c 0 G c 0 ,c 0 0 1 id   ω c 0 F c 0 ,c 0 0 1 F c 0 0 ,c 00 0 1 α c 00 0 ω c 0 id id Λ c 00 0   ω c 0 F c 0 ,c 0 0 1 α c 0 0 G c 0 0 ,c 00 0 1 ω c 0 id α c 0 0 ,c 00 0 1 x x x x x x x x x x x x 7 ? x x x x x x x x x x ω c 0 id Λ c 0 0 id   ω c 0 Λ c 0 ,c 0 0 1 id x 5 D x x x x x x x x x x x x x x x x x x x x x ω c 0 F c 0 ,c 0 0 1 β c 0 0 G c 0 0 ,c 00 0 1 ω c 0 id β c 0 0 ,c 00 0 1 F F F F F F F F F F F F  ' F F F F F F F F F F ω c 0 id Λ c 0 0 ,c 00 0 1 x 5 D x x x x x x x x x x x x x x x x x x x x x E c 0 ,c 0 0 1 ω c 0 0 F c 0 0 ,c 00 0 1 β c 00 0 id ω c 0 0 ,c 00 0 1 β c 00 0 G G G G G G G G G G  ' G G G G G G G G G G ω c 0 β c 0 G c 0 ,c 0 0 1 G c 0 0 ,c 00 0 1 ω c 0 β c 0 ,c 0 0 1 id x x x x x x x x x x x x 7 ? x x x x x x x x x x ω c 0 F c 0 ,c 0 0 1 F c 0 0 ,c 00 0 1 β c 00 0 ω c 0 ,c 0 0 1 id β c 00 0 x x x x x x x x x x x x 7 ? x x x x x x x x x x E c 0 ,c 0 0 1 E c 0 0 ,c 00 0 1 ω c 00 0 β c 00 0 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 131 No w let us consider the pasting of the left-hand side of ab ov e diagram, and write it equationally: h  Λ c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  ω c 0 ◦ −  i • 1 h  F c 0 ,c 0 0 1 ◦ β c 0 0 ,c 00 0 1  • 0  ω c 0 ◦ −  i • 2 h  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  ω c 0 ◦ −  i • 1 h  F c 0 ,c 0 0 1 ◦ Λ c 0 0 ,c 00 0 1  • 0  ω c 0 ◦ −  i By whiskering inter change pr op erty ( LRW ) this equals to h  Λ c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 1  F c 0 ,c 0 0 1 ◦ β c 0 0 ,c 00 0 1  i • 0  ω c 0 ◦ −  • 2 h  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 1  F c 0 ,c 0 0 1 ◦ Λ c 0 0 ,c 00 0 1  i • 0  ω c 0 ◦ −  and b y functoriality of right 0-whiskering ( R 4) 00  h  Λ c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 1  F c 0 ,c 0 0 1 ◦ β c 0 0 ,c 00 0 1  i • 2 h  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 1  F c 0 ,c 0 0 1 ◦ Λ c 0 0 ,c 00 0 1  i  • 0  ω c 0 ◦ −  ﬁnally b y functoriality w.r.t. comp osition of 3-morphism Λ Λ c 0 ,c 00 0 1 • 0 ( ω c 0 ◦ − ) = ω c 0 ◦ Λ c 0 ,c 00 0 1 Concerning composite 2-morphism on the righ t-hand side, w e can apply whisk ering prop erties and comp osition axiom for 2-morphisms  ω c 0 ,c 0 0 1 ◦ F c 0 ,c 00 0 1 ◦ β c 00 0  • 1  E c 0 ,c 0 0 1 ◦ ω c 0 0 ,c 00 0 1 ◦ β c 00 0  = ω c 0 ,c 00 0 1 ◦ β c 00 0 and diagram ab o ve can b e re-drawn as follows ω c 0 α c 0 G c 0 ,c 00 0 1 ω c 0 α c 0 ,c 00 0 1 + 3 ω c 0 Λ c 0 id   ω c 0 F c 0 ,c 00 0 1 α c 00 0 ω c 0 id Λ c 00 0   ω c 0 β c 0 G c 0 ,c 00 0 1 ω c 0 Λ c 0 ,c 00 0 1 { 6 E { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { ω c 0 β c 0 ,c 00 0 1 + 3 ω c 0 F c 0 ,c 00 0 1 β c 00 0 ω c 0 ,c 00 0 1 β c 00 0 + 3 E c 0 ,c 00 0 1 ω c 00 0 β c 00 0 and this conclude the pro of of comp osition axiom. 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 132 Concerning units, for an ob ject c 0 of C one has ω c 0 α c 0 G c 0 ,c 0 1 ( u ( c 0 )) ω c 0 α c 0 ,c 0 1 + 3 ω c 0 Λ c 0 id   ω c 0 F c 0 ,c 0 1 ( u ( c 0 )) α c 0 ω c 0 id Λ c 0   ω c 0 β c 0 G c 0 ,c 0 1 ( u ( c 0 )) ω c 0 Λ c 0 ,c 0 1 x 5 D x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ω c 0 β c 0 ,c 0 1 + 3 ω c 0 F c 0 ,c 0 1 ( u ( c 0 )) β c 0 ω c 0 ,c 0 1 β c 0 + 3 E c 0 ,c 0 1 ( u ( c 0 )) ω c 0 β c 0 then by functoriality w.r.t. units of 3-morphisms of (n-1)categories (and also b y functoriality w.r.t. units of 2-morphisms and 1-morphisms) w e get ω c 0 [ α c 0 ] ω c 0 id + 3 ω c 0 [Λ c 0 ]   ω c 0 [ α c 0 ] ω c 0 [Λ c 0 ]   ω c 0 [ β c 0 ] ω c 0 id ~ 7 F ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ω c 0 id + 3 ω c 0 [ β c 0 ] ω c 0 id + 3 ω c 0 [ β c 0 ] hence the result. 5.5.2 Reduced righ t-comp osition The 3-morphism Λ • 1 σ : α • 1 σ _ * 4 β • 1 σ : F + 3 H : C / / D is giv en by the data b elow • ( on obje cts ) F or an ob ject c 0 of C [Λ • 1 σ ] 0 : c 0 7→ F c 0 αc 0 & & β c 0 x x Λ c 0 + 3 Gc 0 σ c 0   H c 0 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 133 • ( on homs ) F or ob jects c 0 , c 0 0 of C [Λ • 1 σ ] c 0 ,c 0 0 1 =  σ c 0 ,c 0 0 1 • 0 ( αc 0 ◦ − )  • 1  Λ c 0 ,c 0 0 1 • 0 ( − ◦ σ c 0 0 )  = ( α c 0 ◦ σ c 0 ,c 0 0 1 ) • 1 (Λ c 0 ,c 0 0 1 ◦ σ c 0 0 ) αc 0 ◦ σ c 0 ◦ H c 0 ,c 0 0 1 ( − ) [ α • 1 σ ] c 0 ,c 0 0 1 + 3 αc 0 ◦ σ c 0 ,c 0 0 1 N N N N N N N N N N N N N N N N # + N N N N N N N N N N N N N N [Λ • 1 σ ] c 0 ◦ H c 0 ,c 0 0 1 = Λ c 0 ◦ σ c 0 ◦ H c 0 ,c 0 0 1   F c 0 ,c 0 0 1 ( − ) ◦ αc 0 0 ◦ σ c 0 0 F c 0 ,c 0 0 1 ◦ [Λ • 1 σ ] c 0 0 = F c 0 ,c 0 0 1 ◦ Λ c 0 0 ◦ σ c 0 0   αc 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ σ c 0 0 α c 0 ,c 0 0 1 ◦ σ c 0 0 p p p p p p p p p p p p p p p p 3 ; p p p p p p p p p p p p p p Λ c 0 ◦ G c 0 ,c 0 0 1 ◦ σ c 0 0   β c 0 ◦ G c 0 ,c 0 0 1 ( − ) ◦ σ c 0 0 Λ c 0 ,c 0 0 1 ◦ σ c 0 0 p 2 ? p p p p p p p p p p p p p p p p p p p p p p p p p p p β c 0 ,c 0 0 1 ◦ σ c 0 0 N N N N N N N N N N N N N N N N # + N N N N N N N N N N N N N N β c 0 ◦ σ c 0 ◦ H c 0 ,c 0 0 1 ( − ) β c 0 ◦ σ c 0 ,c 0 0 1 3 ; [ β • 1 σ ] c 0 ,c 0 0 1 + 3 F c 0 ,c 0 0 1 ( − ) ◦ β c 0 0 ◦ σ c 0 0 The pair < [Λ • 1 σ ] 0 , [Λ • 1 σ ] − , − 1 > forms indeed a 3-morphism of n-categories. The pro of is a straightforw ard v ariation of the pro of for reduced right- comp osition ab o v e, hence it is omitted. 5.5.3 Prop erties In this section w e giv e some prop erties of left/right 1-comp osition of a 3- morphism with a 2-morphism. They are mo deled on similar prop erties giv en in the deﬁnition of a sesqui-category , and they are extremely useful in dealing with calculations. Let us consider the diagram E 0 ω 0 + 3 E ω + 3 F α   β + 3 γ C K G σ + 3 H σ 0 + 3 H 0 Λ    Σ    5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 134 as a reference for the follo wing Prop osition 5.6. (2-c omp osition (i.e. vertic al) c omp osition of 3-morphisms w.r.t. (r e duc e d) 1-c omp osition with a 2-morphism) ( L 1) 0 id F • 1 L Λ = Λ ( R 1) 0 Λ • 1 R id G = Λ ( L 2) 0 ( ω 0 • 1 ω ) • 1 L Λ = ω 0 • 1 L ( ω • 1 L Λ) ( R 2) 0 Λ • 1 R ( σ • 1 σ 0 ) = (Λ • 1 R σ ) • 1 R σ 0 ( L 3) 0 ω • 1 L id α = id ω α ( R 3) 0 id α • 1 R σ = id ασ ( L 4) 0 ω • 1 L (Λ • 2 Σ) = ( ω • 1 L Λ) • 2 ( ω • 1 L Σ) ( R 4) 0 (Λ • 2 Σ) • 1 R σ = (Λ • 1 R σ ) • 2 (Σ • 1 R σ ) ( LR 5) 0 ( ω • 1 L Λ) • 1 R σ = ω • 1 L (Λ • 1 R σ ) Pr o of. W e only pro v e statemen ts ( L 1) 0 to ( L 4) 0 and statement ( LR 5). Pro of of prop erties ( R 1) 0 to ( R 4) 0 is just a straightforw ard v ariation of pro of of ( L 1) 0 to ( L 4) 0 , hence it will b e omitted. • ( L 1) 0 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then [ id F • 1 Λ] c 0 = [ id F ] c 0 ◦ Λ c 0 = id F c 0 ◦ Λ c 0 = Λ c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then [ id F • 1 Λ] c 0 ,c 0 0 1 = ([ id F ] c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ([ id F ] c 0 ,c 0 0 1 ◦ β c 0 0 ) = ( id F c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( id F c 0 ,c 0 0 1 ◦ β c 0 0 ) ( ♣ ) = Λ c 0 ,c 0 0 1 • 1 id F c 0 ,c 0 0 1 ◦ β c 0 0 ( ♠ ) = Λ c 0 ,c 0 0 1 where ( ♣ ) holds b y ( R 3) and ( ♠ ) holds by ( L 1) 0 in dimension n − 1. • ( L 2) 0 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then [( ω 0 • 1 ω ) • 1 Λ] c 0 = [ ω 0 • 1 ω ] c 0 ◦ Λ c 0 = ω 0 c 0 ◦ ω c 0 ◦ Λ c 0 = ω 0 c 0 ◦ [ ω • 1 Λ] c 0 = [ ω 0 c 0 • 1 ( ω • 1 Λ)] c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then [( ω 0 • 1 ω ) • 1 Λ] c 0 ,c 0 0 1 = 5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 135 = ([ ω 0 • 1 ω ] c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ([ ω 0 • 1 ω ] c 0 ,c 0 0 1 ◦ β c 0 0 ) = ( ω 0 c 0 ◦ ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1   ( ω 0 c 0 ◦ ω c 0 ,c 0 0 1 ) • 1 ( ω 0 1 c 0 ,c 0 0 ◦ ω c 0 0 )  ◦ β c 0 0  = ( ω 0 c 0 ◦ ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω 0 c 0 ◦ ω c 0 ,c 0 0 1 ◦ β c 0 0 ) • 1 ( ω 0 1 c 0 ,c 0 0 ◦ ω c 0 0 ◦ β c 0 0 ) ( ♠ ) =  ω 0 c 0 ◦  ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 )   • 1 ( ω 0 1 c 0 ,c 0 0 ◦ ( ω c 0 0 ◦ β c 0 0 )) = ( ω 0 c 0 ◦ [ ω • 1 Λ] c 0 ,c 0 0 1 ) • 1 ( ω 0 1 c 0 ,c 0 0 ◦ [ ω • 1 β ] c 0 0 ) = [ ω 0 • 1 ( ω • 1 Λ)] c 0 ,c 0 0 1 where expression ( ♠ ) is unambiguous for the same prop erty in dimension n − 1. • ( L 3) 0 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then quite plainly [ ω • 1 id α ] c 0 = ω c 0 ◦ [ id α ] c 0 = ω c 0 ◦ id α c 0 = id ω c 0 ◦ α c 0 = id [ ω • 1 α ] c 0 = [ id ω • 1 α ] c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then [ ω • 1 id α ] c 0 ,c 0 0 1 = ( ω c 0 ◦ [ id α ] c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ α c 0 0 ) = [ id ω c 0 ◦ α ] c 0 ,c 0 0 1 • 1 ( ω c 0 ,c 0 0 1 ◦ α c 0 0 ) = ( id ω c 0 ◦ α c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ α c 0 0 ) ( ♣ ) = id ( ω c 0 ◦ α c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ α c 0 0 ) = id [ ω • 1 α ] c 0 ,c 0 0 1 = [ id ω • 1 α ] c 0 ,c 0 0 1 where ( ♣ ) holds b y ( L 3) 0 in dimension n − 1. • ( L 4) 0 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then [ ω • 1 (Λ • 2 Σ)] c 0 = ω c 0 ◦ 0 [Λ • 2 Σ] c 0 = ω c 0 ◦ 0 (Λ c 0 ◦ 1 Σ c 0 ) ( ♥ ) = ( ω c 0 ◦ 0 Λ c 0 ) ◦ 1 ( ω c 0 ◦ 0 Σ c 0 ) = [ ω • 1 Λ] c 0 ◦ 1 [ ω • 1 Σ] c 0 = [( ω • 1 Λ) • 2 ( ω • 1 Σ)] c 0 where all equalities follo w straigh tforward from deﬁnitions, but ( ♥ ) that is the strict inter change pr op erty of ◦ 0 and ◦ 1 . ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then by deﬁnition [ ω • 1 (Λ • 2 Σ)] c 0 ,c 0 0 1 =  ω c 0 ◦ [Λ • 2 Σ] c 0 ,c 0 0 1  • 1  ω c 0 ,c 0 0 1 ◦ γ c 0 0  5.5 n Cat ( C , D ) : the sesqui-c ate goric al structur e 136 = ω c 0 ◦   (Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 Σ c 0 ,c 0 0 1  • 2  Λ c 0 ,c 0 0 1 • 1 ( F c 0 ,c 0 0 1 ◦ Σ c 0 0 )   ! • 1  ω c 0 ,c 0 0 1 ◦ γ c 0 0  b y 0-whiskering of a morphism with a 2-comp osition ( L 4) 00 this gets  ω c 0 ◦  (Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 Σ c 0 ,c 0 0 1   • 2  ω c 0 ◦  Λ c 0 ,c 0 0 1 • 1 ( F c 0 ,c 0 0 1 ◦ Σ c 0 0 )   ! • 1  ω c 0 ,c 0 0 1 ◦ γ c 0 0  b y 1-whiskering of a 2-morphism with a 2-comp osition ( R 4) 0  ω c 0 ◦  (Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 Σ c 0 ,c 0 0 1   • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) • 2  ω c 0 ◦  Λ c 0 ,c 0 0 1 • 1 ( F c 0 ,c 0 0 1 ◦ Σ c 0 0 )   • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) and b y ( L 4) and asso ciativity of 1-comp osition ( LR whiskering prop erty) ( ω c 0 ◦ Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 ( ω c 0 ◦ Σ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) • 2 ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ◦ F c 0 ,c 0 0 1 ◦ Σ c 0 0 ) • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) rearranging the terms (since in terchange holds for 0-composition with a constan t transformation) ( ω c 0 ◦ Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 ( ω c 0 ◦ Σ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) • 2 ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ Λ c 0 0 ) rearranging the terms again ( ω c 0 ◦ Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1 ( ω c 0 ◦ Σ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 ) • 2 ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 ) • 1 ( E c 0 ,c 0 0 1 ◦ ω c 0 0 ◦ Σ c 0 0 ) just adding brac kets ( ω c 0 ◦ Λ c 0 ◦ G c 0 ,c 0 0 1 ) • 1  ( ω c 0 ◦ Σ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ γ c 0 0 )  • 2  ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 )  • 1 ( E c 0 ,c 0 0 1 ◦ ω c 0 0 ◦ Σ c 0 0 ) b y deﬁnition of 1-whiskering with a 3-morphism, this can b e rewritten more con venien tly  ([ ω • 1 Λ] c 0 ◦ G c 0 ,c 0 0 1 ) • 1 [ ω • 1 Σ] c 0 ,c 0 0 1  • 2  [ ω • 1 Λ] c 0 ,c 0 0 1 • 1 ( E c 0 ,c 0 0 1 ◦ [ ω • 1 Σ] c 0 0 )  5.6 0-whiskering of 3-morphisms 137 and ﬁnally , b y deﬁnition of 2-comp osition of 3-morphisms [( ω • 1 Λ) • 2 ( ω • 1 Σ)] c 0 ,c 0 0 1 • ( LR 5) 0 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then [( ω • 1 Λ) • 1 σ ] c 0 = [ ω • 1 Λ] c 0 ◦ σ c 0 = ω c 0 ◦ Λ c 0 ◦ σ c 0 = ω c 0 ◦ [Λ • 1 σ ] c 0 = [ ω • 1 (Λ • 1 σ )] c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then [( ω • 1 Λ) • 1 σ ] c 0 ,c 0 0 1 = = ([ ω • 1 α ] c 0 ◦ σ c 0 ,c 0 0 1 ) • 1 ([ ω • 1 Λ] c 0 ,c 0 0 1 ◦ σ c 0 0 ) = ( ω c 0 ◦ α c 0 ◦ σ c 0 ,c 0 0 1 ) • 1   ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 )  ◦ σ c 0 0  = ( ω c 0 ◦ α c 0 ◦ σ c 0 ,c 0 0 1 ) • 1 ( ω c 0 ◦ Λ c 0 ,c 0 0 1 ◦ σ c 0 0 ) • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 ◦ σ c 0 0 ) =  ω c 0 ◦  ( α c 0 ◦ σ c 0 ,c 0 0 1 ) • 1 (Λ c 0 ,c 0 0 1 ◦ σ c 0 0 )   • 1 ( ω c 0 ,c 0 0 1 ◦ β c 0 0 ◦ σ c 0 0 ) = ( ω c 0 ◦ [Λ • 1 Σ] c 0 ,c 0 0 1 ) • 1 ( ω c 0 ,c 0 0 1 ◦ [ β • 1 σ ] c 0 0 ) = [ ω • 1 (Λ • 1 σ )] c 0 ,c 0 0 1 5.6 0-whisk ering of 3-morphisms In this section we deﬁne reduced left/right 1-comp osition of a 3-morphism with a 1-morphism, according to the follo wing reference diagram. B E / / C F   G B B D H / / E α   β   Λ _ * 4 5.6 0-whiskering of 3-morphisms 138 5.6.1 Reduced left-composition The 3-morphism E • 0 Λ : E • 0 α _ * 4 E • 0 β : E • 0 F + 3 E • 0 G : B / / D is giv en by the data b elow • ( on obje cts ) F or an ob ject b 0 of B [ E • 0 Λ] 0 : b 0 7→ F ( E b 0 ) α ( E b 0 ) β ( E b 0 ) ~ ~ G ( E b 0 ) Λ( E b 0 ) + 3 • ( on homs ) F or ob jects b 0 , b 0 0 of B [ E • 0 Λ] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 ( − ) • 0 Λ E b 0 ,E b 0 0 1 α ( E b 0 ) ◦ G E b 0 ,E b 0 0 1  E b 0 ,b 0 0 1 ( − )  Λ( E b 0 ) ◦ id + 3 [ E • 0 α ] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 • 0 α E b 0 ,E b 0 0 1   β ( E b 0 ) ◦ G E b 0 ,E b 0 0 1  E b 0 ,b 0 0 1 ( − )  [ E • 0 β ] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 • 0 β E b 0 ,E b 0 0 1   [ E • 0 Λ] b 0 ,b 0 0 1 q r  q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q F E b 0 ,E b 0 0 1  E b 0 ,b 0 0 1 ( − )  ◦ α ( E b 0 0 ) id ◦ Λ( E b 0 0 ) + 3 F E b 0 ,E b 0 0 1  E b 0 ,b 0 0 1 ( − )  ◦ β ( E b 0 0 ) The pair < [ E • 0 α ] 0 , [ E • 0 α ] − , − 1 > forms indeed a 3-morphism of n-categories. Pr o of. W e hav e to show that it satisﬁes comp osition and unit axioms. Let us b egin with comp osition, and ﬁx a triple c 0 , c 0 0 , c 00 0 of C . Notice that, in order to k eep wide diagrams in the page we write ◦ 0 -comp osition b y juxtap osition, and subscripts for transformations on ob jects are used. 5.6 0-whiskering of 3-morphisms 139 α E b 0 [ E G ] b 0 ,b 0 0 1 [ E G ] b 0 0 ,b 00 0 1 Λ E b 0 id id + 3 [ E • 0 α ] b 0 ,b 0 0 1 id   β E b 0 [ E G ] b 0 ,b 0 0 1 [ E G ] b 0 0 ,b 00 0 1 [ E • 0 β ] b 0 ,b 0 0 1 id   [ E • 0 Λ] b 0 ,b 0 0 1 id s s  s s s s s s s s s s s s s s s s s s s s s s s s s s s [ E F ] b 0 ,b 0 0 1 α E b 0 0 [ E G ] b 0 0 ,b 00 0 1 id Λ E b 0 0 id + 3 id [ E • 0 α ] b 0 0 ,b 00 0 1   [ E F ] E b 0 ,E b 0 0 1 β E b 0 0 [ E G ] b 0 0 ,b 00 0 1 id [ E • 0 β ] b 0 0 ,b 00 0 1   id [ E • 0 Λ] b 0 0 ,b 00 0 1 s s  s s s s s s s s s s s s s s s s s s s s s s s s s s s [ E F ] b 0 ,b 0 0 1 [ E G ] b 0 0 ,b 00 0 1 α E b 00 0 id id Λ E b 00 0 + 3 [ E F ] E b 0 ,E b 0 0 1 [ E G ] b 0 0 ,b 00 0 1 β E b 00 0 This can b e written equationally h E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1  ◦  E b 0 0 ,b 00 0 1 • 0 G E b 0 0 ,E b 00 0 1 i • 1 h E b 0 ,b 0 0 1 • 0 F E b 0 ,E b 0 0 1  ◦  E b 0 0 ,b 00 0 1 • 0 β E b 0 0 ,E b 00 0 1 i • 2 h E b 0 ,b 0 0 1 • 0 α E b 0 ,E b 0 0 1  ◦  E b 0 0 ,b 00 0 1 • 0 G E b 0 0 ,E b 00 0 1 i • 1 h E b 0 ,b 0 0 1 • 0 F E b 0 ,E b 0 0 1  ◦  E b 0 0 ,b 00 0 1 • 0 Λ E b 0 0 ,E b 00 0 1 i b y pro duct interc hange b efore ◦ 0 -comp osition this turns to b e h E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0  Λ E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1 i • 1 h E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0  F E b 0 ,E b 0 0 1 ◦ β E b 0 0 ,E b 00 0 1 i • 2 h E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0  α E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1 i • 1 h E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0  F E b 0 ,E b 0 0 1 ◦ Λ E b 0 0 ,E b 00 0 1 i b y whiskering inter change pr op erty ( LR ) this gives  E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0 h Λ E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1  • 1  F E b 0 ,E b 0 0 1 ◦ β E b 0 0 ,E b 00 0 1 i • 2  E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0 h α E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1 ) • 1  F E b 0 ,E b 0 0 1 ◦ Λ E b 0 0 ,E b 00 0 1 i hence b y ( L 4) 00 5.6 0-whiskering of 3-morphisms 140  E b 0 ,b 0 0 1 × E b 0 0 ,b 00 0 1  • 0      Λ E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1  • 1  F E b 0 ,E b 0 0 1 ◦ β E b 0 0 ,E b 00 0 1  • 2  α E b 0 ,E b 0 0 1 ◦ G E b 0 0 ,E b 00 0 1 ) • 1  F E b 0 ,E b 0 0 1 ◦ Λ E b 0 0 ,E b 00 0 1      By comp osition functoriality of Λ, the second row changes to ( − ◦ − ) • 0 Λ E b 0 ,E b 00 0 1 th us giving by asso ciativity of 0-whiskering  E b 0 ,b 0 0 1 ◦ E b 0 0 ,b 00 0 1  • 0 Λ E b 0 ,E b 00 0 1 = E b 0 ,b 00 0 1 • 0 Λ E b 0 ,E b 00 0 1 where the last equalit y follows from comp osition axiom for 1-morphisms. T urning to units axiom, let an ob ject b 0 of B b e giv en. Then u ( b 0 ) • 0 E b 0 ,b 0 1 • 0 Λ E b 0 ,E b 0 1 = u ( E b 0 ) • 0 Λ E b 0 ,E b 0 1 = id [Λ( E b 0 )] = id [( E • 0 Λ)( b 0 )] where the ﬁrst expression is unambiguous for • 0 -asso ciativit y ( L 2) 00 , ﬁrst equalit y holds by units axiom for 1-morphism E , second b y units axiom for 3-morphism Λ, last is the deﬁnition. 5.6.2 Reduced righ t-comp osition The 3-morphism Λ • 0 H : α • 0 _ * 4 β • 0 H : F • 0 H + 3 G • 0 H : C / / E is giv en by the data b elow • ( on obje cts ) F or an ob ject c 0 of C [Λ • 0 H ] 0 : c 0 7→ H ( F c 0 ) H ( αc 0 ) H ( β c 0 ) ~ ~ H ( Gc 0 ) H (Λ c 0 ) + 3 • ( on homs ) F or ob jects c 0 , c 0 0 of C [Λ • 0 H ] c 0 ,c 0 0 1 = Λ c 0 ,c 0 0 1 • 0 H F c 0 ,Gc 0 0 1 5.6 0-whiskering of 3-morphisms 141 H F c 0 ,Gc 0 1 ( α c 0 ) ◦ H Gc 0 ,Gc 0 0 1 ( G c 0 ,c 0 0 1 ( − )) H F c 0 ,Gc 0 1 (Λ c 0 ) ◦ id + 3 H F c 0 ,Gc 0 0 1 ( α c 0 ,c 0 0 1 )   H F c 0 ,Gc 0 1 ( β c 0 ) ◦ H Gc 0 ,Gc 0 0 1 ( G c 0 ,c 0 0 1 ( − )) H F c 0 ,Gc 0 0 1 ( β c 0 ,c 0 0 1 )   [Λ • 0 H ] c 0 ,c 0 0 1 n q ~ n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n H F c 0 ,F c 0 0 1 ( F c 0 ,c 0 0 1 ) ◦ H F c 0 0 ,Gc 0 0 1 ( α c 0 0 ) id ◦ H F c 0 0 ,Gc 0 0 1 (Λ c 0 0 ) + 3 H F c 0 ,F c 0 0 1 ( F c 0 ,c 0 0 1 ) ◦ H F c 0 0 ,Gc 0 0 1 ( β c 0 0 ) The pair < [Λ • 0 H ] 0 , [Λ • 0 H ] − , − 1 > forms indeed a 3-morphism of n-categories. The pro of is a straightforw ard v ariation of the pro of for reduced right- comp osition ab o v e, hence it is omitted. 5.6.3 Prop erties As we did in describing the sesqui-categorical structure for homs in n Cat , w e use again a left-and-right approach to describ e prop erties of the 0-whisk ering of of a 3-morphism with a morphism. Let us consider the diagram A E 0 / / B E / / C F   G B B D H / / E H 0 / / F α   β   γ   Λ _ * 4 Σ _ * 4 as a reference for the follo wing Prop osition 5.7 (2-comp osition (i.e. vertical) comp osition of 3-morphisms w.r.t. (reduced) 0-comp osition with a (1-)morphism) . ( L 1) 00 id C • 0 L Λ = Λ ( R 1) 00 Λ • 0 R id D = Λ ( L 2) 00 ( E 0 • 0 L E ) • 0 L Λ = E 0 • 0 L ( E • 0 L Λ) ( R 2) 00 Λ • 0 R ( H • 0 R H 0 ) = (Λ • 0 R H ) • 0 R H 0 ( L 3) 00 E • 0 L id α = id E • 0 L α ( R 3) 00 id α • 0 R H = id α • 0 R H ( L 4) 00 E • 0 L (Λ • 2 Σ) = ( E • 0 L Λ) • 2 ( E • 0 L Σ) ( R 4) 00 (Λ • 2 Σ) • 0 R H = (Λ • 0 R H ) • 2 (Σ • 0 R H ) ( LR 5) 00 ( E • 0 L Λ) • 0 R H = E • 0 L (Λ • 0 R H ) Pr o of. W e prov e statements ( L 1) 00 to ( L 4) 00 and ( LR 5). Pro ofs of statements ( R 1) 00 to ( R 4) 00 is similar, hence it is omitted. 5.6 0-whiskering of 3-morphisms 142 • ( L 1) 00 ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then [ id C • 0 Λ] c 0 = Λ( id C ( c 0 )) = Λ c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then [ id C • 0 Λ] c 0 c 0 0 1 = [ id C ] c 0 ,c 0 0 1 • 0 Λ c 0 c 0 0 1 = id C 1 ( c 0 ,c 0 0 ) • 0 Λ c 0 c 0 0 1 = Λ c 0 c 0 0 1 • ( L 2) 00 ( on obje cts) L et an obje ct a 0 of A b e given. Then [( E 0 • 0 E ) • 0 Λ] a 0 = Λ ( E 0 • 0 E ) a 0 = Λ E ( E 0 a 0 ) = [ E • 0 Λ] E 0 a 0 = [ E 0 • 0 ( E • 0 Λ)] a 0 ( on homs ) L et obje cts a 0 , a 0 0 of A b e given. Then [( E 0 • 0 E ) • 0 Λ] a 0 ,a 0 0 1 = [ E 0 • 0 E ] a 0 ,a 0 0 1 • 0 Λ E ( E 0 a 0 ) ,E ( E 0 a 0 0 ) 1 = E 0 1 a 0 ,a 0 0 • 0 E E 0 a 0 ,E 0 a 0 0 1 • 0 Λ E ( E 0 a 0 ) ,E ( E 0 a 0 0 ) 1 = E 0 1 a 0 ,a 0 0 • 0 [ E • 0 Λ] E 0 a 0 ,E 0 a 0 0 1 = [ E 0 • 0 ( E • 0 Λ)] a 0 ,a 0 0 1 • ( L 3) 00 ( on ob jects ) L et an obje ct b 0 of B b e given. Then [ E • 0 id α ] b 0 = [ id α ] E b 0 = id α E b 0 = id [ E • 0 α ] b 0 = [ id E • 0 α ] b 0 ( on homs ) L et obje cts b 0 , b 0 0 of B b e given. Then [ E • 0 id α ] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 • 0 [ id α ] E b 0 ,E b 0 0 1 = E b 0 ,b 0 0 1 • 0 id α E b 0 ,E b 0 0 1 ( ♣ ) = id E b 0 ,b 0 0 1 • 0 α E b 0 ,E b 0 0 1 = id [ E • 0 α ] b 0 ,b 0 0 1 = [ id E • 0 α ] b 0 ,b 0 0 1 wher e ( ♣ ) holds for the same pr op erty in dimension n − 1 . • ( L 4) 00 ( on ob jects ) L et an obje ct b 0 of B b e given. Then [ E • 0 (Λ • 2 Σ)] b 0 = [(Λ • 2 Σ] E b 0 = Λ E b 0 ◦ 1 Σ E b 0 = [ E • 0 Λ] b 0 ◦ 1 [ E • 0 Σ] b 0 = [( E • 0 Λ) • 2 ( E • 0 Σ)] b 0 5.6 0-whiskering of 3-morphisms 143 ( on homs ) L et obje cts b 0 , b 0 0 of B b e given. Then [ E • 0 (Λ • 2 Σ)] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 • 0 [Λ • 2 Σ] E b 0 ,E b 0 0 1 = E b 0 ,b 0 0 1 • 0    (Λ E b 0 ◦ G E b 0 ,E b 0 0 1 ) • 1 Σ E b 0 ,E b 0 0 1 • 2 Λ E b 0 ,E b 0 0 1 • 1 ( F E b 0 ,E b 0 0 1 ◦ Σ E b 0 0 )    ( ♥ ) =    E b 0 ,b 0 0 1 • 0  (Λ E b 0 ◦ G E b 0 ,E b 0 0 1 ) • 1 Σ E b 0 ,E b 0 0 1  • 2 E b 0 ,b 0 0 1 • 0  Λ E b 0 ,E b 0 0 1 • 1 ( F E b 0 ,E b 0 0 1 ◦ Σ E b 0 0 )     ( ♠ ) =     Λ E b 0 ◦ ( E b 0 ,b 0 0 1 • 0 G E b 0 ,E b 0 0 1 )  • 1 ( E b 0 ,b 0 0 1 • 0 Σ E b 0 ,E b 0 0 1 ) • 2 ( E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1 ) • 1  ( E b 0 ,b 0 0 1 • 0 F E b 0 ,E b 0 0 1 ) ◦ Σ E b 0 0     =     [ E • 0 Λ] b 0 ◦ [ E • 0 G ] b 0 ,b 0 0 1  • 1 [ E • 0 Σ] b 0 ,b 0 0 1 • 2 [ E • 0 Λ] b 0 ,b 0 0 1 • 1  [ E • 0 F ] b 0 ,b 0 0 1 ◦ [ E • 0 Σ] b 0 0     = [( E • 0 Λ) • 2 ( E • 0 Σ)] b 0 ,b 0 0 1 wher e ( ♥ ) holds by the same pr op erty in dimension n − 1 , and ( ♣ ) holds by whisk ering interc hange prop erty . • ( LR 5) 00 ( on ob jects ) L et an obje ct b 0 of B b e given. Then [( E • 0 Λ) • 0 H ] b 0 = H ([ E • 0 Λ] b 0 = = H (Λ E b 0 ) = [Λ • 0 H ] E b 0 = [ E • 0 (Λ • 0 H )] b 0 ( on homs ) L et obje cts b 0 , b 0 0 of B b e given. Then [( E • 0 Λ) • 0 H ] b 0 ,b 0 0 1 = [ E • 0 Λ] b 0 ,b 0 0 1 • 0 H F ( E b 0 ) ,G ( E b 0 0 ) 1 = E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1 • 0 H F ( E b 0 ) ,G ( E b 0 0 ) 1 = E b 0 ,b 0 0 1 • 0 [Λ • 0 H ] E b 0 ,E b 0 0 1 = [ E • 0 (Λ • 0 H )] b 0 ,b 0 0 1 Before switc hing to next section, let us give a last prop erty that express at once functoriality of left and right 0-comp osition with a morphism. T o this end, let us b e giv en also 2-morphisms ω : M ⇒ F and σ : G ⇒ N , as 5.6 0-whiskering of 3-morphisms 144 represen ted in the diagram b elow B E   C M , , F ( ( G v v N r r D ω + 3 α + 3 β + 3 σ + 3 Λ    H   E Left/righ t 0-comp osition of a 3-morphism with a morphism satisﬁes also the follo wing prop erty that relates 0-whiskering w.r.t. 1-whisk ering: Prop osition 5.8 (Whisk ering interc hange prop erty) . ( LRW ) E • 0 L  ω • 1 L Λ • 1 R σ  • 0 R H = ( E • 0 L ω • 0 R H ) • 1 L ( E • 0 L Λ • 0 R H ) • 1 R ( E • 0 L σ • 0 R H ) Pr o of. Without loss of generality it suﬃces to prov e the follo wing tw o equal- ities: ( LRW ) 1 E • 0 L  ω • 1 L Λ  = ( E • 0 L ω ) • 1 L ( E • 0 L Λ) ( LRW ) 2 E • 0 L  Λ • 1 R σ  = ( E • 0 L Λ) • 1 R ( E • 0 L σ ) • ( LRW ) 1 ( on obje cts ) Let an ob ject b 0 of B b e giv en. Then [ E • 0  ω • 1 Λ  ] b 0 = [ ω • 1 Λ] E b 0 = ω E b 0 ◦ Λ E b 0 = [ E • 0 ω ] b 0 ◦ [ E • 0 Λ] b 0 = [( E • 0 ω ) • 1 ( E • 0 Λ)] b 0 ( on homs ) Let ob jects b 0 , b 0 0 of B b e giv en. Then  E • 0  ω • 1 Λ  b 0 ,b 0 0 1 = 5.6 0-whiskering of 3-morphisms 145 = E b 0 ,b 0 0 1 • 0 [ ω • 1 Λ] E b 0 ,E b 0 0 1 = E b 0 ,b 0 0 1 • 0   ω E b 0 ◦ Λ E b 0 ,E b 0 0 1  • 1  ω E b 0 ,E b 0 0 1 ◦ β E b 0 0   ( ♣ ) =  E b 0 ,b 0 0 1 • 0  ω E b 0 ◦ Λ E b 0 ,E b 0 0 1   • 1  E b 0 ,b 0 0 1 • 0  ω E b 0 ,E b 0 0 1 ◦ β E b 0 0   =  E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1 • 0  ω E b 0 ◦ −   • 1  E b 0 ,b 0 0 1 • 0 ω E b 0 ,E b 0 0 1 • 0  − ◦ β E b 0 0   =  ω E b 0 ◦  E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1   • 1   E b 0 ,b 0 0 1 • 0 ω E b 0 ,E b 0 0 1  ◦ β E b 0 0  =  [ E • 0 ω ] b 0 ◦ [ E • 0 Λ] b 0 ,b 0 0 1  • 1  [ E • 0 ω ] b 0 ,b 0 0 1 ◦ [ E • 0 β ] b 0 0  = [( E • 0 ω ) • 1 ( E • 0 Λ)] b 0 ,b 0 0 1 where equalit y ( ♣ ) holds by same property in dimension n − 1, and the follo wing by asso ciativity of 0-comp osition. • ( LRW ) 2 ( on obje cts ) Let an ob ject b 0 of B b e giv en. Then [ E • 0  Λ • 1 σ  ] b 0 = [Λ • 1 σ ] E b 0 = Λ E b 0 ◦ σ E b 0 = [ E • 0 Λ] b 0 ◦ [ E • 0 σ ] b 0 = [( E • 0 Λ) • 1 ( E • 0 σ )] b 0 ( on homs ) Let ob jects b 0 , b 0 0 of B b e giv en. Then [ E • 0  Λ • 1 σ  ] b 0 ,b 0 0 1 = E b 0 ,b 0 0 1 • 0 [Λ • 1 σ ] E b 0 ,E b 0 0 1 = E b 0 ,b 0 0 1 • 0   α E b 0 ◦ σ E b 0 ,E b 0 0 1  • 1  Λ E b 0 ,E b 0 0 1 ◦ σ E b 0 0   =  E b 0 ,b 0 0 1 • 0 σ E b 0 ,E b 0 0 1  α E b 0 ◦ −   • 1  E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1  − ◦ σ E b 0 0   =  E b 0 ,b 0 0 1 • 0  α E b 0 ◦ σ E b 0 ,E b 0 0 1   • 1  E b 0 ,b 0 0 1 • 0  Λ E b 0 ,E b 0 0 1 ◦ σ E b 0 0   =  α E b 0 ◦  E b 0 ,b 0 0 1 • 0 σ E b 0 ,E b 0 0 1   • 1   E b 0 ,b 0 0 1 • 0 Λ E b 0 ,E b 0 0 1  ◦ σ E b 0 0  =  [ E • 0 α ] b 0 ◦ [ E • 0 σ ] b 0 ,b 0 0 1  • 1  [ E • 0 Λ] b 0 ,b 0 0 1 ◦ [ E • 0 σ ] b 0 0  = [( E • 0 Λ) • 1 ( E • 0 σ )] b 0 ,b 0 0 1 5.7 Dimension r aising 0-c omp osition of 2-morphisms 146 5.7 Dimension raising 0-comp osition of 2-morphisms Let t wo 0-intersecting 2-morphisms of n-categories b e given. C F   G E E α   D H   K E E β   E It is easy to v erify that in general α \ β := ( F • 0 β ) • 1 ( α • 0 K ) 6 = ( α • 0 H ) • 1 ( G • 1 β ) =: α/β C F   D H   K E E β   E C F   G E E α   D K E E E 6 = C F   G E E α   D H   E C G E E D H   K E E β   E (5.5) More in terestingly they constitute a 3-morphism α ∗ β : α \ β _ * 4 α/β In fact, for ev ery ob ject c 0 of C one deﬁnes [ α ∗ β ] 0 : c 0 7→ H ( F c 0 ) β F c 0 y y s s s s s s s s H αc 0 % % K K K K K K K K K ( F c 0 ) β 1 ( αc 0 ) + 3 K αc 0 % % K K K K K K K K H ( Gc 0 ) β Gc 0 y y s s s s s s s s K ( Gc 0 ) Moreo ver for every pair of ob jects c 0 , c 0 0 of C one deﬁnes [ α ∗ β ] c 0 ,c 0 0 1 = α c 0 ,c 0 0 1 ∗ β F c 0 ,Gc 0 0 1 Claim 5.9. The p air < [ α ∗ β ] 0 , [ α ∗ β ] − , − 1 > is inde e d a 3-morphism of n-c ate gories. 5.7 Dimension r aising 0-c omp osition of 2-morphisms 147 Pr o of. W e m ust show that domain and co domain of [ α ∗ β ] − , − 1 are compatible with those of the deﬁnition of 3-morphism. Moreov er the pair ab ov e must satisfy unit and comp osition axioms. In order to prov e the ﬁrst fact, w e write the diagram that represen ts the 3-morphism of (n-1)categories [ α ∗ β ] c 0 ,c 0 0 1 : α c 0 ,c 0 0 1 \ β F c 0 ,Gc 0 0 1 _ * 4 α c 0 ,c 0 0 1 /β F c 0 ,Gc 0 0 1 i.e. the comp osition [ c 0 , c 0 0 ] F 1 w w n n n n n n n n n G 1 ' ' P P P P P P P P P [ F c 0 , F c 0 0 ] −◦ αc 0 0 ' ' P P P P P P P P P [ Gc 0 , Gc 0 0 ] αc 0 ◦− w w n n n n n n n n n α c 0 ,c 0 0 1 k s [ F c 0 , Gc 0 0 ] H 1 w w n n n n n n n n n K 1 ' ' P P P P P P P P P [ H ( F c 0 ) , H ( Gc 0 0 )] −◦ β Gc 0 0 ' ' P P P P P P P P P [ K ( F c 0 ) , K ( Gc 0 0 )] β F c 0 ◦− w w n n n n n n n n n β F c 0 ,Gc 0 0 1 k s [ H ( F c 0 ) , K ( Gc 0 0 )] Its domain is computed b elo w [ c 0 , c 0 0 ] [ F H ] 1 w w o o o o o o o o o o o o o o o o o o o o o [ GH ] 1   αc 0 ◦ G 1 ( − ) ' ' O O O O O O O O O O O O O O O O O O O O O [ H ( F c 0 ) , H ( F c 0 0 )] −◦ H αc 0 0   [ H ( Gc 0 ) , H ( Gc 0 0 )] [ αH ] c 0 ,c 0 0 1 k s H αc 0 ◦− w w o o o o o o o o o o o o o o o o o o o o o [ F c 0 , Gc 0 0 ] H 1 w w o o o o o o o o o o o o o o o o o o o o o H 1   [ H ( F c 0 ) , H ( Gc 0 0 )] −◦ β Gc 0 0 ' ' O O O O O O O O O O O O O O O O O O O O O [ H ( F c 0 ) , H ( Gc 0 0 )] −◦ β Gc 0 0   [ K ( F c 0 ) , K ( Gc 0 0 )] β F c 0 ,Gc 0 0 1 k s β F c 0 ◦− w w o o o o o o o o o o o o o o o o o o o o o [ H ( F c 0 ) , K ( Gc 0 0 )] No w, by functoriality w.r.t 0-comp osition, with constant left comp osite one has ( αc 0 ◦ − ) • 0 β F c 0 ,Gc 0 0 1 =  K Gc 0 ,Gc 0 0 1 • 0 ( β 1 ( αc 0 ) ◦ − )  • 1  β Gc 0 ,Gc 0 0 1 • 0 ( H αc 0 ◦ − )  5.7 Dimension r aising 0-c omp osition of 2-morphisms 148 and b y deﬁnition of ∗ -comp osition on ob jects, =  K Gc 0 ,Gc 0 0 1 • 0 ([ α ∗ β ] c 0 ◦ − )  • 1  β Gc 0 ,Gc 0 0 1 • 0 ( H αc 0 ◦ − )  Hence w e can redraw the domain [ c 0 , c 0 0 ] [ F H ] 1 w w o o o o o o o o o o o o o o o o o o o o o [ GH ] 1   [ GK ] 1 ' ' O O O O O O O O O O O O O O O O O O O O O [ H ( F c 0 ) , H ( F c 0 0 )] −◦ H αc 0 0   [ H ( Gc 0 ) , H ( Gc 0 0 )] [ αH ] c 0 ,c 0 0 1 k s −◦ β Gc 0 0   H αc 0 ◦− w w o o o o o o o o o o o o o o o o o o o o o [ K ( Gc 0 ) , K ( Gc 0 0 )] [ Gβ ] c 0 ,c 0 0 1 k s β Gc 0 ◦− o o o o o o o o o w w o o o o o o o o o K αc 0 ◦−   [ H ( F c 0 ) , H ( Gc 0 0 )] −◦ β Gc 0 0 ' ' O O O O O O O O O O O O O O O O O O O O O [ H ( Gc 0 ) , K ( Gc 0 0 )] −◦ β Gc 0 0   [ K ( F c 0 ) , K ( Gc 0 0 )] [ α ∗ β ] c 0 ◦− k s β F c 0 ◦− w w o o o o o o o o o o o o o o o o o o o o o [ H ( F c 0 ) , K ( Gc 0 0 )] And this completes the domain-part. Concerning the co domain, the calcula- tion is similar, but on the left side of diagrams. T urning to functorialit y axioms, we start with functoriality w.r.t. units. T o this end, let us supp ose an ob ject c 0 of C b een giv en. Then u ( c 0 ) • 0 [ α ∗ β ] c 0 ,c 0 1 = u ( c 0 ) • 0 ( α c 0 ,c 0 ∗ β F c 0 ,Gc 0 1 ) ( i ) = ( u ( c 0 ) • 0 α c 0 ,c 0 ) ∗ β F c 0 ,Gc 0 1 ( ii ) = id [ α ( c 0 )] ∗ β F c 0 ,Gc 0 1 ( iii ) = id [ α ( c 0 )] • 0 β F c 0 ,Gc 0 1 ( iv ) = id [ β F c 0 ,Gc 0 1 ]( α ( c 0 )) = id [ α ∗ β ] 0 ( c 0 ) where ﬁrst and last equation are deﬁnitions, ( i ) holds by ∗ -asso ciativit y , ( ii ) b y units axioms for α , ( iii ) b y ∗ -iden tity prop ert y , ( iv ) is simply the application of a 2-morphism to a costan t 1-morphism. T o prov e comp osition axiom, we start by ﬁxing arbitrary ob jects c 0 , c 0 0 , c 00 0 of C . Then it is easier to start from the result and bac k-trac k the c hain of equalities as sho wn b elow. By ∗ -asso ciativit y one has ( − ◦ − ) • 0 ( α c 0 ,c 00 0 1 ∗ β F c 0 ,Gc 00 0 1 ) = (( − ◦ − ) • 0 α c 0 ,c 00 0 1 ) ∗ β F c 0 ,Gc 00 0 1 5.7 Dimension r aising 0-c omp osition of 2-morphisms 149 applying comp osition coherence of α h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 1  F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1 i ∗ β F c 0 ,Gc 00 0 1 b y ∗ -functoriality h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  ∗ β F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  • 0  H F c 0 ,Gc 00 0 1 ◦ β Gc 00 0 i • 2 h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  ∗ β F c 0 ,Gc 00 0 1 i that is h  α c 0 ,c 0 0 1 × G c 0 0 ,c 00 0 1  • 0 ( − ◦ − )  ∗ β F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  • 0  H F c 0 ,Gc 00 0 1 ◦ β Gc 00 0 i • 2 h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1 i • 1 h  F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  • 0 ( − ◦ − )  ∗ β F c 0 ,Gc 00 0 1 i Then b y ∗ -asso ciativity we obtain h α c 0 ,c 0 0 1 × G c 0 0 ,c 00 0 1  ∗  ( − ◦ − ) • 0 β F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  • 0  H F c 0 ,Gc 00 0 1 ◦ β Gc 00 0 i • 2 h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  ∗  ( − ◦ − ) • 0 β F c 0 ,Gc 00 0 1 i applying no w comp osition axiom of β this turns in h α c 0 ,c 0 0 1 × G c 0 0 ,c 00 0 1  ∗  ( β F c 0 ,Gc 0 0 1 ◦ K Gc 0 0 ,Gc 0 0 1 ) • 1 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ,Gc 00 0 1 ) i • 1 h F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  • 0  H F c 0 ,Gc 00 0 1 ◦ β Gc 00 0 i • 2 h α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1 i • 1 h F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  ∗  ( β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1 ) • 1 ( H F c 0 ,F c 0 0 1 ◦ β F c 0 0 ,Gc 00 0 1 ) i i.e.      α c 0 ,c 0 0 1 × G c 0 0 ,c 00 0 1  ∗  ( β F c 0 ,Gc 0 0 1 ◦ K Gc 0 0 ,Gc 0 0 1 ) • 1 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ,Gc 00 0 1 )  • 1  F c 0 ,c 0 0 1 ◦ α c 0 0 ,c 00 0 1  • 0  H F c 0 ,Gc 00 0 1 ◦ β Gc 00 0      • 2 (5.6)      α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1  • 1  F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  ∗  ( β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1 ) • 1 ( H F c 0 ,F c 0 0 1 ◦ β F c 0 0 ,Gc 00 0 1 )      5.7 Dimension r aising 0-c omp osition of 2-morphisms 150 Let us fo cus our atten tion on the second comp osite (w.r.t. • 2 -comp osition). In fact the calculations on the ﬁrst comp onen t are precisely symmetrical. Applying ∗ -functorialit y to this we get  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1  • 1       F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  ∗  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1   • 1   F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1 ◦ α c 00 0  • 0  H F c 0 ,F c 0 0 1 ◦ β F c 0 0 ,Gc 00 0 1   • 2   F c 0 ,c 0 0 1 × α c 0 0 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1   • 1   F c 0 ,c 0 0 1 × α c 0 0 ,c 00 0 1  ∗  H F c 0 ,F c 0 0 1 ◦ β F c 0 0 ,Gc 00 0 1       By comp osing on pro duct comp onen ts w e notice that upp er ∗ -comp osition giv es indeed an identit y  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1  • 1      ( F c 0 ,c 0 0 1 • 0 β F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 • 0 K F c 0 0 ,Gc 00 0 1 )  • 1   F c 0 ,c 0 0 1 × F c 0 0 ,c 00 0 1 ◦ α c 00 0  • 0  H F c 0 ,F c 0 0 1 ◦ β F c 0 0 ,Gc 00 0 1   • 2   F c 0 ,c 0 0 1 × α c 0 0 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1   • 1  ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 )      hence all the middle ro w is an iden tity 2-morphism, and the whole simpliﬁes to the follo wing  α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1  • 1   F c 0 ,c 0 0 1 × α c 0 0 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1   • 1  ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 )  b y • 1 -whisk ering asso ciativity this can b e rearranged   α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1   • 1   F c 0 ,c 0 0 1 × α c 0 0 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 00 0 1   • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) By functorialit y of K 5.7 Dimension r aising 0-c omp osition of 2-morphisms 151   α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1   • 1   F c 0 ,c 0 0 1 × G c 0 0 ,c 00 0 1  • 0  β F c 0 ,F c 0 0 1 ◦ K F c 0 0 ,Gc 0 0 1 ( α c 0 0 ) ◦ K Gc 0 0 ,Gc 00 0 1   • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) b y functoriality of − ◦ −   α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1  • 0  β F c 0 ◦ K F c 0 ,Gc 00 0 1   • 1  ( F c 0 ,c 0 0 1 • 0 β F c 0 ,F c 0 0 1 ) ◦ K F c 0 0 ,Gc 0 0 1 ( α c 0 0 ) ◦ ( G c 0 0 ,c 00 0 1 • 0 K Gc 0 0 ,Gc 00 0 1 )  • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) that is  β F c 0 ◦  ( α c 0 ,c 0 0 1 ◦ G c 0 0 ,c 00 0 1 ) • 0 K F c 0 ,Gc 00 0 1   • 1  ( F c 0 ,c 0 0 1 • 0 β F c 0 ,F c 0 0 1 ) ◦ K F c 0 0 ,Gc 0 0 1 ( α c 0 0 ) ◦ ( G c 0 0 ,c 00 0 1 • 0 K Gc 0 0 ,Gc 00 0 1 )  • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) again b y functoriality of K w e can write the result as β F c 0 ◦  α c 0 ,c 0 0 1 • 0 K F c 0 ,Gc 0 0 1  ◦  G c 0 0 ,c 00 0 1 • 0 K Gc 0 0 ,Gc 00 0 1  • 1 ( F c 0 ,c 0 0 1 • 0 β F c 0 ,F c 0 0 1 ) ◦ K F c 0 0 ,Gc 0 0 1 ( α c 0 0 ) ◦ ( G c 0 0 ,c 00 0 1 • 0 K Gc 0 0 ,Gc 00 0 1 ) • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) or more simply  β F c 0 ◦  α c 0 ,c 0 0 1 • 0 K F c 0 ,Gc 0 0 1   • 1  ( F c 0 ,c 0 0 1 • 0 β F c 0 ,F c 0 0 1 ) ◦ K F c 0 0 ,Gc 0 0 1 ( α c 0 0 )  ◦ ( G c 0 0 ,c 00 0 1 • 0 K Gc 0 0 ,Gc 00 0 1 ) • 1 ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ ( α c 0 0 ,c 00 0 1 ∗ β F c 0 0 ,Gc 00 0 1 ) i.e.  ( F • 0 β ) • 1 ( α • 0 K )  c 0 ,c 0 0 1 ◦  G • 0 K  c 0 0 ,c 00 0 1 • 1  F • 0 H  c 0 ,c 0 0 1 ◦  α ∗ β  c 0 0 ,c 00 0 1 5.7 Dimension r aising 0-c omp osition of 2-morphisms 152 Carrying on the analogous calculations on the ﬁrst comp onent, (5.6) equals to   α ∗ β  c 0 ,c 0 0 1 ◦  G • 0 K  c 0 0 ,c 00 0 1  • 1   F • 0 H  c 0 ,c 0 0 1 ◦  ( F • 0 β ) • 1 ( α • 0 K )  c 0 0 ,c 00 0 1  • 2   ( F • 0 β ) • 1 ( α • 0 K )  c 0 ,c 0 0 1 ◦  G • 0 K  c 0 0 ,c 00 0 1  • 1   F • 0 H  c 0 ,c 0 0 1 ◦  α ∗ β  c 0 0 ,c 00 0 1  that is   α ∗ β  c 0 ,c 0 0 1 ◦  G • 0 K  c 0 0 ,c 00 0 1  • 1   F • 0 H  c 0 ,c 0 0 1 ◦  dom ( α ∗ β )  c 0 0 ,c 00 0 1  • 2   co d ( α ∗ β )  c 0 ,c 0 0 1 ◦  G • 0 K  c 0 0 ,c 00 0 1  • 1   F • 0 H  c 0 ,c 0 0 1 ◦  α ∗ β  c 0 0 ,c 00 0 1  and this conclude the pro of. R emark 5.10 . W e hav e adopted the ∗ -sym b ol instead of the more ob vious • 0 in order to emphasize the dimension-raising prop erty of this comp osition. Nev ertheless ∗ -prop erties w.r.t. other • 0 -comp ositions are someho w b etter understo o d thinking only in terms of • 0 . Lemma 5.11. Given the c ase C F   G E E α   D H   K E E β   E If α is a lax natur al n -tr ansformation and β is a strict natur al n -tr ansformation, the c omp osition α ∗ β is an identity. In this c ase it is p ossible to de al with dimension pr eserving 0-c omp osition of 2-morphisms, by letting α ˜ ∗ β = dom ( α ∗ β ) = co d ( α ∗ β ) Pr o of. Let us supp ose α is a lax natural n -transformation and β is a strict natural n -transformation. Then for an ob ject c 0 of C , β 1 ( α c 0 ) is the commu- tativ e square β F c 0 ◦ K ( α c 0 ) = H ( α c 0 ) ◦ β Gc 0 . Moreo ver once ob jects c 0 , c 0 0 of C are ﬁxed, since β is strict β F c 0 ,Gc 0 0 1 = id H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 = id α F c 0 ◦ K F c 0 ,Gc 0 0 1 5.7 Dimension r aising 0-c omp osition of 2-morphisms 153 then [ α ∗ β ] c 0 ,c 0 0 1 = α c 0 ,c 0 0 1 ∗ β F c 0 ,Gc 0 0 1 = α c 0 ,c 0 0 1 ∗ id H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 = id α c 0 ,c 0 0 1 • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ) No w the whiskering α c 0 ,c 0 0 1 • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ) has domain dom  α c 0 ,c 0 0 1 • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 )  = ( F c 0 ,c 0 0 1 ◦ α c 0 0 ) • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ) = ( F c 0 ,c 0 0 1 • 0 H F c 0 ,F c 0 0 1 ) ◦ H ( α c 0 0 ) ◦ β Gc 0 0 = [ F • 0 H ] c 0 ,c 0 0 1 ◦ [ α ˜ ∗ β ] c 0 0 Similarly the co domain is co d  α c 0 ,c 0 0 1 • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 )  = ( α c 0 ◦ G c 0 ,c 0 0 1 ) • 0 ( H F c 0 ,Gc 0 0 1 ◦ β Gc 0 0 ) = ( α c 0 ◦ G c 0 ,c 0 0 1 ) • 0 ( β F c 0 ◦ K F c 0 ,Gc 0 0 1 ) = β F c 0 ◦ K ( α c 0 ) ◦ ( G c 0 ,c 0 0 1 • 0 K F c 0 ,Gc 0 0 1 ) = [ α ˜ ∗ β ] c 0 ◦ [ G • 0 K ] c 0 ,c 0 0 1 hence the result is (an iden tity ov er) a 2-morphism. Imp ortance of L emma ab ov e is in that it allows to right-0-compose freely with constan t transformations, such as − ◦ c 2 or c 2 ◦ − for a 2-cell c 2 : c 1 ⇒ c 0 1 : c 0 → c 0 0 . Notice that L emma do es not hold for α strict and β lax, since in this case the result is a strict 3-morphism. 5.7.1 Prop erties The follo wing prop ositions conclude the description of dimension-rising com- p osition in the sesqui 2 category of strict n -categories. Giv en the situation B E / / C F   G E E α   D H   K E E β   E L / / F one has the follo wing 5.7 Dimension r aising 0-c omp osition of 2-morphisms 154 Prop osition 5.12 ( ∗ -asso ciativit y 1) . ( L ∗ A ) ( E • 0 L α ) ∗ β = E • 0 L ( α ∗ β ) ( R ∗ A ) α ∗ ( β • 0 R L ) = ( α ∗ β ) • 0 R L Pr o of. W e prov e ( L ∗ A ). The pro of of ( R ∗ A ) is similar hence it is omitted. ( on obje cts ) Let an ob ject b 0 of C b e giv en. Then [( E • 0 α ) ∗ β ] b 0 = β ([ E • 0 α ] b 0 ) = β ( α E b 0 ) = [ α ∗ β ] E b 0 = [ E • 0 ( α ∗ β )] b 0 ( on homs ) Let ob jects b 0 , b 0 0 of B b e giv en. Then [( E • 0 α ) ∗ β ] b 0 ,b 0 0 1 = [ E • 0 α ] b 0 ,b 0 0 1 ∗ β F ( E b 0 ) ,G ( E b 0 0 ) 1 = ( E b 0 ,b 0 0 1 • 0 α E b 0 ,E b 0 0 1 ) ∗ β F ( E b 0 ) ,G ( E b 0 0 ) 1 ( ♠ ) = E b 0 ,b 0 0 1 • 0 ( α E b 0 ,E b 0 0 1 ∗ β F ( E b 0 ) ,G ( E b 0 0 ) 1 ) = E b 0 ,b 0 0 1 • 0 [ α 1 ∗ β ] E b 0 ,E b 0 0 = [ E • 0 ( α ∗ β )] b 0 ,b 0 0 1 where ( ♠ ) holds b y ∗ -asso ciativity in dimension n − 1. Prop osition 5.13 ( ∗ -iden tity) . ( L ) id E ∗ α = id E • 0 L α ( R ) α ∗ id H = id α • 0 R H Pr o of. W e prov e ( L ). The pro of of ( R ) is similar hence it is omitted. ( on obje cts ) Let an ob ject b 0 of B b e giv en. Then the follo wing equalities are straigh tforward [ id E ∗ α ] b 0 = α ([ id E ] b 0 ) = α ( id E b 0 ) = id α ( E b 0 ) = [ id E • 0 L α ] b 0 ( on homs ) Let ob jects b 0 , b 0 0 of B b e giv en. Then [ id E ∗ α ] b 0 ,b 0 0 1 = [ id E ] b 0 ,b 0 0 1 ∗ α E b 0 ,E b 0 0 1 = id E b 0 ,b 0 0 1 ∗ α E b 0 ,E b 0 0 1 ( ♠ ) = id E b 0 ,b 0 0 1 • 0 α E b 0 ,E b 0 0 1 = [ id E • 0 L α ] b 0 ,b 0 0 1 where ( ♠ ) is ∗ -iden tity in dimension n − 1. 5.7 Dimension r aising 0-c omp osition of 2-morphisms 155 In the situation C F   G E E α   D M / / D 0 H   K E E β   E one has the follo wing Prop osition 5.14 ( ∗ -asso ciativit y 2) . α ∗ ( M • 0 L β ) = ( α • 0 R M ) ∗ β Pr o of. ( on obje cts ) Let an ob ject c 0 of C b e given. Then the following equalities are straigh tforward [ α ∗ ( M • 0 β )] c 0 = [ M • 0 β ]( α c 0 ) = β  M ( α c 0 )  = β  [ α • 0 M ] c 0  = [( α • 0 M ) ∗ β ] c 0 ( on homs ) Let ob jects c 0 , c 0 0 of C b e giv en. Then  α ∗ ( M • 0 β )  c 0 ,c 0 0 1 = α c 0 ,c 0 0 1 ∗ [ M • 0 β ] F c 0 ,Gc 0 0 1 = α c 0 ,c 0 0 1 ∗  M F c 0 ,Gc 0 0 1 • 0 β M ( F c 0 ) ,M ( Gc 0 0 ) 1  ( ♠ ) =  α c 0 ,c 0 0 1 • 0 M F c 0 ,Gc 0 0 1  ∗ β M ( F c 0 ) ,M ( Gc 0 0 ) 1 = [ α • 0 M ] c 0 ,c 0 0 1 ∗ β M ( F c 0 ) ,M ( Gc 0 0 ) 1 =  ( α • 0 M ) ∗ β  c 0 ,c 0 0 1 where ( ♠ ) is ∗ -asso ciativit y in dimension n − 1. In the situation b elo w B D   E E E ω   C F   G / / H H H α   β   D K   L E E γ   E one has the follo wing 5.7 Dimension r aising 0-c omp osition of 2-morphisms 156 Prop osition 5.15 ( ∗ -functorialit y) . ( a ) ( α • 1 β ) ∗ γ =  ( α ∗ γ ) • 1 ( β • 0 L )  • 2  ( α • 0 K ) • 1 ( β ∗ γ )  ( b ) ω ∗ ( α • 1 β ) =  ( ω ∗ α ) • 1 ( E • 0 β )  • 2  ( D • 0 α ) • 1 ( ω ∗ β )  Pr o of. W e prov e ( a ). The pro of of ( b ) is similar, hence it is omitted. ( on obje cts ) Let an ob ject c 0 of C b e giv en. Then   ( α ∗ γ ) • 1 ( β • 0 L ) • 2 ( α • 0 K ) • 1 ( β ∗ γ )   c 0 =  ( α ∗ γ ) • 1 ( β • 0 L )  c 0 ◦ 1  ( α • 0 K ) • 1 ( β ∗ γ )  c 0 ( ♠ ) =   γ ( α c 0 ) • 0 L ( β c 0 ) ◦ 1 K ( α c 0 ) • 0 γ ( β c 0 )   = γ ( α c 0 ◦ β c 0 ) = [( α • 1 β ) ∗ γ ] c 0 where all the equalities are just deﬁnitions, but ( ♠ ) that is given b y functori- alit y w.r.t. 0-comp osition of γ . ( on homs ) Let ob jects c 0 , c 0 0 of C b e given. Applying the deﬁnition of 2- comp osition of 3-morphisms   ( α ∗ γ ) • 1 ( β • 0 L ) • 2 ( α • 0 K ) • 1 ( β ∗ γ )   c 0 ,c 0 0 1 = = h ( α • 0 K ) • 1 ( β ∗ γ ) i c 0 ,c 0 0 1 • 1  [ α • 0 K ] c 0 ,c 0 0 1 ◦ co d ( β ∗ γ )  • 2  dom ( α ∗ γ ) ◦ [ β • 0 L ] c 0 ,c 0 0 1  • 1 h ( α ∗ γ ) • 1 ( β • 0 L ) i c 0 ,c 0 0 1 b y deﬁnition of whiskering of 3-morphisms and 2-morphisms  [ α • 0 K ] c 0 ◦ [ β ∗ γ ] c 0 ,c 0 0 1  • 1  [ α • 0 K ] c 0 ,c 0 0 1 ◦ co d ( β ∗ γ )  • 2  dom ( α ∗ γ ) ◦ [ β • 0 L ] c 0 ,c 0 0 1  • 1  [ α ∗ γ ] c 0 ,c 0 0 1 ◦ [ β • 0 L ] c 0 0  that is  [ α • 0 K ] c 0 ◦ [ β ∗ γ ] c 0 ,c 0 0 1  • 1  [ α • 0 K ] c 0 ,c 0 0 1  ◦ K ( β c 0 0 ) ◦ γ H c 0 0  • 2  γ F c 0 ◦ L ( α c 0 ) ◦ [ β • 0 L ] c 0 ,c 0 0 1  • 1  [ α ∗ γ ] c 0 ,c 0 0 1 ◦ [ β • 0 L ] c 0 0  5.7 Dimension r aising 0-c omp osition of 2-morphisms 157 b y deﬁnition of ∗ -comp osition on homs (and of 0-whiskering for 2-morphisms)  K ( α c 0 ) ◦  β c 0 ,c 0 0 1 ∗ γ Gc 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 • 0 K F c 0 ,Gc 0 0 1  ◦ K ( β c 0 0 ) ◦ γ H c 0 0  • 2  γ F c 0 ◦ L ( α c 0 ) ◦  β c 0 ,c 0 0 1 • 0 L Gc 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ∗ γ F c 0 ,Gc 0 0 1  ◦ L ( β c 0 0 )  this can b e rearranged  K ( α c 0 ) ◦  β c 0 ,c 0 0 1 ∗ γ Gc 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 • 0  K F c 0 ,Gc 0 0 1 ◦ K ( β c 0 0 ) ◦ γ H c 0 0  • 2  β c 0 ,c 0 0 1 • 0  γ F c 0 ◦ L ( α c 0 ) ◦ L Gc 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ∗ γ F c 0 ,Gc 0 0 1  ◦ L ( β c 0 0 )  b y functoriality of L and K  K ( α c 0 ) ◦  β c 0 ,c 0 0 1 ∗ γ Gc 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ◦ β c 0 0  • 0  K F c 0 ,H c 0 0 1 ◦ γ H c 0 0  • 2  α c 0 ◦ β c 0 ,c 0 0 1  • 0  γ F c 0 ◦ L F c 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ∗ γ F c 0 ,Gc 0 0 1  ◦ L ( β c 0 0 )  b y functoriality (and ∗ -asso ciativity)  α c 0 ◦ β c 0 ,c 0 0 1  ∗ γ F c 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ◦ β c 0 0  • 0  K F c 0 ,H c 0 0 1 ◦ γ H c 0 0  • 2  α c 0 ◦ β c 0 ,c 0 0 1  • 0  γ F c 0 ◦ L F c 0 ,H c 0 0 1  • 1  α c 0 ,c 0 0 1 ◦ β c 0 0  ∗ γ F c 0 ,H c 0 0 1  and ﬁnally b y ∗ -functoriality  ( α c 0 ◦ β c 0 ,c 0 0 1 ) • 1 ( α c 0 ,c 0 0 1 ◦ β c 0 0 )  ∗ γ F c 0 ,H c 0 0 1 that is the result: h ( α • 1 β ) ∗ γ i c 0 ,c 0 0 1 . Chapter 6 h -Pullbac ks revisited and the long exact sequence 6.1 2-dimensional h -pullbac ks in n Cat W e in tro duce here a notion of 2-dimensional h-pullbac k in the sesqui 2 -category n Cat . It will b e shown that our construction of the standar d h -pullbac k of n -categories is an instance of suc h a 2-dimensional one. In order to ﬁx notation, let us consider the follo wing diagram in n Cat C G   A F / / B A h -2pullbac k of F and G is a four-tuple ( P , P , Q, ε ) P Q / / P   C G   A F / / B ε ; C         that satisﬁes the follo wing 2-dimensional universal prop erty: Univ ersal Prop ert y 6.1 ( h -2pullbacks) . F or any other two four-tuple 6.1 2-dimensional h -pul lb acks in n Cat 159 ( X , M , N , ω ) X N / / M   C G   A F / / B ω ; C         and ( X , ˆ M , ˆ N , ˆ ω ) X ˆ N / / ˆ M   C G   A F / / B ˆ ω ; C         2 − morphism α, β X M   ˆ M n n α   2 2 2 2 2 2 2 2 N   ˆ N 0 0 β           A C and 3 − morphism Σ M • 0 F α • 0 F + 3 ω   ˆ M • 0 F ˆ ω   N • 0 G Σ l 0 < l l l l l l l l l l l l l l l l l l l l l l l l β • 0 G + 3 ˆ N • 0 G ther e exists a unique λ : L ⇒ ˆ L : X → P such that (UP) 1. λ • 0 P = α 2. λ • 0 Q = β 3. λ ∗ ε = Σ As an immediate consequence of the deﬁnition, w e state the following Prop osition 6.2. 2 - Univ ersal Prop ert y of h -2pul lb acks implies 1 -dimensional one. Henc e h -2pul lb acks ar e deﬁne d up to isomorphism. Pr o of. Just put α , β and Σ identities. Let us notice that Pr op osition 6.2 holds in ev ery sesqui 2 -category . More in terestingly in n Cat a kind of con verse to this prop osition also holds. Prop osition 6.3. Given the diagr am C G   A F / / B in n Cat , the standar d h -pul lb ack satisﬁes also Universal prop ert y 6.1 . 6.1 2-dimensional h -pul lb acks in n Cat 160 Pr o of. Firstly we remark that 1-dimensional Universal Pr op erty 2.12 of h -pullbac ks applied to the four-tuple ( X , M , N , ω ) yields an L : X → P , while applied to ( X , ˆ M , ˆ N , ˆ ω ), a ˆ L : X → P . Those ha ve to b e domain and c o-domain of the 2-cell pro vided by the universal property , namely λ : L ⇒ ˆ L . W e recall the constructions in order to ﬁx notation. • L 0 : X 0 → P 0 is the map x 0 7→ ( M x 0 , F ( M x 0 ) ω ( x 0 ) / / G ( N x 0 ) , N x 0 ) =: p 0 • for ev ery pair of ob jects x 0 , x 0 0 of X , L x 0 ,x 0 0 1 : X 1 ( x 0 , x 0 0 ) → P 1 ( L ( x 0 ) , L ( x 0 0 )) is giv en by the universal prop erty in dimension n − 1, and is suc h that L x 0 ,x 0 0 1 • 0 P Lx 0 ,Lx 0 0 1 = M x 0 ,x 0 0 1 L x 0 ,x 0 0 1 • 0 Q Lx 0 ,Lx 0 0 1 = N x 0 ,x 0 0 1 L x 0 ,x 0 0 1 • 0 ε Lx 0 ,Lx 0 0 1 = ω x 0 ,x 0 0 1 The pair L = < L 0 , L − , − 1 > is a 1-morphism. Similarly one determines ˆ L = < ˆ L 0 , ˆ L − , − 1 > . No w w e show that remaining data (namely , α , β and Σ) of the hypothesis pro vide a 2-morphism λ : L ⇒ ˆ L that satisﬁes required prop ert y . T o this end, let us consider the follo wing assignments: • F or ev ery ob ject x 0 of X , λ x 0 = ( α x 0 , Σ x 0 , β x 0 ) : Lx 0 → ˆ Lx 0 i.e. M x 0 α x 0 / / ˆ M x 0 F ( M x 0 ) F ( α x 0 ) / / ω x 0   F ( ˆ M x 0 ) ˆ ω x 0   G ( N x 0 ) Σ x 0 2 : n n n n n n n n n n n n n n n n G ( β x 0 ) / / G ( ˆ N ) x 0 N x 0 β x 0 / / ˆ N x 0 6.1 2-dimensional h -pul lb acks in n Cat 161 • F or ev ery pair of ob jects x 0 , x 0 0 of X , X 1 ( x 0 , x 0 0 ) L x 0 ,x 0 0 1 { { v v v v v v v v v ˆ L x 0 ,x 0 0 1 # # H H H H H H H H H P 1 ( Lx 0 , Lx 0 0 ) −◦ λx 0 0 # # G G G G G G G G G P 1 ( ˆ Lx 0 , ˆ Lx 0 0 ) λx 0 ◦− { { w w w w w w w w w λ x 0 ,x 0 0 1 k s P 1 ( Lx 0 , ˆ Lx 0 0 ) is giv en by the universal prop erty for (n-1)categories. In fact the 0-co domain of λ x 0 ,x 0 0 1 , namely P 1 ( Lx 0 , ˆ Lx 0 0 ) is deﬁned inductiv ely as a h -2pullbac k in (n-1) Cat : P 1 ( Lx 0 , ˆ Lx 0 0 ) Q Lx 0 , ˆ Lx 0 0 1 / / P Lx 0 , ˆ Lx 0 0 1   C 1 ( N x 0 , ˆ N x 0 0 ) G N x 0 , ˆ N x 0 0 1   ε Lx 0 , ˆ Lx 0 0 1 p x i i i i i i i i i i i i i i i i i i i i i i i i i i B 1 ( G ( N x 0 ) , G ( ˆ N x 0 0 )) ω x 0 ◦−   A 1 ( M x 0 , ˆ M x 0 0 ) F M x 0 , ˆ M x 0 0 1 / / B 1 ( F ( M x 0 ) , F ( ˆ M x 0 0 )) −◦ ˆ ω x 0 0 / / B 1 ( F ( M x 0 ) , G ( ˆ N x 0 0 )) Ov er the same base are also deﬁned X 1 ( x 0 , x 0 0 ) N x 0 ,x 0 0 1 ◦ β x 0 0 / / M x 0 ,x 0 0 1 ◦ αx 0 0   C 1 ( N x 0 , ˆ N x 0 0 ) G N x 0 , ˆ N x 0 0 1   θ = ( ω x 0 ,x 0 0 1 ◦ G ( β x 0 0 )) • 1 ([ M F ] x 0 ,x 0 0 1 ◦ Σ x 0 0 ) i i i i i i i i i i i i i i i i i i p x i i i i i i i i i i i i i i i i B 1 ( G ( N x 0 ) , G ( ˆ N x 0 0 )) ω x 0 ◦−   A 1 ( M x 0 , ˆ M x 0 0 ) F M x 0 , ˆ M x 0 0 1 / / B 1 ( F ( M x 0 ) , F ( ˆ M x 0 0 )) −◦ ˆ ω x 0 0 / / B 1 ( F ( M x 0 ) , G ( ˆ N x 0 0 )) 6.1 2-dimensional h -pul lb acks in n Cat 162 and X 1 ( x 0 , x 0 0 ) β x 0 ◦ ˆ N x 0 ,x 0 0 1 / / αx 0 ◦ ˆ M x 0 ,x 0 0 1   C 1 ( N x 0 , ˆ N x 0 0 ) G N x 0 , ˆ N x 0 0 1   ˆ θ = (Σ x 0 ◦ [ ˆ N G ] x 0 ,x 0 0 1 ) • 1 ( F ( αx 0 ) ◦ ˆ ω x 0 ,x 0 0 1 ) i i i i i i i i i i i i i i i i i i p x i i i i i i i i i i i i i i i i B 1 ( G ( N x 0 ) , G ( ˆ N x 0 0 )) ω x 0 ◦−   A 1 ( M x 0 , ˆ M x 0 0 ) F M x 0 , ˆ M x 0 0 1 / / B 1 ( F ( M x 0 ) , F ( ˆ M x 0 0 )) −◦ ˆ ω x 0 0 / / B 1 ( F ( M x 0 ) , G ( ˆ N x 0 0 )) Moreo ver we can consider 2-morphisms: α x 0 ,x 0 0 1 : αx 0 ◦ ˆ M x 0 ,x 0 0 1 ⇒ M x 0 ,x 0 0 1 ◦ αx 0 0 : X 1 ( x 0 , x 0 0 ) → A 1 ( M x 0 , ˆ M x 0 0 ) β x 0 ,x 0 0 1 : β x 0 ◦ ˆ N x 0 ,x 0 0 1 ⇒ N x 0 ,x 0 0 1 ◦ β x 0 0 : X 1 ( x 0 , x 0 0 ) → A 1 ( M x 0 , ˆ M x 0 0 ) and the 3-morphism ( β x 0 ◦ ˆ N x 0 ,x 0 0 1 ) • 0 ( ω x 0 ◦ G N x 0 , ˆ N x 0 0 1 ) β x 0 ,x 0 0 1 • 0 id + 3 ˆ θ   ( N x 0 ,x 0 0 1 ◦ β x 0 0 ) • 0 ( ω x 0 ◦ G N x 0 , ˆ N x 0 0 1 ) θ   Σ x 0 ,x 0 0 1 k k k k k k k k k k k k k k k k k k k k k k k k k p { k k k k k k k k k k k k k k k k k k k k k ( αx 0 ◦ ˆ M x 0 ,x 0 0 1 ) • 0 ( F M x 0 , ˆ M x 0 0 1 ◦ ˆ ωx 0 0 ) β x 0 ,x 0 0 1 • 0 id + 3 ( M x 0 ,x 0 0 1 ◦ αx 0 0 ) • 0 ( F M x 0 , ˆ M x 0 0 1 ◦ ˆ ωx 0 0 ) Finally w e can apply the universal property , in order to get a unique 2- morphism λ x 0 ,x 0 0 1 : L x 0 ,x 0 0 1 ◦ λx 0 0 ⇒ λx 0 ◦ ˆ L x 0 ,x 0 0 1 suc h that λ x 0 ,x 0 0 1 • 0 Q Lx 0 , ˆ Lx 0 0 1 = β x 0 ,x 0 0 1 (6.1) λ x 0 ,x 0 0 1 • 0 P Lx 0 , ˆ Lx 0 0 1 = α x 0 ,x 0 0 1 (6.2) λ x 0 ,x 0 0 1 ∗ ε Lx 0 , ˆ Lx 0 0 1 = Σ x 0 ,x 0 0 1 (6.3) That the pair λ = < λ 0 , λ − , − 1 > is a 2-morphism of n-categories is pro ved in the follo wing (quite technical) L emma 6.4 . 6.1 2-dimensional h -pul lb acks in n Cat 163 Moreo ver it satisﬁes b y construction Universal Pr op erty 6.1 . In fact for any ob ject x 0 of X [ λ • 0 P ] x 0 = P ( λ x 0 ) = P  ( α x 0 , Σ x 0 , β x 0 )  = α x 0 and for an y pair of x 0 , x 0 0 [ λ • 0 P ] x 0 ,x 0 0 1 = λ x 0 ,x 0 0 1 • 0 P Lx 0 , ˆ Lx 0 0 1 = α x 0 ,x 0 0 1 th us λ • 0 P = α . Similarly [ λ • 0 Q ] x 0 = Q ( λ x 0 ) = Q  ( α x 0 , Σ x 0 , β x 0 )  = β x 0 and [ λ • 0 Q ] x 0 ,x 0 0 1 = λ x 0 ,x 0 0 1 • 0 Q Lx 0 , ˆ Lx 0 0 1 = β x 0 ,x 0 0 1 th us λ • 0 Q = β . Finally [ λ ∗ ε ] x 0 = ε ( λ x 0 ) = ε  ( α x 0 , Σ x 0 , β x 0 )  = Σ x 0 and [ λ ∗ ε ] x 0 ,x 0 0 1 = λ x 0 ,x 0 0 1 ∗ ε Lx 0 , ˆ Lx 0 0 1 = Σ x 0 ,x 0 0 1 T o conclude the pro of we still need to prov e uniqueness. But this will easily b e achiev ed. Indeed the ob ject part of 2-morphism λ satisfying the univ ersal prop ert y is univ o cally determined by the fact that P 0 , Q 0 and ε 0 are pro jection, and once that is determined, uniqueness in dimension n − 1 guaran ties the homs part. Lemma 6.4. The p air λ = < λ 0 , λ − , − 1 > is inde e d a 2-morphism. Pr o of. W e ha ve to sho w that functoriality axioms for 2-morphisms are satis- ﬁed. Let us start with units axiom. T o this end let us ﬁx an arbitrary ob ject x 0 of X . If w e denote by u ( x 0 ) the iden tity I ( n − 1) → X 1 ( x 0 , x 0 ) then u ( x 0 ) • 0 λ x 0 ,x 0 1 • 0 Q Lx 0 , ˆ Lx 0 1 ( i ) = u ( x 0 ) • 0 β x 0 ,x 0 1 ( ii ) = id [ β x 0 ] ( iii ) = id [ Q ( λx 0 )] ( iv ) = id [ λx 0 ] • 0 Q Lx 0 , ˆ Lx 0 1 where ( i ) holds b y (6.1) ab o ve, ( ii ) b y unit functoriality of β , ( iii ) b y deﬁnition of λ 0 , ( iv ) is a whiskering identit y axiom. Similarly one can prov e u ( x 0 ) • 0 λ x 0 ,x 0 1 • 0 P Lx 0 , ˆ Lx 0 1 = id [ λx 0 ] • 0 P Lx 0 , ˆ Lx 0 1 6.1 2-dimensional h -pul lb acks in n Cat 164 Moreo ver, for the formally analogous prop erty w.r.t ∗ -comp osition u ( x 0 ) • 0 λ x 0 ,x 0 1 ∗ ε Lx 0 , ˆ Lx 0 1 = u ( x 0 ) • 0 Σ x 0 ,x 0 1 = id [Σ x 0 ] = id [ ε ( λx 0 )] = id [ λx 0 ] ∗ ε Lx 0 , ˆ Lx 0 1 Notice that ﬁrst comp osites are unam biguous by asso ciativity axioms. Calculations show that both id [ λx 0 ] and u ( x 0 ) • 0 λ x 0 ,x 0 1 satisfy equations prescrib ed b y the univ ersal prop ert y , hence b y uniqueness they must b e equal, and unit axiom is pro ved. T urning to comp osition coherence, let ob jects x 0 , x 0 0 , x 00 0 b e giv en. Then  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  • 0 Q Lx 0 ˆ Lx 00 0 1 = ( i ) =  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 0 Q Lx 0 , ˆ Lx 00 0 1  • 1  ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 ) • 0 Q Lx 0 , ˆ Lx 00 0 1  ( ii ) =  ( λ x 0 ,x 0 0 1 • 0 Q Lx 0 , ˆ Lx 0 0 1 ) ◦ ( ˆ L x 0 0 ,x 00 0 1 • 0 Q ˆ Lx 0 0 , ˆ Lx 00 0 1 )  • 1 • 1  ( L x 0 ,x 0 0 1 • 0 Q Lx 0 ,Lx 0 0 1 ) ◦ ( λ x 0 0 ,x 00 0 1 • 0 Q Lx 0 0 ˆ Lx 00 0 1 )  ( iii ) = ( β x 0 ,x 0 0 1 ◦ ˆ N x 0 0 ,x 00 0 1 ) • 1 ( N x 0 ,x 0 0 1 ◦ β x 0 0 ,x 00 0 1 ) ( iv ) = ( − ◦ − ) • 0 β x 0 ,x 00 0 1 ( v ) = ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 • 0 Q x 0 ,x 00 0 1 where ( i ) holds by (sesqui)functoriality of − • 0 Q Lx 0 ˆ Lx 00 0 1 , ( ii ) b y functoriality w.r.t. ◦ -comp osition of Q , ( iii ) and ( v ) b y (6.1), ( iv ) b y functorialit y w.r.t. comp osition of β . Similarly one can pro ve  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  • 0 P Lx 0 ˆ Lx 00 0 1 = ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 • 0 P x 0 ,x 00 0 1 No w let us compute  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  ∗ ε Lx 0 , ˆ Lx 00 0 1 b y ∗ -functoriality this equals to 6.1 2-dimensional h -pul lb acks in n Cat 165  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) ∗ ε Lx 0 , ˆ Lx 00 0  | {z } • 1  ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 ) • 0 ([ P F ] Lx 0 , ˆ Lx 00 0 1 ◦ ˆ ωx 00 0 )  • 2  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 0 ( ω x 0 ◦ [ QG ] Lx 0 , ˆ Lx 00 0 1 )  • 1  ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 ) ∗ ε Lx 0 , ˆ Lx 00 0 1  (6.4) In order to simplify the expression, let us analyze ﬁrst under-braced one. This can b e re-written explicitly and pro cessed b y ∗ -asso ciativit y ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) ∗ ε Lx 0 , ˆ Lx 00 0 = =  ( λ x 0 ,x 0 0 1 × ˆ L x 0 0 ,x 00 0 1 ) • 0 ( − ◦ − )  ∗ ε Lx 0 , ˆ Lx 00 0 = ( λ x 0 ,x 0 0 1 × ˆ L x 0 0 ,x 00 0 1 ) ∗  ( − ◦ − ) • 0 ε Lx 0 , ˆ Lx 00 0  = ( λ x 0 ,x 0 0 1 × ˆ L x 0 0 ,x 00 0 1 ) ∗  ( ε Lx 0 , ˆ Lx 0 0 1 ◦ [ QG ] ˆ Lx 0 0 , ˆ Lx 00 0 1 ) • 1 ([ P F ] Lx 0 , ˆ Lx 0 0 1 ◦ ε ˆ Lx 0 0 , ˆ Lx 00 0 1 )  where the last is giv en by comp osition coherence of ε . Applying again ∗ -functorialit y , this turns to b e  ( λ x 0 ,x 0 0 1 × ˆ L x 0 0 ,x 00 0 1 ) ∗ ( ε Lx 0 , ˆ Lx 0 0 1 ◦ [ QG ] ˆ Lx 0 0 , ˆ Lx 00 0 1 )  • 1  (( L x 0 ,x 0 0 1 ◦ λx 0 0 ) × ˆ L x 0 0 ,x 00 0 1 ) • 0 ([ P F ] Lx 0 , ˆ Lx 0 0 1 ◦ ε ˆ Lx 0 0 , ˆ Lx 00 0 1 )  • 2  (( λx 0 ◦ ˆ L x 0 ,x 0 0 1 ) × ˆ L x 0 0 ,x 00 0 1 ) • 0 ( ε Lx 0 , ˆ Lx 0 0 1 ◦ [ QG ] ˆ Lx 0 0 , ˆ Lx 00 0 1 )  • 1  ( λ x 0 ,x 0 0 1 × ˆ L x 0 0 ,x 00 0 1 ) ∗ ([ P F ] Lx 0 , ˆ Lx 0 0 1 ◦ ε ˆ Lx 0 0 , ˆ Lx 00 0 1 )  No w, all the second row is clearly an identit y 3-morphism, b eing the ∗ - comp osition on separate comp onen ts of a pro duct, hence it can b e canceled. What remains can b e re-written as  ( λ x 0 ,x 0 0 1 ∗ ε Lx 0 , ˆ Lx 0 0 1 ) ◦ [ ˆ LQG ] x 0 0 ,x 00 0 1  • 1  [ LP F ] x 0 ,x 0 0 1 ◦ F ( P ( λx 0 0 )) ◦ ( ˆ L x 0 0 ,x 00 0 1 • 0 ε ˆ Lx 0 0 , ˆ Lx 00 0 1 )  and with the help of (6.3) this is simply (Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1 ) • 1 ([ M F ] x 0 ,x 0 0 1 ◦ F ( P ( λx 0 0 )) ◦ ˆ ω x 0 0 ,x 00 0 1 ) Substituting the ﬁrst line of (6.4) b ecomes a triple • 1 -comp osition  Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦ F ( P ( λx 0 0 )) ◦ ˆ ω x 0 0 ,x 00 0 1  • 1  ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 ) • 0 ([ P F ] Lx 0 , ˆ Lx 00 0 1 ◦ ˆ ω x 00 0 )  (6.5) No w F ( P ( λx 0 0 )) = F ( αx 0 0 ) = [ α • 0 F ] x 0 0 , furthermore P F ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 ) = P F ( L x 0 ,x 0 0 1 ) ◦ P F ( λ x 0 0 ,x 00 0 1 ) = [ M F ] x 0 ,x 0 0 1 ◦ [ α • 0 F ] x 0 0 ,x 00 0 1 . Hence (6.5) is equal to 6.2 Ω and a se c ond deﬁnition of π 1 166  Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦ [ α • 0 F ] x 0 0 ◦ ˆ ω x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦ [ α • 0 F ] x 0 0 ,x 00 0 1 ◦ ˆ ωx 00 0  i.e.  Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦   [ α • 0 F ] x 0 0 ◦ ˆ ω x 0 0 ,x 00 0 1  • 1  [ α • 0 F ] x 0 0 ,x 00 0 1 ◦ ˆ ωx 00 0   By deﬁnition of 1-comp osition of 2-morphisms this is also  Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦  ( α • 0 F ) • 1 ˆ ω  x 0 0 ,x 00 0 1  Symmetrical calculations can b e made on the second 2-comp osite of (6.4), giving the comp osite  Σ x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦  ( α • 0 F ) • 1 ˆ ω  x 0 0 ,x 00 0 1  • 2  [ ω • 1 ( β • 0 G )] x 0 ,x 0 0 1 ◦ [ ˆ N G ] x 0 0 ,x 00 0 1  • 1  [ M F ] x 0 ,x 0 0 1 ◦ Σ x 0 0 ,x 00 0 1  By comp osition coherence for Σ w e get ( − ◦ − ) • 0 Σ x 0 ,x 00 0 1 and b y (6.3) again ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 ∗ ε Lx 0 , ˆ Lx 00 0 1 Concluding, for every choice of three ob jects x 0 , x 0 0 , x 00 0 of X the following three equations hold           ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  • 0 Q Lx 0 ˆ Lx 00 0 1 = ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 • 0 Q x 0 ,x 00 0 1  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  • 0 P Lx 0 ˆ Lx 00 0 1 = ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 • 0 P x 0 ,x 00 0 1  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  ∗ ε Lx 0 , ˆ Lx 00 0 1 = ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 ∗ ε Lx 0 , ˆ Lx 00 0 1 Hence b oth  ( λ x 0 ,x 0 0 1 ◦ ˆ L x 0 0 ,x 00 0 1 ) • 1 ( L x 0 ,x 0 0 1 ◦ λ x 0 0 ,x 00 0 1 )  and ( − ◦ − ) • 0 λ x 0 ,x 00 0 1 satisfy equations prescrib ed by universal prop erty , hence by uniqueness they m ust b e equal, and comp osition coherence is pro ved. 6.2 Ω and a second deﬁnition of π 1 W e use the h -2pullbac k deﬁned ab ov e to give an alternativ e description of the sesqui-functor π ( n ) 1 . A key observ ation is the analogy b etw een the 6.2 Ω and a se c ond deﬁnition of π 1 167 hom-( n − 1)-group oid of a n -group oid C and the paths of a top ological space. Giv en a n -group oid ( n -category) C and t w o ob jects c 0 , c 0 0 , w e deﬁne P c 0 ,c 0 0 ( C ) b y means of the following h -pullback: P c 0 ,c 0 0 ( C ) ! / / !   I ( n ) [ c 0 0 ]   I ( n ) [ c 0 ] / / C ε c 0 ,c 0 0 C 7 ? x x x x x x x x x x x x (6.6) This deﬁnition easily extends to morphisms. In fact for F : C → D one deﬁnes P c 0 ,c 0 0 ( F ) : P c 0 ,c 0 0 ( C ) → P c 0 ,c 0 0 ( D ) b y means of the universal prop erty of h -pullbac ks yielding P c 0 ,c 0 0 ( D ), for the four-tuple P c 0 ,c 0 0 ( C ) ! / / !   ε c 0 ,c 0 0 C 9 A | | | | | | | | | | I ( n ) [ c 0 0 ]                [ F c 0 0 ]   C F   > > > > > > > > I ( n ) [ c 0 ] 7 7 n n n n n n n n n n n n n n n n [ F c 0 ] / / D This mak es P c 0 ,c 0 0 ( − ) “someho w” functorial: in fact for H : D → E , P c 0 ,c 0 0 ( F ) • 0 P c 0 ,c 0 0 ( H ) = P c 0 ,c 0 0 ( F • 0 H ) , P c 0 ,c 0 0 ( id C ) = id P c 0 ,c 0 0 ( C ) Unfortunately this do es not extend straightforw ard to 2-morphisms. In fact for a pair of parallel morphisms F , G : C → D , P c 0 ,c 0 0 ( F ) and P c 0 ,c 0 0 ( G ) are no longer parallel, this making it diﬃcult to extend P c 0 ,c 0 0 ( − ) to natural n -transformations. Indeed in applying the same argument as for deﬁning P c 0 ,c 0 0 ( − ) on mor- phisms, the corresp onding diagram (shown b elow) suggests to consider the 0-comp osition of 2-morphisms ε c 0 ,c 0 0 C ∗ α : ε c 0 ,c 0 0 C \ α _ * 4 ε c 0 ,c 0 0 C /α 6.2 Ω and a se c ond deﬁnition of π 1 168 where as usual ε c 0 ,c 0 0 C \ α = ([ c 0 ] • 0 α ) • 1 ( ε c 0 ,c 0 0 C • 0 G ) = [ α c 0 ] • 1 ( ε c 0 ,c 0 0 C • 0 G ) and ε c 0 ,c 0 0 C /α = ( ε c 0 ,c 0 0 C • 0 F ) • 1 ([ c 0 0 ] • 0 α ) = ( ε c 0 ,c 0 0 C • 0 F ) • 1 [ α c 0 0 ] P c 0 ,c 0 0 ( C ) ! / / !   ε c 0 ,c 0 0 C 9 A | | | | | | | | | | I ( n ) [ c 0 0 ]                [ Gc 0 0 ]   C G   F , , α ; C ~ ~ ~ ~ ~ ~ I ( n ) [ c 0 ] 7 7 n n n n n n n n n n n n n n n n [ F c 0 ] / / D Hence w e can consider the four-ples P c 0 ,c 0 0 ( C ) ! / / !   I ( n ) [ Gc 0 0 ]   I ( n ) ε c 0 ,c 0 0 C \ α z z z z z z z z 8 @ z z z z z z [ F c 0 ] / / D P c 0 ,c 0 0 ( C ) ! / / !   I ( n ) [ Gc 0 0 ]   I ( n ) ε c 0 ,c 0 0 C /α z z z z z z z z 8 @ z z z z z z [ F c 0 ] / / D together with id ! : ! ⇒ ! (taken tw o times) and the 3-morphism ε c 0 ,c 0 0 C ∗ α . Applying the univ ersal prop erty of h -2pullbacks we get a 2-morphism P c 0 ,c 0 0 ( α ) : P [ α c 0 ] ◦ P c 0 ,c 0 0 ( G )) ⇒ P c 0 ,c 0 0 ( F ) ◦ P [ α c 0 0 ] : P c 0 ,c 0 0 ( C ) → P F c 0 ,Gc 0 0 ( D ) suc h that P c 0 ,c 0 0 ( α ) • 0 ! = id ! P c 0 ,c 0 0 ( α ) • 0 ! = id ! P c 0 ,c 0 0 ( α ) ∗ ε F c 0 ,Gc 0 0 D = ε c 0 ,c 0 0 C ∗ α (6.7) Notice that w e ha ve denoted by P [ α c 0 ] ◦ P c 0 ,c 0 0 ( G )) and P c 0 ,c 0 0 ( F ) ◦ P [ α c 0 0 ] the morphisms obtained by applying one-dimensional the universal prop ert y to ε c 0 ,c 0 0 C \ α and ε c 0 ,c 0 0 C /α resp ectiv ely . Therefore the symbol ◦ in volv ed should b e considered just a typographical suggestion. Indeed it can b e sho wn that 6.2 Ω and a se c ond deﬁnition of π 1 169 it is a 0-comp osition of morphisms, but this w ould lead us far from the p oin t. F urthermore it is inessential with resp ect to our purp oses. F or this reasons its further dev eloping is left to the curious reader. Purp ose of the rest of the section is to giv e a c haracterization of π 1 as a consequence of the follo wing Theorem 6.5. F or every n-c ate gory C , and every two obje cts c 0 , c 0 0 in C , ther e exists a c anonic al isomorphism S c 0 ,c 0 0 C : D ( C 1 ( c 0 , c 0 0 )) → P c 0 ,c 0 0 ( C ) In the c ase of p ointe d n-gr oup oids, this gives a natur al isomorphism with c omp onents S ∗ , ∗ C : D ( π 1 ( C )) → Ω( C ) wher e Ω( C ) = P ∗ , ∗ ( C ) W e start b y making explicit h -pullbac k of (6.6), but ﬁrst we need to b e more precise on units. R emark 6.6 . Let C b e a n-category . F or a ﬁxed ob ject c 0 of C , let us consider the unit (n-1)functor giv en by the n-category structure of C : C u 0 ( c 0 ) : I ( n − 1) → C 1 ( c 0 , c 0 ) W e can mak e it explicit as a pair [ C u 0 ( c 0 )] 0 : ∗ 7→ id ( c 0 ) ∈ [ C 1 ( c 0 , c 0 0 )] 0 [ C u 0 ( c 0 )] 1 : I ( n − 2) 7→ [ C 1 ( c 0 , c 0 0 )] 1 ( id ( c 0 ) , id ( c 0 )) No w, by functoriality we get the interc hange [ C u 0 ( c 0 )] 1 = C u 1 ( id ( c 0 )) = C 1 ( c 0 ,c 0 ) u 0 ( id ( c 0 )) and this allo ws the following explicit deﬁnition: C u 0 ( c 0 ) = where u ( k ) ( c 0 ) is the iden tity k -cell ov er c 0 . In the rest of this section, in order to simplify notation, the n-category P c 0 ,c 0 0 ( C ) will b e denoted b y Q . 6.2 Ω and a se c ond deﬁnition of π 1 170 Prop osition 6.7. Given the h -pul lb ack of n -c ate gories Q ! / / !   I ( n ) [ c 0 0 ]   I ( n ) [ c 0 ] / / C ε < D           (6.8) the hom- ( n − k ) -c ate gory Q k  u ( k − 1) ( ∗ ) , c k − 1 c k / / c 0 k − 1 , u ( k − 1) ( ∗ )  ,  u ( k − 1) ( ∗ ) , c k − 1 c 0 k / / c 0 k − 1 , u ( k − 1) ( ∗ )  ! is wel l deﬁne d and it is given by h -pul lb ack over the diagr am I ( n − k ) [ c 0 k ]   I ( n − k ) [ c k ] / / C k ( c k − 1 , c 0 k − 1 ) Pr o of. By ﬁnite induction ov er k . k = 1 W e recall the deﬁnition of h -pullbac k: Q 0 is giv en by the limit in Set Q 0 ! u u ε 0   ! ) ) {∗} = [ I ] 0 [ c 0 ] 0 $ $ I I I I I I I I I I [ C 1 ] 0 d | | z z z z z z z z c " " D D D D D D D D [ I ] 0 = {∗} [ c 0 0 ] 0 z z u u u u u u u u u u C 0 C 0 Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) is giv en by a h -pullback Q 1 (  ,  ) ! / / !   I ( n − 1) [ c 0 0 ] 1   C 1 ( c 0 0 , c 0 0 ) c 1 ◦−   I ( n − 1) [ c 0 ] 1 / / C 1 ( c 0 , c 0 ) −◦ c 0 1 / / C 1 ( c 0 , c 0 0 ) ε  ,  1 t | r r r r r r r r r r r r r r = Q 1 (  ,  ) ! / / !   I ( n − 1) [ c 0 1 ]   I ( n − 1) [ c 1 ] / / C 1 ( c 0 , c 0 0 ) ε  ,  1 > F         6.2 Ω and a se c ond deﬁnition of π 1 171 with the sym b ol  when the substitute is clear from the context. k > 1 Induction h yp othesis gives the following deﬁnition for Q k − 1  ( ∗ , c k − 2 c k − 1 / / c 0 k − 2 , ∗ ) , ( ∗ , c k − 2 c 0 k − 1 / / c 0 k − 2 , ∗ )  Q k − 1 (  ,  ) ! / / !   I ( n − k +1) [ c 0 k − 1 ]   I ( n − k +1) [ c k − 1 ] / / C k − 1 ( c k − 2 , c 0 k − 2 ) ε  ,  k − 1 2 : n n n n n n n n More explicitly , w e get the following set-theoretical limit [ Q k − 1 (  ,  )] 0 ! t t [ ε  ,  k − 1 ] 0   ! * * {∗} = [ I ( n − k − 1) ] 0 [ c 0 ] 0 % % J J J J J J J J J [ C k − 1 ( c k − 2 , c 0 k − 2 )] 1 d x x q q q q q q q q q q q c & & M M M M M M M M M M M [ I ( n − k − 1) ] 0 = {∗} [ c 0 0 ] 0 y y t t t t t t t t t [ C k − 1 ( c k − 2 , c 0 k − 2 )] 0 [ C k − 1 ( c k − 2 , c 0 k − 2 )] 0 i.e. the set { ( ∗ , c k − 2 c k − 1 / / c 0 k − 2 , ∗ ) } . Hence Q k (( ∗ , c k − 1 c k / / c 0 k − 1 , ∗ ) , ( ∗ , c k − 1 c 0 k / / c 0 k − 1 , ∗ )) has (inductively) w ell deﬁned domain and c odomain, namely ( ∗ , c k − 1 c k / / c 0 k − 1 , ∗ ) and ( ∗ , c k − 1 c 0 k / / c 0 k − 1 , ∗ ) are legitimate ob jects of a Q k − 1 (  ,  ). By deﬁnition of h -pullbac k, we can sp ell it out: Q k (( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )) ! / / !   h I ( n − k +1) i 1 [ c 0 k − 1 ] 1    C k − 1 ( c k − 2 , c 0 k − 2 )  1 ( c k − 1 0 , c 0 k − 1 ) c k ◦−   h I ( n − k +1) i 1 [ c k − 1 ] 1 / /  C k − 1 ( c k − 2 , c 0 k − 2 )  1 ( c k − 1 , c k − 1 ) −◦ c 0 k / /  C k − 1 ( c k − 2 , c 0 k − 2 )  1 ( c k − 1 0 , c 0 k − 1 ) [ ε  ,  k − 1 ] ( ∗ ,c k , ∗ ) , ( ∗ ,c 0 k , ∗ ) 1 o w h h h h h h h h h h h h h h h h h h h h h h h h h h h h 6.2 Ω and a se c ond deﬁnition of π 1 172 that ma y b e rewritten Q k (( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )) ! / / !   I ( n − k ) [ c 0 k ]   I ( n − k ) [ c k ] / / C k ( c k − 1 , c 0 k − 1 ) ε ( ∗ ,c k , ∗ ) , ( ∗ ,c 0 k , ∗ ) k 5 = s s s s s s s s s s s s s s Pro of of Pr op osition 6.7 gives immediately the following Corollary 6.8. The 2-morphism ε is given explicitly by ε = < ε 0 , [ ε − , − 1 ] 0 , . . . , [ ε − , − n − 1 ] 0 , = > wher e [ ε ( ∗ ,c k − 1 , ∗ ) , ( ∗ ,c 0 k − 1 , ∗ ) k ] 0 : Q k (( ∗ , c k − 1 , ∗ ) , ( ∗ , c 0 k − 1 , ∗ )) → C k ( c k − 1 , c 0 k − 1 ) ( ∗ , c k − 1 c k / / c 0 k − 1 , ∗ ) 7→ c k Next Corollary states that h -pullbac ks along tw o constants are n-discrete. Corollary 6.9. With notation as ab ove, D ( π 0 ( Q )) = Q Pr o of. It suﬃces to let k = n in the ab ov e. Q n (( ∗ , c n , ∗ ) , ( ∗ , c 0 n , ∗ )) is given b y the following pullback in Set : Q n (( ∗ , c n , ∗ ) , ( ∗ , c 0 n , ∗ )) ! / / !   {∗} [ c 0 n ]   {∗} [ c n ] / / C n ( c n − 1 , c 0 n − 1 ) Hence, Q n (( ∗ , c n , ∗ ) , ( ∗ , c 0 n , ∗ )) = {∗} if c n = c 0 n , the empty-set otherwise. 6.2.1 0 -comp osition in P c 0 ,c 0 0 ( C ) W e hav e to describe 0-composition in Q = P c 0 ,c 0 0 ( C ) , being all k -comp ositions (with k > 0) implicit in the inductive deﬁnition of Q . Let us start with functoriality of 2-morphism ε with resp ect to 0-comp osition. F or ev ery triple ( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ ) and ( ∗ , c 00 1 , ∗ ), diagram (3.5) may b e writ- ten 6.2 Ω and a se c ond deﬁnition of π 1 173 Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) × Q 1 (( ∗ , c 0 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) id × [ c 00 1 ]   id × [ c 0 1 ] u u [ c 0 1 ] × id ) ) [ c 00 1 ] × id   Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) × C 1 ( c 0 , c 0 0 ) P r 2 . . C 1 ( c 0 , c 0 0 ) × Q 1 (( ∗ , c 0 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) P r 1 p p C 1 ( c 0 , c 0 0 ) id × ε ( ∗ ,c 0 0 , ∗ )( ∗ ,c 00 0 , ∗ ) 1 g o V V V V V V V V ε ( ∗ ,c 0 , ∗ )( ∗ ,c 0 0 , ∗ ) 1 × id o w h h h h h h h h that reduces to Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) × Q 1 (( ∗ , c 0 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) [ c 00 1 ] , , [ c 0 1 ]   [ c 1 ] r r C 1 ( c 0 , c 0 0 ) P r 1 ◦ ε ( ∗ ,c 0 , ∗ )( ∗ ,c 0 0 , ∗ ) 1 k s P r 2 ◦ ε ( ∗ ,c 0 0 , ∗ )( ∗ ,c 00 0 , ∗ ) 1 k s (6.9) On the other side, the last term of equalit y (3.5) may b e written Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) × Q 1 (( ∗ , c 0 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) Q ◦ 0   Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) [ c 00 1 ] & & [ c 1 ] x x ε ( ∗ ,c 0 , ∗ )( ∗ ,c 00 0 , ∗ ) 1 k s C 1 ( c 0 , c 0 0 ) (6.10) Comparing diagrams (6.9) and (6.10), the very deﬁnition of 0-comp ositions in h -pullbac ks prov es the following 6.2 Ω and a se c ond deﬁnition of π 1 174 Prop osition 6.10. L et c 0 , c 0 0 , c 00 0 : c 0 → c 0 0 b e ﬁxe d in C . Given c k : c 1 + 3 _ _ _ _ c 0 1 , c 0 k : c 1 + 3 _ _ _ _ c 0 1 with 1 < k ≤ n , the fol lowing e quation holds: ( ∗ , c k , ∗ ) Q ◦ 0 ( ∗ , c k , ∗ ) = ( ∗ , c k C ◦ 1 c 0 k , ∗ ) Notation 6.11. We use the notation c k : c h + 3 _ _ _ _ c 0 h with h < k , to me an that k -c el l c k has h -domain c h and h -c o domain c 0 h , i.e. ther e exist c el ls c k − 1 , c 0 k − 1 . . . c h , c 0 h such that c k : c k − 1 → c 0 k − 1 : c k − 2 → c 0 k − 2 : · · · : c h +1 → c 0 h +1 : c h → c 0 h Pr o of. (of Prop osition 6.10) By deﬁnition of vertical comp osition of 2- morphisms and whisk ering, diagram (6.9) gives  h P r 1 ◦ 0 ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 0 1 , ∗ ) 1  ◦ 1  P r 2 ◦ 0 ε ( ∗ ,c 0 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) 1 i k − 1  0  ( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )  = = h ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 0 1 , ∗ ) k i 0  [ P r 1] k − 1  ( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )  C ◦ 1 · · · · · · C ◦ 1 h ε ( ∗ ,c 0 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) k i 0  [ P r 2] k − 1  ( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )  = h ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 0 1 , ∗ ) k i 0  ( ∗ , c k , ∗ )  C ◦ 1 h ε ( ∗ ,c 0 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) k i 0  ( ∗ , c 0 k , ∗ )  = c k C ◦ 1 c 0 k where the last equalit y holds by Cor ol lary 6.8 . Next, diagram (6.10) gives  hh − Q ◦ 0 − i ◦ 0 ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) 1 i k − 1  0  ( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )  = = h ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) k i 0  ( ∗ , c k , ∗ ) Q ◦ 0 ( ∗ , c 0 k , ∗ )  = h ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 00 1 , ∗ ) k i 0  ( ∗ , ¯ c k , ∗ )  = ¯ c k Then, b y comparison we get ( ∗ , c k , ∗ ) Q ◦ 0 ( ∗ , c 0 k , ∗ ) = ( ∗ , ¯ c k , ∗ ) if, and only if, ¯ c k = c k C ◦ 1 c 0 k and this completes the pro of. 6.2 Ω and a se c ond deﬁnition of π 1 175 6.2.2 0 -units in P c 0 ,c 0 0 ( C ) W e ha ve to describ e 0-units in Q = P c 0 ,c 0 0 ( C ) , b eing all k -units (with k > 0) implicit in the inductiv e deﬁnition of Q . Let us start with functorialit y of 2-morphism ε with resp ect to 0-units, for ev ery ( ∗ , c 1 , ∗ ) in Q 0 one can consider Q u 0 (( ∗ , c 1 , ∗ )) : I ( n − 1) / / Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 1 , ∗ )) Unit coherence (3.6) is then the equalit y I ( n − 1) Q u 0 (( ∗ ,c 1 , ∗ ))   Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 1 , ∗ )) [ c 1 ] ' ' [ c 1 ] w w C 1 ( c 0 , c 0 0 ) ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 1 , ∗ ) 1 k s = I ( n − 1) [ c 1 ]   [ c 1 ]   C 1 ( c 0 , c 0 0 ) id k s This comparison, with the explicit description of ε giv en in Cor ol lary 6.8 , pro ves the following Prop osition 6.12. L et c 0 : c 0 → c 0 0 b e ﬁxe d in C . F or 1 < k ≤ n , the fol lowing e quation holds h Q u 0 (( ∗ , c 1 , ∗ )) i k =  ∗ , h C u 1 ( c 1 ) i , ∗  6.2.3 Comparison isomorphism S What we are going to state pro vides an extremely p ow erful to ol in developing the theory . Lemma 6.13. L et c 0 , c 0 0 b e obje cts of an n-c ate gory C . The assignment S c 0 ,c 0 0 C = S : D ( C 1 ( c 0 , c 0 0 )) → P c 0 ,c 0 0 ( C ) = Q given explicitly by S = < S 0 , S 1 , . . . , S n > with S i − 1 := c i 7→ ( ∗ , c i , ∗ ) , i = 1 , 2 , . . . , n S n := S n − 1 is an isomorphism of n-discr ete n-c ate gories. 6.2 Ω and a se c ond deﬁnition of π 1 176 Pr o of. By induction on n . n = 1 The map S ( c 1 ) = ( ∗ , c 1 , ∗ ) is trivially an isomorphism b etw een discrete categories D ( C 1 ( c 0 , c 0 0 )) and P c 0 ,c 0 0 ( C ). n > 1 Let us denote S = < S 0 , { S 1 , . . . , S n } > . In order for S to b e an isomorphism of n-categories the follo wing facts hav e to b e c heck ed: 1. S 0 is an isomorphism 2. for ev ery pair c 1 , c 0 1 : c 0 → c 0 0 , { S 1 , . . . , S n } c 1 ,c 0 1 is an isomorphism of ( n − 1)-categories 3. ab o ve data satisfy usual functoriality axioms 1. Since n > 1, [ D ( C 1 ( c 0 , c 0 0 ))] 0 = [ C 1 ( c 0 , c 0 0 )] 0 . Y et, by Pr op osition 6.6 one has [ P c 0 ,c 0 0 ( C )] 0 = {∗} × [ C 1 ( c 0 , c 0 0 )] 0 × {∗} . Hence the assignmen t S 0 ( c 1 ) = ( ∗ , c 1 , ∗ ) is clearly an isomorphism. 2. F or any pair c 1 , c 0 1 : c 0 → c 0 0 , induction h yp othesis guaran ties the existence of an isomorphism T c 1 ,c 0 1 :  D  C 1 ( c 0 , c 0 0 )  1 ( c 1 , c 0 1 ) / / h P c 0 ,c 0 0 ( C ) i 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) D  C 1 ( c 0 , c 0 0 )  1 ( c 1 , c 0 1 )  T c 1 ,c 0 1 / / P c 1 ,c 0 1 ( C 1 ( c 0 , c 0 0 )) deﬁned b y T c 1 ,c 0 1 k − 1 ( c k ) = ( ∗ , c k , ∗ ) , k = 2 , . . . n T c 1 ,c 0 1 n = T c 1 ,c 0 1 n − 1 Hence w e let S k − 1 = a c 1 ,c 0 1 ∈ C 1 ( c 0 ,c 0 0 ) T c 1 ,c 0 1 k − 1 , k = 2 , . . . n S n = S n − 1 6.2 Ω and a se c ond deﬁnition of π 1 177 so that the isomorphism T c 1 ,c 0 1 is exactly { S 1 , . . . , S n } c 1 ,c 0 1 . 3. W e wan t to pro ve that < S 0 , T c 1 ,c 0 1 > is an (iso)morphism of n-categories, i.e. it satisﬁes usual coherence axioms. • Let c 1 , c 0 1 , c 00 1 : c 0 → c 0 0 b e given. Coherence w.r.t. comp osition amoun ts to the comm utativity of the following diagram:  D  C 1 ( c 0 , c 0 0 )  1 ( c 1 , c 0 1 ) ×  D  C 1 ( c 0 , c 0 0 )  1 ( c 0 1 , c 00 1 ) D ( C 1 ( c 0 ,c 0 0 )) ◦ 0 / / ( i )  D  C 1 ( c 0 , c 0 0 )  1 ( c 1 , c 00 1 ) D  C 2 ( c 1 , c 0 1 ) × C 2 ( c 0 1 , c 00 1 )  ( ii ) D ( C ◦ 1 ) / / D  C 2 ( c 1 , c 00 1 )  T c 1 ,c 00 1   D  C 2 ( c 1 , c 0 1 )  × D  C 2 ( c 0 1 , c 00 1 )  T c 1 ,c 0 1 × T c 0 1 ,c 00 1   Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) × Q 1 (( ∗ , c 0 1 , ∗ ) , ( ∗ , c 00 1 , ∗ )) Q ◦ 0 / / Q 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) Here ( i ) comm utes by deﬁnition, while ( ii ) comm utes p oint-wise . In fact, for an y k = 2 , . . . , n and for c k : c 1 + 3 _ _ _ _ c 0 1 and c 0 k : c 0 1 + 3 _ _ _ _ c 00 1 Prop osition 6.10 giv es ( c k , c 0 k )  D ( C ◦ 1 ) / / _ T c 1 ,c 0 1 × T c 0 1 ,c 00 1   c k C ◦ 1 c 0 k _ T c 1 ,c 00 1    ( ∗ , c k , ∗ ) , ( ∗ , c 0 k , ∗ )   C ◦ 0 / / ( ∗ , c k C ◦ 1 c 0 k , ∗ ) • Let c 1 : c 0 → c : 0 0 b e giv en. Coherence w.r.t. units amounts to the comm utativity of the following diagram: I ( n − 1) D ( C 1 ( c 0 ,c 0 0 )) u 0 ( c 1 ) / / C u 1 ( c 1 ) * * U U U U U U U U U U U U U U U U U U U U U Q u 0 ( ∗ ,c 1 , ∗ ) " " D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D  D  C 1 ( c 0 , c 0 0 )  D  C 2 ( c 1 , c 1 )  T c 1   Q 1  ( ∗ , c 1 , ∗ ) , ( ∗ , c 1 , ∗ )  6.2 Ω and a se c ond deﬁnition of π 1 178 Upp er triangle commutes by deﬁnition, lo wer triangle commutes p oint-wise . In fact, for an y k = 2 , . . . , n Pr op osition 6.12 gives ∗  [ C u 1 ( c 1 )] k / /  [ Q u 0 ( ∗ ,c 1 , ∗ )] k ' ' O O O O O O O O O O O O O O O O O O O O O O O [ C u 1 ( c 1 )] k ( ∗ ) _ T c 1 ,c 1   ( ∗ , [ C u 1 ( c 1 )] k ( ∗ ) , ∗ ) Finally , in the proof we did not explicit the lev el of S n , as n-categories considered are n -discrete and sesqui-functor D is a full inclusion. 6.2.4 Bac k to the Theorem No w that w e ha ve developed the mac hinery , w e are able to prov e the main theorem of the section. Pr o of of Theorem 6.5 . The previous Lemma guaran ties precisely the exis- tence of a canonical isomorphism of n-categories S c 0 ,c 0 0 C : D ( C 1 ( c 0 , c 0 0 )) → P c 0 ,c 0 0 ( C ) = Q for any pair of ob jects c 0 , c 0 0 . F urther, for a n-functor F : C → D w e get a ( c 0 , c 0 0 )-indexed family of comm utative squares: D ( C 1 ( c 0 , c 0 0 )) D ( F c 0 ,c 0 0 1 ) / / S c 0 ,c 0 0 C   D ( D 1 ( F c 0 , F c 0 0 )) S F c 0 ,F c 0 0 D   P c 0 ,c 0 0 ( C ) P c 0 ,c 0 0 ( F ) / / P F c 0 ,F c 0 0 ( D ) (6.11) W e pro ve this by induction. F or n = 1 it is just a diagram of discrete categories. It suﬃces to verify comm utativity on ob jects. T o this end, let us choose a c 1 : c 0 → c 0 0 . Equations b elo w complete the case: S D ( D F ( c 1 )) = S D ( F c 1 ) = ( ∗ , F c 1 , ∗ ) P F ( S C ( c 1 )) = P F ( ∗ , c 1 , ∗ ) = ( ∗ , F c 1 , ∗ ) . Hence let us consider a generic n > 1. First w e ha ve to show that diagram (6.11) commutes on ob jects, but this amounts exactly to what we hav e just sho wn for n = 1. 6.2 Ω and a se c ond deﬁnition of π 1 179 Th us we ﬁx c 1 , c 0 1 : c 0 → c 0 0 and consider homs: [ D ( C 1 ( c 0 , c 0 0 ))] 1 ( c 1 , c 0 1 ) [ D ( F c 0 ,c 0 0 1 )] c 1 ,c 0 1 1 / / [ S c 0 ,c 0 0 C ] c 1 ,c 0 1 1   [ D ( D 1 ( F c 0 , F c 0 0 ))] 1 ( F c 1 , F c 0 1 ) [ S F c 0 ,F c 0 0 D ] F c 1 ,F c 0 1 1   [ P c 0 ,c 0 0 ( C )] 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) [ P c 0 ,c 0 0 ( F )]  ,  1 / / [ P F c 0 ,F c 0 0 ( D )] 1 (( ∗ , F c 1 , ∗ ) , ( ∗ , F c 0 1 , ∗ )) The deﬁnition of discretizer functor D allo ws us to re-formulate the diagram as follo ws: D  [ C 1 ( c 0 , c 0 0 )] 1 ( c 1 , c 0 1 )  D  [ F c 0 ,c 0 0 1 ] c 1 ,c 0 1 1  / / [ S c 0 ,c 0 0 C ] c 1 ,c 0 1 1   D  [ D 1 ( c 0 , c 0 0 )] 1 ( F c 1 , F c 0 1 )  [ S F c 0 ,F c 0 0 D ] F c 1 ,F c 0 1 1   [ P c 0 ,c 0 0 ( C )] 1 (( ∗ , c 1 , ∗ ) , ( ∗ , c 0 1 , ∗ )) [ P c 0 ,c 0 0 ( F )]  ,  1 / / [ P F c 0 ,F c 0 0 ( D )] 1 (( ∗ , F c 1 , ∗ ) , ( ∗ , F c 0 1 , ∗ )) and the previous discussion turns it in D  [ C 1 ( c 0 , c 0 0 )] 1 ( c 1 , c 0 1 )  D  [ F c 0 ,c 0 0 1 ] c 1 ,c 0 1 1  / / T c 1 ,c 0 1 C   D  [ D 1 ( c 0 , c 0 0 )] 1 ( F c 1 , F c 0 1 )  T F c 1 ,F c 0 1 D   P c 1 ,c 0 1  C 1 ( c 0 , c 0 0 )  P c 1 ,c 0 1 ( F c 0 ,c 0 0 1 ) / / P F c 1 ,F c 0 1  D 1 ( F c 0 , F c 0 0 )  No w, as T ’s are just S ’s given for n − 1, i.e. T c 1 ,c 0 1 C = S c 1 ,c 0 1 C 1 ( c 0 ,c 0 0 ) and T F c 1 ,F c 0 1 D = S F c 1 ,F c 0 1 D 1 ( F c 0 ,F c 0 0 ) the last diagram comm utes by induction hypothesis. It is clear that all this restricts to n-group oids. Moreo ver, in p ointed case w e obtain a 2-contra-v ariant natural isomorphism of sesqui-functors, i.e. a strict natural transformation of sesqui-functors that rev erses the direction of 2-morphisms and in whic h the assignments on ob jects are isomorphisms: ( n − 1) Gp d ∗ D   n Gp d ∗ π 1 4 4 Ω @ @ n Gp d ∗ S   6.2 Ω and a se c ond deﬁnition of π 1 180 Indeed in n Gp d ∗ ( i.e. in n Gp d with c 0 = ∗ = c 0 0 ), for 2-morphism α : F ⇒ G : C → D , w e can express the (strict) naturality condition D ( C 1 ( ∗ , ∗ )) D ( G ∗ , ∗ 1 ) ( ( D ( F ∗ , ∗ 1 ) 6 6 S ∗ , ∗ C   D ( D 1 ( ∗ , ∗ )) S ∗ , ∗ D   P ∗ , ∗ ( C ) P ∗ , ∗ ( G ) ( ( P ∗ , ∗ ( F ) 6 6 P ∗ , ∗ ( D ) D ( α ∗ , ∗ 1 )   P ∗ , ∗ ( α )   D ( α ∗ , ∗ 1 ) • 0 S ∗ , ∗ D = S ∗ , ∗ C • 0 P ∗ , ∗ ( α ) The pro of that this condition indeed holds is a corollary to the following L emma , that therefore concludes the pro of. Lemma 6.14. Given the 2-morphism of n-gr oup oids α : F ⇒ G : C → D , then the fol lowing e quation holds. D ( C 1 ( c 0 , c 0 0 )) D ( α c 0 ◦ G c 0 ,c 0 0 1 ) ) ) D ( F c 0 ,c 0 0 1 ◦ α c 0 0 ) 5 5 S c 0 ,c 0 0 C   D ( D 1 ( F c 0 , Gc 0 0 )) S F c 0 ,Gc 0 0 D   P c 0 ,c 0 0 ( C ) P [ α c 0 ] ◦ P c 0 ,c 0 0 ( G ) ) ) P c 0 ,c 0 0 ( F ) ◦ P [ α c 0 0 ] 5 5 P F c 0 ,Gc 0 0 ( D ) D ( α c 0 ,c 0 0 1 )   P c 0 ,c 0 0 ( α )   D ( α c 0 ,c 0 0 1 ) • 0 S F c 0 ,Gc 0 0 D = S c 0 ,c 0 0 C • 0 P c 0 ,c 0 0 ( α ) Pr o of. W e will prov e the L emma by means of the universal prop ert y of h - 2pullbac k deﬁning P F c 0 ,Gc 0 0 ( D ). T o this end let us ﬁrst consider the follo wing quite trivial c hain of equalities (taken tw o times): D ( α c 0 ,c 0 0 1 ) • 0 S F c 0 ,Gc 0 0 D • 0 ! = D ( α c 0 ,c 0 0 1 ) • 0 ! = [ id ! ] = S c 0 ,c 0 0 C • 0 [ id ! ] = S c 0 ,c 0 0 C • 0 P c 0 ,c 0 0 ( α ) • 0 ! Less trivially w e wan t to prov e ( D ( α c 0 ,c 0 0 1 ) • 0 S F c 0 ,Gc 0 0 D ) ∗ ε F c 0 ,Gc 0 0 D = ( S c 0 ,c 0 0 C • 0 P c 0 ,c 0 0 ( α )) ∗ ε F c 0 ,Gc 0 0 D By equation (6.7) this can b e rewritten D ( α c 0 ,c 0 0 1 ) ∗ ( S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D ) = ( S c 0 ,c 0 0 C • 0 ε c 0 ,c 0 0 C ) ∗ α 6.2 Ω and a se c ond deﬁnition of π 1 181 Let us prov e directly the equality on ob jects. T o this end let us ﬁx an arbitrary “ob ject” c 1 of D ( C c 0 ,c 0 0 1 ). Then applying deﬁnitions we get h D ( α c 0 ,c 0 0 1 ) ∗  S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D i 0 ( c 1 ) = = h S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D i 1 h D ( α c 0 ,c 0 0 1 ) i 0 ( c 1 )  = h S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D i 1 F c 0 α c 0 ◦ Gc 1 ! ! F c 1 ◦ α c 0 0 } } α c 1 + 3 Gc 0 0 ! = h ε F c 0 ,Gc 0 0 D i 1  S F c 0 ,Gc 0 0 D ( α c 1 )  = h ε F c 0 ,Gc 0 0 D i 1  ( ∗ , α c 1 , ∗ )  = α c 1 = α 1 h ε c 0 ,c 0 0 C i 0  ( ∗ , c 1 , ∗ )   = α 1 h ε c 0 ,c 0 0 C i 0  S c 0 ,c 0 0 C ( c 1 )   = α 1 h S c 0 ,c 0 0 C • 0 ε c 0 ,c 0 0 C i 0 ( c 1 )  = h ( S c 0 ,c 0 0 C • 0 ε c 0 ,c 0 0 C ) ∗ α i 0 ( c 1 ) On homs w e will pro ceed by induction. Hence let us ﬁx arbitrary “ob jects” c 1 , c 0 1 of D ( C c 0 ,c 0 0 1 ). Then 6.2 Ω and a se c ond deﬁnition of π 1 182 h D ( α c 0 ,c 0 0 1 ) ∗  S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D i c 1 ,c 0 1 1 = ( i ) = h D ( α c 0 ,c 0 0 1 ) i c 1 ,c 0 1 1 ∗ h S F c 0 ,Gc 0 0 D • 0 ε F c 0 ,Gc 0 0 D i α c 0 ◦ Gc 1 ,F c 0 1 ◦ α c 0 0 1 ( ii ) = h D ( α c 0 ,c 0 0 1 ) i c 1 ,c 0 1 1 ∗  h S F c 0 ,Gc 0 0 D i α c 0 ◦ Gc 1 ,F c 0 1 ◦ α c 0 0 1 • 0 h ε F c 0 ,Gc 0 0 D i ( ∗ ,α c 0 ◦ Gc 1 , ∗ ) , ( ∗ ,F c 0 1 ◦ α c 0 0 , ∗ ) 1  ( iii ) = h D ( α c 0 ,c 0 0 1 ) i c 1 ,c 0 1 1 ∗  S α c 0 ◦ Gc 1 ,F c 0 1 ◦ α c 0 0 D F c 0 ,Gc 0 0 1 • 0 ε ( ∗ ,α c 0 ◦ Gc 1 , ∗ ) , ( ∗ ,F c 0 1 ◦ α c 0 0 , ∗ ) D F c 0 ,Gc 0 0 1  ( iv ) = D  h α c 0 ,c 0 0 1 i c 1 ,c 0 1 1  ∗  S α c 0 ◦ Gc 1 ,F c 0 1 ◦ α c 0 0 D F c 0 ,Gc 0 0 1 • 0 ε ( ∗ ,α c 0 ◦ Gc 1 , ∗ ) , ( ∗ ,F c 0 1 ◦ α c 0 0 , ∗ ) D F c 0 ,Gc 0 0 1  ( v ) =  S c 1 ,c 0 1 C c 0 ,c 0 0 1 • 0 ε ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 0 1 , ∗ ) C c 0 ,c 0 0 1  ∗ α c 0 ,c 0 0 1 ( vi ) =  h S c 0 ,c 0 0 C i c 1 ,c 0 1 1 • 0 h ε c 0 ,c 0 0 C i ( ∗ ,c 1 , ∗ ) , ( ∗ ,c 0 1 , ∗ ) 1  ∗ α c 0 ,c 0 0 1 ( vii ) = h S c 0 ,c 0 0 C • 0 ε c 0 ,c 0 0 C i c 1 ,c 0 1 1 ∗ α c 0 ,c 0 0 1 ( viii ) = h ( S c 0 ,c 0 0 C • 0 ε c 0 ,c 0 0 C ) ∗ α i c 1 ,c 0 1 1 where ( i ) and ( v iii ) hold by deﬁnition of ∗ -comp osition, ( ii ) and ( v ii ) by deﬁnition of 0-whisk ering for 2-morphisms, ( iii ) and ( v i ) b y the inductive deﬁnition of S and ε , ( iv ) b y deﬁnition of discretizer and ﬁnally ( v ) b y induction h yp othesis. Uniqueness pro vided by the universal prop erty completes the pro of. A t last the promised alternative description of π 1 . Corollary 6.15. L et C b e a p ointe d n-gr oup oid. Then ther e exists a natur al isomorphism of sesqui-functors with c omp onents π ( n ) 0 ( S ∗ , ∗ C ) : π ( n ) 1 ( C ) → π ( n ) 0 (Ω( C )) Pr o of. Since S ∗ , ∗ C is n -discrete, π ( n ) 0 ( S ∗ , ∗ C ) is still an isomorphism. R emark 6.16 . F rom now on, as a consequence of Cor ol lary 6.15 w e will often iden tify the sesqui-functors π 1 ( − ) and π 0 (Ω( − )). 6.2.5 Final remark on S As the reader may guess, S is of a richer nature than we hav e sho wn in previous sections. 6.2 Ω and a se c ond deﬁnition of π 1 183 As a matter of fact we hav e dev elop ed the theory as far as our purp oses require. Nevertheless we urge to give a hint of the big picture b ehind. It is p ossible to deﬁne a sesqui-functor ˜ Σ ( n ) = ˜ Σ : n Cat → ( n + 1) Cat that assigns to a n -category C the ( n + 1)category ˜ Σ ( C ) with tw o distinguished ob jects ∗ 0 and ∗ 1 , and the follo wing hom- n -categories • [ ˜ Σ( C )] 1 ( ∗ 0 , ∗ 0 ) = I ( n ) • [ ˜ Σ( C )] 1 ( ∗ 0 , ∗ 1 ) = C • [ ˜ Σ( C )] 1 ( ∗ 1 , ∗ 0 ) = ∅ • [ ˜ Σ( C )] 1 ( ∗ 1 , ∗ 1 ) = I ( n ) with trivial comp ositions and ob vious units. Then our comparison giv es natural isomorphism S ∗ 0 , ∗ 1 ˜ Σ( C ) : D ( C ) → P ∗ 0 , ∗ 1 ( ˜ Σ( C )) . If we restrict to equiv alence sesqui-categories Cat eq , this allows to represen t an y n -category as a ( π 0 of a) sp eciﬁed h -pullbac k. Moreov er our S c 0 ,c 0 0 C are indeed hom- n -categories h P ∗ 0 , ∗ 1 ( ˜ Σ( C )) i 1  ( ∗ , c 0 , ∗ ) , ( ∗ , c 0 0 , ∗ )  (this justifying the notation adopted) of this representation, and it is easy to c heck that comp ositions and units are provided b y the universal prop ert y of h -pullbac ks. All this suggests to further dev elop the theory in the direction of homotopy theory . In fact the construction developed for P − , − ( − ) is indeed obtained b y the more general pro duct preserving p ath-functor P : n Cat → n Cat deﬁned b y the h -pullback P ( C ) C / / D   C C ε C 8 @ z z z z z z z z z z C 6.3 Monoidal structur e on Ω( C ) 184 This is a univ ersal 2-morphisms representor, in that it represen ts ev ery 2-cell α : F ⇒ G : B → C as a functor ˜ α : B → P ( C ) such that D ( ˜ α ) = F , C ( ˜ α ) = G and ε C ( ˜ α ) = α (compare with univ ersal prop ert y of h -pullbac k). Moreo ver the existence of a path-functor p ermits to obtain all h -pullbac ks as ordinary limits. In fact it is easy to show that the h -pullback ov er A F / / B C G o o is indeed the usual categorical limit o ver A F / / B P ( B ) D o o C / / B C G o o All this can b e made absolutely precise and algebraic by considering the notion of cubic al c omonad and related structures on it [Gra97]. 6.3 Monoidal structure on Ω( C ) Let C b e a p ointed n -group oid. In applying the loop sesqui-functor Ω to C one notices that the new structure has no memory of the 0-comp osition in C . In fact, every cell of Ω( C ), i.e. ev ery cell of C with the ob ject “ ∗ ” as its 0- domain and 0-co domain is 0-comp osable. Hence we can reco ver the forgotten structure, in order to get a many sorte d strict monoidal structure  Ω( C ) , ⊗ , I  with ⊗ = ◦ 0 , and I = ( ∗ , 1 ∗ , ∗ ). Moreov er cells are weakly in vertible w.r.t. this structure, th us giving a gr oup like structure to the n -group oid Ω( C ). R emark 6.17 . This is indeed the same as considering Ω( C ) as a ( n + 1)- group oid with one ob ject. n -F unctorialit y axioms prov e the following Lemma 6.18. The fol lowing t wo statements hold for n -gr oup oids, n > 0 . 1. L et n -functor F : C → D b e given. Then Ω( F ) is a strict monoidal n -functor. 2. L et natur al n -tr ansformation α : F ⇒ G : C → D b e given. Then Ω( F ) is a strict monoidal natur al n -tr ansformation. F urthermore if we apply Ω once more, w e get another monoidal structure on the n -discrete ( n − 1)-discrete n -group oid Ω(Ω( C )). This new structure corresp onds to 0-comp osition of Ω( C ), i.e. 1-comp osition of C . F unctoriality of units and comp ositions allow us to apply an Hec kmann-Hilton like ar- gumen t, this showing that comp ositions coincide and are indeed commutativ e. 6.4 Ω and π 1 pr eserve exactness 185 In terms of monoidal structures, w e ca resume the previous discussion:  Ω(Ω( C )) , ⊗ , I  is a comm utative strict monoidal structure on the n -group oid Ω(Ω( C )). Notice that monoidal structure is automatically preserved by π 0 , hence all this can b e said for sesqui-functor π 1 : Prop osition 6.19. L et C b e a n -gr oup oid. Then the ( n − 1) -gr oup oid π 1 ( C ) is natur al ly endowe d with we akly invertible strict monoidal structur e. This structur e is c ommutative when we c onsider ( π 1 ) 2 ( C ) . 6.4 Ω and π 1 preserv e exactness In the following paragraphs w e will show that, giv en a three-term exact sequence in n Gp d , the sesqui-functor Ω ( n ) pro duces a three-term exact sequence in n Gp d . As a consequence, we obtain a similar result for π ( n ) 1 Lemma 6.20. Sesqui-functor Ω ( n ) pr eserves h -pul lb acks. Pr o of. W e consider a slightly more general setting, in order to get the pro of of the statemen t as a consequence. Notice that we will omit the sup erscripts ( n ). Let us consider a h -pullbac k Q Q / / P   C G   A F / / φ 4 < p p p p p p p p p p p p B in n Gp d . Moreo v er let us ﬁx ob jects α ∈ A 0 and γ ∈ C 0 suc h that F ( α ) = β = G ( γ ). Next let us apply the univ ersal prop ert y of h -2pullbac k to get P ( φ ) as in the diagram P q ,q ( Q ) P ( P ) / / P ( Q )   P α,α ( A ) P ( F )   P γ ,γ ( C ) P ( G ) / / P ( φ ) 2 : l l l l l l l l l l l l l l l l l l P β ,β ( B ) where q = ( α, 1 β , γ ). F urther let us consider the diagram X M / / N   P α,α ( A ) P ( F )   P γ ,γ ( C ) P ( G ) / / ω 2 : l l l l l l l l l l l l l l l l l l P β ,β ( B ) 6.4 Ω and π 1 pr eserve exactness 186 that b y The or em 6.5 can b e re-dra wn X M / / N   D ( A 1 ( α, α )) D ( F α,α 1 )   D ( C 1 ( γ , γ )) D ( G γ ,γ 1 ) / / ω 1 9 k k k k k k k k k k k k k k k k k k D ( B 1 ( β , β )) Applying π 0 one gets π 0 ( X ) π 0 ( M ) / / π 0 ( N )   A 1 ( α, α ) F α,α 1   C 1 ( γ , γ ) G γ ,γ 1 / / π 0 ( ω ) 1 9 l l l l l l l l l l l l l l l l l l B 1 ( β , β ) Since Q 1 ( q , q ) is deﬁned as a h -pullbac k (see Section 3.6), the universal prop ert y yields a unique L : π 0 ( X ) → Q 1 ( q , q ) suc h that ( i ) L • 0 P q ,q 1 = π 0 ( M ) ( ii ) L • 0 Q q ,q 1 = π 0 ( N ) ( iii ) L • 0 φ q ,q 1 = π 0 ( ω ) whic h in turn implies that ther e exists a unique X η X / / D π 0 ( X ) DL / / D ( Q 1 ( q , q )) = P q ,q ( Q ) such that η X • 0 D L • 0 P ( P ) = η X • 0 D L • 0 D P q ,q 1 = η X • 0 D ( L • 0 P q ,q 1 ) = η X • 0 D ( π 0 ( M )) = M where the last equalit y holds by universalit y of adjunctions. Similarly one gets η X • 0 D L • 0 P ( Q ) = N and η X • 0 D L • 0 P ( φ ) = ω and this concludes the pro of. Lemma 6.21. Sesqui-functor Ω ( n ) pr eserves h -surje ctive morphisms. 6.4 Ω and π 1 pr eserve exactness 187 Pr o of. This is absolutely straightforw ard. Let L : K → A b e an h -surjectiv e morphism. Then, for a ﬁxed ob ject κ of K P κ,κ ( L ) = D ( L κ,κ 1 ) No w L κ,κ 1 is h -surjectiv e b y deﬁnition since L is, D preserv es trivially h - surjectiv e morphisms. Prop osition 6.22. L et the exact se quenc e of p ointe d n -gr oup oids A F / / 0   B G / / C λ   b e given. Then the se quenc e Ω A Ω F / / 0 = = Ω B Ω G / / Ω C Ω λ   is an exact se quenc e of p ointe d n -gr oup oids. Pr o of. Let L : A → K b e the comparison with the kernel of G , that is h -surjectiv e by deﬁnition. By L emma 6.20 ab o v e, Ω L is the comparison with the k ernel of Ω G , and it is h -surjective by L emma 6.21 . Besides w e get the following for fr e e Corollary 6.23. Sesqui-functor π ( n ) 1 pr eserves exact se quenc es, r eversing the dir e ction of the 2-morphism. Similar results hold for the non p oin ted case, when we ﬁx suitable p oints. 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 188 6.5 Fibration sequence of a n -functor and the Ziqqurath of exact sequences Purp ose of this and next sections is to establish the setting in order to get the main result. Let b e giv en a morphism of n -group oids F : B → C If w e ﬁx an ob ject β of B , the ﬁb er diagram ov er F β K K / / [ F β ]   B F / / C ϕ         (6.12) is an exact sequence of n -group oids. In the following sections we will show that this pro duces a canonical exact sequence P β ,β ( B ) P β ,β ( F ) / / [( ∗ , 1 β ,β )] " " P Fβ,Fβ ( C ) [ β ] < < ∇ / / K K / / [ F β ] " " B F / / C σ   ϕ   i.e. the sequence represented ab ov e is exact in P FβFβ ( C ), in K and in B . 6.5.1 Connecting morphism ∇ Although ∇ is easily obtained by means of the universal property of h - pullbac k of K , we will consider a slightly more general situation in order to apply induction prop erly in the construction of the exact sequence. Let us consider the (past) h -ﬁb er K = F ( p ) F,F β , β ∈ B 0 . Then for any other β 0 ∈ B 0 , the univ ersal prop erty yields a ∇ = ∇ ( p ) F β ,β 0 ,F : P Fβ,Fβ 0 ( C ) → F ( p ) F,F β = K P Fβ,Fβ 0 ( C ) / /   ∇ $ $ I I I I I I I I I I ε F β ,Fβ 0 C I ( n ) [ β 0 ]   6 > K K / /   B F   I ( n ) id I ( n ) [ F β ] / / ϕ ; C           C 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 189 By L emma 2.14 this is b etter understo o d as P Fβ,Fβ 0 ( C ) / / ∇   I ( n ) [ β 0 ]   K ( † ) K / /   B F   I ( n ) [ F β ] / / ϕ 6 > u u u u u u u u u u u u C (6.13) Where upp er comm uting square ( † ) is a pullback, i.e. ∇ is the strict ﬁb er of K = K ( p ) F,F β o ver β 0 . Prop osition 6.24. The se quenc e < ∇ , id [ β 0 ] , K > ab ove is exact. Pr o of. What we must prov e is that the comparison L : P Fβ,Fβ 0 ( C ) → F ( f ) K,β 0 = H is h -surjectiv e. W e will pro ve the statement by induction. F or n = 1 the functor D ( C 1 ( F β , F β 0 )) → F ( f ) K,β 0 is clearly essentially surjectiv e on ob jects and full. In fact it is deﬁned on ob jects as in the general case L 0 b elo w, hence it is essen tially surjective on ob jects for the same pro of. Moreov er it is also full: c hose tw o ob jects c 1 , c 0 1 in the domain, then w e get ob jects  ( ∗ , c 1 , β 0 ) , 1 β 0 , ∗  ,  ( ∗ , c 0 1 , β 0 ) , 1 β 0 , ∗  An arro w b etw een them is a 1-cell b 1 : β 0 → β 0 suc h that (1) c 0 1 ◦ F b 1 = c 1 (2) b 1 ◦ 1 β 0 = 1 β 0 Condition (2) forces b 1 = 1 β 0 . This mak es condition (1) true only for c 1 = c 0 1 , i.e. the image hom-set is a singleton, implying that functor L on homs is trivially surjectiv e. Henceforth let us supp ose n > 1. In order to ﬁx notation w e recall the h -ﬁb er H is a triple ( H , H , ψ ). Equations L • 0 H = ∇ , L • 0 ϕ = id [ β 0 ] 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 190 giv e L on ob jects: L 0 : ( ∗ , F β c 1 / / F β 0 , ∗ ) 7→  ( ∗ , F β c 1 / / F β 0 , β 0 ) , β 0 1 β 0 / / β 0 , ∗  This is indeed an essen tially surjective map. In fact, giv en h 0 =  ( ∗ , F β c 1 / / F b 0 , b 0 ) , b 0 b 1 / / β 0 , ∗  ∈ H 0 there exist a p 0 and a h 1 : h 0 → L ( p 0 ): simply let p 0 = ( ∗ , F β c 1 / / F b 0 F b 1 / / F β 0 , ∗ ) and h 1 as b elo w h 0 h 1   =  ( ∗ , F β c 1 / / = F b 0 F b 1   b 0 ) b 1   , b 0 b 1 / / b 1   = β 0 , ∗  L ( p 0 ) =  ( ∗ , F β c 1 F b 1 / / F b 0 , b 0 ) , b 0 b 1 / / β 0 , ∗  Then w e ﬁx a pair of ob jects p 0 , p 0 0 of P Fβ,Fβ 0 ( C ) p 0 = ( ∗ , F β c 1 / / F β 0 , ∗ ) , p 0 0 = ( ∗ , F β c 0 1 / / F β 0 , ∗ ) . As w e hav e shown in proving the univ ersal prop erty of h -pullbac ks, the comparison on homs L p 0 ,p 0 0 1 : [ P Fβ,Fβ 0 ( C )] 1 ( p 0 , p 0 0 ) − → [ F ( f ) K,β 0 ] 1 ( F p 0 , F p 0 0 ) is giv en by universal prop erty on homs ( i.e. for ( n − 1)group oids) L 0 : P c 1 ,c 0 1 ( C 1 ( F β , F β 0 )) − → F ( p ) K H h 0 ,H h 0 0 1 ,c 0 1 No w c 1 = c 1 ◦ F β 0 ,β 0 1 (1 β 0 ), and b y deﬁnition K H h 0 ,H h 0 0 1 = K ( f ) c 1 ◦ F β 0 ,β 0 1 ,c 0 1 . Hence w e started with a comparison of the kind P x,F y ( Z ) − → F ( f ) K ( p ) F,x (6.14) and w e obtain its homs part as a comparison of the kind P F y ,x ( Z ) − → F ( p ) K ( f ) F,x (6.15) This situation requires to chec k also that comparison (6.15) is essentially surjectiv e and then calculate it on homs. W e obtain a comparison (6.14) 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 191 that terminates a t wo-lev el induction pro cess and gives at once that b oth comparisons are h -surjectiv e. In fact the same calculation as abov e sho ws that (6.15) is essen tially surjective, but reverses the direction. Nev ertheless this is not a serious obstruction, as all cells of an n -group oid are equiv alences. In order to understand this fully , it may b e in teresting to mak e the con- struction explicit for 1-cells. Let 1-cell h 1 : Lp 0 → Lp 0 0 as in the following diagram Lp 0 h 1   =  ( ∗ , F β c 1 / / F β 0 F b 1   c 2 y  { { { { { { { { { { β 0 ) b 1   , β 0 1 β 0 / / b 1   β 0 b 2 |            , ∗  Lp 0 0 =  ( ∗ , F β c 0 1 / / F β 0 , β 0 ) , β 0 1 β 0 / / β 0 , ∗  i.e. h 1 =  (1 ∗ , c 1 ◦ F b 1 c 2 + 3 1 F β ◦ c 0 1 , b 1 ) , 1 β 0 ◦ 1 β 0 b 2 + 3 b 1 ◦ 1 β 0 , 1 ∗  then there exist a p 1 : p 0 → p 0 0 and a h 2 : L ( p 1 ) → h 1 . In fact such a p 1 should b e of the form ( ∗ , c 1 c 2 + 3 c 0 1 , ∗ ). Hence L ( p 1 ) is of the form Lp 0 L ( p 1 )   =  ( ∗ , F β c 1 / / F β 0 c 2 y  { { { { { { { { { { β 0 ) , β 0 1 β 0 / / β 0 = , ∗  Lp 0 0 =  ( ∗ , F β c 0 1 / / F β 0 , β 0 ) , β 0 1 β 0 / / β 0 , ∗  i.e. L ( p 1 ) =  (1 ∗ , c 1 ◦ 1 F β 0 c 2 + 3 1 F β ◦ c 0 1 , 1 β 0 ) , id 1 β 0 , 1 ∗  Then it suﬃces to let c 2 = c 1 c 1 F b 2 + 3 c 1 F b 1 c 2 + 3 c 0 1 and get the w anted 2-cell: L ( p 1 )   =  (1 ∗ , c 1 ◦ 1 F β 0 ( c 1 F b 2 ) c 2 + 3 c 1 F b 2   1 F β ◦ c 0 1 ≡ 1 β 0 ) b 2   , 1 β 0 id 1 β 0 1 β 0 b 2   = , 1 ∗  h 1 =  (1 ∗ , c 1 ◦ F b 1 c 2 + 3 1 F β ◦ c 0 1 , b 1 ) , 1 β 0 b 2 + 3 b 1 , 1 ∗  6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 192 6.5.2 Connecting 2-morphism σ In order to paste diagram (6.13) with P of the original sequence, exactness in P F β , F β 0 ( C ) must b e shown, that means we hav e to ﬁnd the 2-morphism [0] ⇒ P ( F ) • 0 ∇ that realizes exactness. This is done by means of 2- dimensional univ ersal prop erty of h -pullbacks. Let us recognize this fact in the follo wing diagram: P β ,β 0 ( B ) P ( F ) / /   P F β ,F β 0 ( C ) / / ∇   I ( n ) [ β 0 ]   I ( n ) [( ∗ , 1 β ,β )] / / σ ; C K K / /   ( pb ) B F   I ( n ) [ F β ] / / ϕ ; C                   C (6.16) Construction of σ W e can apply 2-dimensional universal prop erty of K = F ( p ) F,F β (ev en if h - pullbac ks regularity is enough) to the following set of data: 2-morphisms P ( B ) [ β ] / /   B F   I id z z z z z z z z [ F β ] / / C and P ( B ) [ β 0 ] / /   B F   I ε B F 7 ? v v v v v v v v [ F β ] / / C o ver the base, 2-morphism P ( B )     I and P ( B ) [ β ]   [ β 0 ]   ε B + 3 I and (identit y) 3-morphism [ F β ] [ β ] F ε B F   [ F β ] ε B F + 3 [ β 0 ] F ≡ , where the second diagram is justiﬁed b y the equalities P ( F ) • 0 ∇ • 0 ϕ = P ( F ) • 0 ε C = ε B • 0 F . Then there exists a unique σ : [( ∗ , 1 β , β )] ⇒ P ( F ) • 0 ∇ suc h that ( i ) σ • 0 K = ε β ,β 0 B ( ii ) σ ∗ ϕ = id ε β ,β 0 B • 0 F 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 193 Comparison morphism Prop osition 6.25. The triple ( P ( F ) , σ, ∇ ) is exact. Pr o of. In order to show that triple ( P ( F ) , σ, ∇ ) is exact, w e must verify that comparison with (past) h -ﬁbre of ∇ is h -surjectiv e. Namely w e construct the h -pullbac k N J / /   P F β ,F β 0 ( C ) ∇   I ξ 6 > t t t t t t t t t t t t [( ∗ , 1 F β ,β )] / / K where K = F ( p ) F,F β ∇ = ∇ ( p ) F β ,β 0 ,F N = F ( p ) ( ∗ , 1 F β ,β ) , ∇ The univ ersal prop erty for N yields a unique N : P β ,β 0 ( B ) → N suc h that ( i ) N • 0 J = P ( F ) ( ii ) N • 0 ξ = σ (6.17) F or n = 1 the result is easily obtained. In fact in this case N is a discrete group oid, and since also P β ,β 0 ( B ) is, it suﬃces to c heck surjectivit y on ob jects, and this is achiev ed in the same manner as h -surjectivit y for the general case. Hence let us supp ose n > 1. W e will need an explicit description of n -functor N = < N 0 , N − , − 1 > . N 0 : ( ∗ , β b 1 / / β 0 , ∗ ) 7→  ∗ , (1 ∗ , F b 1 id F b 1 + 3 F b 1 , b 1 ) , ( ∗ , F b 1 , ∗ )  Moreo ver for any pair of ob jects ( ∗ , b 1 , ∗ ) and ( ∗ , b 0 1 , ∗ ) in P β ,β 0 ( B ), N ( ∗ ,b 1 , ∗ ) , ( ∗ ,b 0 1 , ∗ ) 1 is obtained by the universal prop erty of h -pullbac ks as sho wn by the diagram b elo w, where p 0 = ( ∗ , b 1 , ∗ ) and p 0 0 = ( ∗ , b 0 1 , ∗ ), [ P β ,β 0 ( B )] 1 (( ∗ , b 1 , ∗ ) , ( ∗ , b 0 1 , ∗ )) [ P ( F )] ( ∗ ,b 1 , ∗ ) , ( ∗ ,b 0 1 , ∗ ) 1 # # % % N ( ∗ ,b 1 , ∗ ) , ( ∗ ,b 0 1 , ∗ ) 1   N 1 ( N p 0 , N p 0 0 ) J N p 0 ,N p 0 0 1 / /   [ P F β ,F β 0 ( C )] 1 (( ∗ , F b 1 , ∗ ) , ( ∗ , F b 0 1 , ∗ )) [ ∇ ] ( ∗ ,F b 1 , ∗ ) , ( ∗ ,F b 0 1 , ∗ ) 1   K 1  ( ∗ , F b 1 , β 0 ) , ( ∗ , F b 0 1 , β 0 )  (1 ∗ , 1 F b 1 ,b 1 ) ◦−   I ( n − 1) ξ N p 0 ,N p 0 0 1 r z n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n [(1 ∗ , 1 F b 0 1 ,b 0 1 )] / / K 1  ( ∗ , 1 F β , β ) , ( ∗ , F b 0 1 , β 0 )  (6.18) 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 194 suc h that ( i ) N  ,  1 • 0 J  ,  1 = [ P ( F )]  ,  1 ( ii ) N  ,  1 • 0 ξ  ,  1 = σ  ,  1 (6.19) Claim 6.26. N is essential ly surje ctive on obje cts. Let an ob ject n 0 of N b e giv en n 0 =  ∗ , (1 ∗ , F b 1 c 2 + 3 c 1 , b 1 ) , ( ∗ , F β c 1 / / F β 0 , ∗ )  then it suﬃces to consider the ob ject p 0 = ( ∗ , b 1 , ∗ ) to get an arro w N ( p 0 ) → n 0 , as suggested b y the diagram b elow N ( p 0 ) =    ∗ , ( ∗ , F b 1 id + 3 F b 1 , c 2   b 1 ) , ( ∗ , F β F b 1 / / F β 0 , ∗ )  n 0 =  ∗ , ( ∗ , F b 1 c 2 + 3 ≡ c 1 , b 1 ) , ( ∗ , F β c 1 / / c 2 y  z z z z z z z z z z F β 0 , ∗ )  Claim 6.27. F or any p air ( ∗ , b 1 , ∗ ) , ( ∗ , b 0 1 , ∗ ) , the ( n − 1) -functor N ( ∗ ,b 1 , ∗ ) , ( ∗ ,b 0 1 , ∗ ) 1 is h -surje ctive. Here comes the inductiv e step: w e w ant to get N  ,  1 from a situation in which is indeed a comparison itself, as in the original setting of diagram (6.16). T o this end let us consider the diagram P b 1 ,b 0 1  B 1 ( β , β 0 )  [ P ( F )]  ,  1 / /   P F b 1 ,F b 0 1  C 1 ( F β , F β 0 )  / / (1 ∗ ,id F b 1 ,b 1 ) ◦ [ ∇ ]  ,  1   I ( † ) [ b 1 ]   I σ  ,  1 r z m m m m m m m m m m m m m m m m m m m m m m m m m m m m [(1 ∗ ,id F b 0 1 ,b 0 1 )] / / K 1  ( ∗ , 1 F β , β ) , ( ∗ , F b 0 1 , β 0 )    K  ,  1 / / B 1 ( β , β 0 ) F β ,β 0 1   I ϕ  ,  1 r z m m m m m m m m m m m m m m m m m m m m m m m m m m [ F b 0 1 ] / / C 1 ( F β , F β 0 ) By deﬁnition of h -pullbac k the square ϕ  ,  1 is the h -ﬁb er F ( f ) F β ,β 0 1 ,F b 1 . F ur- thermore the square ( † ) is a pullbac k by L emma 2.14 . In fact this follows by the univ ersal prop erty , as the follo wing equations hold: ( ♣ )  (1 ∗ , id F b 1 , b 1 ) ◦ [ ∇ ] ( ∗ ,F b 1 , ∗ ) , ( ∗ ,F b 0 1 , ∗ ) 1  • 0 K ( ∗ , 1 F β ,β ) , ( ∗ ,F b 0 1 ,β 0 ) 1 = [ b 1 ] ( ♦ )  (1 ∗ , id F b 1 , b 1 ) ◦ [ ∇ ] ( ∗ ,F b 1 , ∗ ) , ( ∗ ,F b 0 1 , ∗ ) 1  • 0 ϕ ( ∗ , 1 F β ,β ) , ( ∗ ,F b 0 1 ,β 0 ) 1 = ε F b 1 ,F b 0 1 C 1 ( F β ,F β 0 ) 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 195 pr o of of ( ♣ ) and of ( ♦ ). They follow easily from the univ ersal prop erty in dimension n − 1:  (1 ∗ , id F b 1 , b 1 ) ◦ ∇  ,  1  • 0 K  ,  1 = K  (1 ∗ , id F b 1 , b 1 )  ◦ ( ∇  ,  1 • 0 K  ,  1 ) = K  (1 ∗ , id F b 1 , b 1 )  ◦ ([ ∇ • 0 K ]  ,  1 ) ( ∗ ) = K  (1 ∗ , id F b 1 , b 1 )  ◦ ([ β 0 ]  ,  1 ) =  K  (1 ∗ , id F b 1 , b 1 )  = [ b 1 ]  (1 ∗ , id F b 1 , b 1 ) ◦ [ ∇ ]  ,  1  • 0 ϕ  ,  1 ( i ) =  1 F b 1 ◦ ( ∇  ,  1 • 0 K F  ,  1 )  • 1  1 F β ◦ ( ∇  ,  1 • 0 ϕ  ,  1 )  =  ∇  ,  1 • 0 K F  ,  1  • 1  ∇  ,  1 • 0 ϕ  ,  1  ( ii ) = ∇  ,  1 • 0 ϕ  ,  1 ( ∗ ) = [ ε F β ,F β 0 C ] F b 1 ,F b 0 1 1 = ε F b 1 ,F b 0 1 C 1 ( F β ,F β 0 ) where equations mark ed ( ∗ ) hold for inductiv e deﬁnition of universal property of h -pullbac ks, while ( i ) is comp osition axiom of 2-morphism, and ( ii ) since the ﬁrst comp onen t of 1-comp osition is a 1-morphism. Hence (1 ∗ , id F b 1 , b 1 ) ◦ h ∇ ( p ) F β ,β 0 ,F i  ,  1 = ∇ ( f ) F β ,β 0 1 b 0 1 ,b 1 ,F β ,β 0 1 i.e. it is a “ ∇ ” itself. In other w ords, we wan ted to prov e that comparison with an h -ﬁb er of ∇ ( p ) F β ,β 0 ,F : P F β ,F β 0 ( C ) − → F ( p ) F,F β is h -surjectiv e, and we ﬁnd out that is equiv alent to asking: 1. its essen tial surjectivity 2. that the c omp arison in dimension (n-1) w.r.t. ∇ ( f ) F β ,β 0 1 b 0 1 ,b 1 ,F β ,β 0 1 : P F b 1 ,F b 0 1  C 1 ( F β , F β 0 )  − → F ( f ) F β ,β 0 1 ,F β ,β 0 1 b 0 1 is h -surjectiv e. In conclusion, w e get the same construction as in dimension n , up to direc- tions. Since we are in a (w eakly) in vertible setting, w e can apply induction prop erly and get the result. 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 196 6.5.3 The ﬁbration sequence of F F rom no w on we will consider h -k ernels in the p ointed setting n Gp d ∗ . Nev ertheless all the constructions plainly apply to h -ﬁb ers in n Gp d , as sho wn in the construction for diagram (6.12). Let F : C → D b e a morphism of p ointed n -group oids. Sp ecializing construc- tions ab o ve with β = ∗ and 0 = [ ∗ ] we can exhibit the exact sequence Ω B Ω F / / 0 # # Ω C 0 < < ∇ / / K K / / 0 " " B F / / C σ   ϕ   Since Ω preserv es exactness, this gives another exact sequence Ω 2 B Ω 2 F / / 0 ; ; Ω 2 C 0 # # Ω ∇ / / Ω K Ω K / / 0 ; ; Ω B Ω F / / Ω C Ω σ   Ω ϕ   Those can b e pasted together in the sev en-term exact sequence Ω 2 B Ω 2 F / / 0 > > Ω 2 C 0 Ω ∇ / / Ω K Ω K / / 0 > > Ω B Ω F / / 0 Ω C 0 > > ∇ / / K K / / 0 B F / / C σ   ϕ   Ω σ   Ω ϕ   Of course the pro cess can b e iterated indeﬁnitely in order to get a longer exact sequence, ev en if it trivializes after n applications. 6.5.4 The Ziqqurath of a morphism of (p oin ted) n -group oids A diﬀerent p ersp ective is gained by considering sesqui-functor π 1 in place of Ω. In fact in the longer exact sequences obtained ab ov e, repeated applications of Ω give structures whic h are discrete in higher dimensional cells. Their exactness may b e fruitfully inv estigated in lo wer dimensional settings, i.e. after rep eated applications of π 0 . T o this end w e state the following 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 197 Lemma 6.28. Sesqui-functor π 0 c ommutes with sesqui-functor π 1 , i.e. for every inte ger n > 1 the fol lowing diagr am is c ommutative n Gp d ∗ π ( n ) 0 / / π ( n ) 1   ( n − 1) Gp d ∗ π ( n − 1) 1   ( n − 1) Gp d ∗ π ( n − 1) 0 / / ( n − 2) Gp d ∗ Pr o of. This can b e pro v ed directly . F or n = 2 the diagram comm utes trivially . Hence let us supp ose n > 2. Let us b e given a p oin ted n -group oid C , then b y direct application of inductive deﬁnitions inv olved one has [ π 0 ( π 1 ( C ))] 0 = [ π 0 ( C 1 ( ∗ , ∗ ))] 0 = [ C 1 ( ∗ , ∗ )] 0 = [ π 0 ( C 1 ( ∗ , ∗ ))] 0 = [[ π 0 ( C )] 1 ( ∗ , ∗ )] 0 = [ π 1 ( π 0 ( C ))] 0 Moreo ver for any pair of “ob jects” c 1 , c 0 1 one has [ π 0 ( π 1 ( C ))] 1 ( c 1 , c 0 1 ) = [ π 0 ( C 1 ( ∗ , ∗ ))] 1 ( c 1 , c 0 1 ) = π 0 ([ C 1 ( ∗ , ∗ )] 1 ( c 1 , c 0 1 )) =  π 0 ( C 1 ( ∗ , ∗ ))  1 ( c 1 , c 0 1 ) =  [ π 0 ( C )] 1 ( ∗ , ∗ )  1 ( c 1 , c 0 1 ) =  π 1 ( π 0 ( C ))  1 ( c 1 , c 0 1 ) Finally this extends plainly to morphisms and 2-morphisms. R emark 6.29 . In the language of lo ops, w e can re-state L emma ab ov e in other terms: π 0 ( π 0 (Ω( − ))) = π 0 (Ω( π 0 ( − ))) Let no w a morphism F : C → D of p oin ted n -group oids b e giv en. Then the h -k ernel exact sequence K K / / 0 B F / / C ϕ   giv es tw o exact sequences of p ointed ( n − 1)-group oids: π 1 K π 1 K / / 0 < < π 1 B π 1 F / / π 1 C π 1 ϕ   π 0 K π 0 K / / 0 " " π 0 B π 0 F / / π 0 C π 0 ϕ   6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 198 Those can b e connected together in order to give a six term exact sequence of p oin ted ( n − 1)-group oids π 1 K π 1 K / / 0 < < π 1 B π 1 F / / 0 " " π 1 C ∆ / / 0 < < π 1 ϕ   δ   π 0 K π 0 K / / 0 " " π 0 B π 0 F / / π 0 C π 0 ϕ   where ∆ = π 0 ( ∇ ) and δ = π 0 ( σ ). Notice that the three leftmost terms are endo wed with strict monoidal structure and weak inv erses. Applying π 0 and π 1 , we get tw o six-term exact sequences. Those can b e pasted b y L emma 6.28 in a nine-term exact sequence of ( n − 2)-group oids (cells to b e pasted are dotted in the diagram): · / /   · / / π 2 1 ϕ   B B ·   / / · / / B B · / / π 1 π 0 ϕ   · π 1 δ   · / / B B · / / π 0 π 1 ϕ     · B B / / · / /   · / / π 2 0 ϕ   · π 0 δ   No w the three leftmost terms are endow ed with a commutativ e strict monoidal structure and weak inv erses and the three middle terms are endow ed with strict monoidal structure and w eak inv erses. Iterating the pro cess w e obtain a sort of to wer, a Ziqqurath, in whic h the lo wer is the level, the low er is the dimension the longer is the length of the sequence. · / / · / /   · n Gp d · / / > > · / /   · / /   > > · / /   · / /   · ( n − 1) Gp d · / / > > · / /   · / /   > > · / /   · / /   > > · / /   · / /   · ( n − 2) Gp d . . . . . . · / / > > · / /   · / /   · · · · · / / > > · / /   · / /   · Gp d · / / > > · / /   · / /   · · · · · / / > > · / /   · / /   · Set 6.5 Fibr ation se quenc e of a n -functor and the Ziqqur ath of exact se quenc es 199 In particular, the last ro w counts 3 · n terms. F rom left to right, there are 3 · n − 6 ab elian groups, 3 groups and 3 p oin ted sets. The row b efore the last counts 3 · ( n − 1) terms. F rom left to right, there are 3 · n − 9 strictly commutativ e categorical groups, 3 categorical groups, 3 p oin ted group oids. Let us observe that categorical groups pro duced in this w ay are strict monoidal weakly inv ertible ones. App endix A n -Group oids, comparing deﬁnitions A.1 n Cat: the globular approac h In this section w e compare the classical globular deﬁnition of n -category with the inductiv ely enriched one presented up to here. The follo wing deﬁnition is freely adapted from the one presented in [BH81]. It is essentially the same presented also in [ KV91 ], and is indeed equiv alent to that of [Str87]. Deﬁnition A.1. A n -c ate gory is a r eﬂexive n -trunc ate d globular set ( C • = {C i } i =0 , ··· ,n , { s i , t i : C i +1 → C i } i , { e i +1 : C i → C i +1 } i ) We wil l often (ab)use the notations s k : C i → C k meaning s i − 1 · s i − 2 · · · · · s k , t k : C i → C k meaning t i − 1 · t i − 2 · · · · · t k , e i : C k → C i meaning e k +1 · e k +2 · · · · · e i . C • is endowe d with op er ations ( m < i ) ? m : C i t m × s m C i → C i such that: 1. for al l c, c 0 ∈ C • and m ≤ k , s k ( c ? m c 0 ) =  s k ( c ) if m = k s k ( c ) ? m s k ( c 0 ) if m < k t k ( c ? m c 0 ) =  t k ( c 0 ) if m = k t k ( c ) ? m t k ( c 0 ) if m < k A.1 n Cat : the globular appr o ach 201 2. for al l c ∈ C i , al l m , e i ( s m ( c )) ? m c = c = c ? m e i ( t m ( c )) 3. for al l c, c 0 ∈ C , al l m , e ( c ? m c 0 ) = e ( c ) ? m e ( c 0 ) 4. for al l c, c 0 , c 00 ∈ C , al l m , c ? m ( c 0 ? m c 00 ) = ( c ? m c 0 ) ? m c 00 5. for al l c, c 0 , d, d 0 ∈ C , al l p < q , ( c ? q c 0 ) ? p ( d ? q d 0 ) = ( c ? p d ) ? q ( c 0 ? p d 0 ) In or der to avoid c onfusion, this wil l b e c al le d globular n-category . Giv en a n -category C , this deﬁnes a globular n -category C . In fact it suﬃces to let C 0 = C 0 and for ev ery 0 < i ≤ n , C i = [ c i − 1 ,c 0 i − 1 ∈C i − 1 [ C i ( c i − 1 , c 0 i − 1 )] 0 Sources targets and iden tities are obtained comp osing the following ones: s i : C i +1 → C i , t i : C i +1 → C i , e i +1 : C i → C i +1 where s i − 1 ( c i : c i − 1 i / / c 0 i − 1 ) = c i − 1 t i − 1 ( c i : c i − 1 i / / c 0 i − 1 ) = c 0 i − 1 e i +1 ( c i ) = h C i ( c i − 1 ,c 0 i − 1 ) u ( c i ) i ( ∗ ) : c i i +1 / / c i A simple calculation shows that this forms a reﬂexive n -truncated globular set. Let no w supp ose we are given a pair ( c i , c 0 i ) ∈ C i t m × s m C i . This means that c i is a cell of C m +1 ( c m , c 0 m ) and that c 0 i is a cell of C m +1 ( c 0 m , c 00 m ), for certain c m , c 0 m , c 00 m . Then we can easily deﬁne the comp osition c i ? m c 0 i = c i ◦ m c 0 i A.1 n Cat : the globular appr o ach 202 where ◦ m is the m -comp osition morphism ( − ) C ◦ m c m ,c 0 m ,c 00 m ( − ) : C m +1 ( c m , c 0 m ) × C m +1 ( c 0 m , c 00 m ) → C m +1 ( c m , c 00 m ) (A.1) This can b e seen as 0-comp osition. In fact, b y the inductiv e deﬁnition of a n -category , there exist c m − 1 , c 0 m − 1 suc h that c m , c 0 m , c 00 m : c m − 1 m / / c 0 m − 1 . Then C ◦ m is indeed ( − ) C m ( c m − 1 ,c 0 m − 1 ) ◦ 0 c m ,c 0 m ,c 00 m ( − ) : C m +1 ( c m , c 0 m ) × C m +1 ( c 0 m , c 00 m ) → C m +1 ( c m , c 00 m ) where, as usual, the v arious C m +1 ( x, y ) are indeed the short form for [ C m ( c m − 1 , c 0 m − 1 )] 1 ( x, y ). These data satisfy axioms for a globular n -category . Pr o of. The pro of is divided into ﬁve parts, according to the ﬁve axioms. 1. The statement will b e pro v ed b y (ﬁnite) induction ov er k , for a ﬁxed m . The base of the induction is giv en by deﬁnition: s n − 1 ( c n ? n − 1 c 0 n ) = s n − 1 ( c n − 1 c n / / c 0 n − 1 c 0 n / / c 00 n − 1 ) = c n − 1 . No w supp ose k ≥ m . Then for ev ery i > k one has s k ( c i ? m c 0 i ) ( i ) = s k ( s k +1 ( c i ? m c 0 i )) ( ii ) = s k ( s k +1 ( c i ) ? m s k +1 ( c 0 i )) ( iii ) = s k ( c k +1 ? m c 0 k +1 ) where ( i ) holds by deﬁnition, ( ii ) by induction, ( iii ) is just a typo- graphical substitution. Indeed what we mean with the expression “ c k +1 ? m c 0 k +1 ” is the image under the morphism (A.1) of the pair ( c k +1 , c 0 k +1 ). This is inductively deﬁned on homs, hence for strictly k > m one can make it explicit: ( c k +1 , c 0 k +1 ) : ( c k , ¯ c k ) k +1 / / ( c 0 k , ¯ c 0 k ) This is a ( k − m )-cell of C m ( c m , c 0 m ) × C m ( c 0 m , c 00 m ), and its image under [ C ◦ m c m ,c 0 m ,c 00 m ] is indeed its image under [ C ◦ m c m ,c 0 m ,c 00 m ] ( c k , ¯ c k ) , ( c 0 k , ¯ c 0 k ) k − m . Then functorialit y ov er the tw o comp onents of a pro duct implies s k ( c k +1 ? m c 0 k +1 ) = s k ( c k +1 ) ? m s k ( c 0 k +1 ) = s k ( c i ) ? m s k ( c 0 i ) . A.1 n Cat : the globular appr o ach 203 Diﬀeren tly for k = m , s k ( c k +1 ? k c 0 k +1 ) = s k ( c k c k +1 / / c 0 k c 0 k +1 / / c 00 k ) = c k . The analogous statemen t relative to targets is dealt similarly . 2. First we observe that for ev ery c k : c k − 1 k / / c 0 k − 1 , k < n , functo- rialit y w.r.t. units forces the morphism C k ( c k − 1 ,c 0 k − 1 ) u to satisfy the equation expressed b y the following diagram I id / / [ C k +1 ( c k ,c k ) u (1 c k ) T T T T T T T T T T T T T T T ) ) T T T T T T T T T T T I C k ( c k − 1 ,c 0 k − 1 u ( c k )] ∗ , ∗ 1   C k +1 ( c k , c k ) that is [ C k ( c k, 1 ,c 0 k − 1 ) u ( c k )] ∗ , ∗ 1 = C k +1 ( c k ,c k ) u (1 c k ) This fact inductively extends and gives relations betw een units. In particular for k < i this implies e i ( c k ) = e i ( e i − 1 ( · · · e k +1 ( e k ( c k )) · · · )) = e i ( e i − 1 ( · · · e k +1 ([ C k ( c k − 1 ,c 0 k − 1 ) u ( c k )]) · · · )) = e i ( e i − 1 ( · · · e k +1 (1 c k ) · · · )) = e i ( e i − 1 ( · · · [ C k +1 ( c k ,c k ) u (1 c k )] · · · )) = e i ( e i − 1 ( · · · [ C k ( c k − 1 ,c 0 k − 1 ) u ( c k )] 1 · · · )) . . . = [ C k ( c k − 1 ,c 0 k − 1 ) u ( c k )] i − k Hence w e can calculate e i ( s k ( c i )) ? m c i = [ C k ( c k − 1 ,c 0 k − 1 ) u ( s k ( c i ))] i − k ◦ m c i = c i as a direct consequence of neutral m -units. The analogous statemen t relative to targets is dealt similarly . 3. F or m < i < n w e wan t to prov e e i +1 ( c i ? m ¯ c i ) = e i +1 ( c i ) ? m e i +1 (¯ c i ) A.1 n Cat : the globular appr o ach 204 or more simply 1 ( c i ◦ m ¯ c i ) = 1 c i ◦ m 1 ¯ c i . This is just functorialit y w.r.t. units of the morphism ( − ) ◦ m ( − ). In fact if c i : c i − 1 → c 0 i − 1 and ¯ c i : ¯ c i − 1 → ¯ c 0 i − 1 , m -comp osition sends the iden tity (1 c i , 1 ¯ c i ) : ( c i , ¯ c i ) i +1 / / ( c i , ¯ c i ) of C i ( c i − 1 , c 0 i − 1 ) × C i (¯ c i − 1 , ¯ c 0 i − 1 ) to the iden tity 1 ( c i ◦ m ¯ c i ) : ( c i ◦ m ¯ c i ) i +1 / / ( c i ◦ m ¯ c i ) of C i ( c i − 1 ◦ m ¯ c i − 1 , c 0 i − 1 ◦ m ¯ c 0 i − 1 ). 4. Let m < i ≤ n . W e hav e to prov e the equalit y c i ? m ( c 0 i ? m c 00 i ) = ( c i ? m c 0 i ) ? m c 00 i i.e. c i ◦ m ( c 0 i ◦ m c 00 i ) = ( c i ◦ m c 0 i ) ◦ m c 00 i This holds b y asso ciativity of m -comp osition. 5. Let us supp ose p < q < i ≤ n . W e hav e to pro ve the equality ( c i ? q ¯ c i ) ? p ( d i ? q ¯ d i ) = ( c i ? p d i ) ? q (¯ c i ? p ¯ d i ) i.e. ( c i ◦ q ¯ c i ) ◦ p ( d i ◦ q ¯ d i ) = ( c i ◦ p d i ) ◦ q (¯ c i ◦ p ¯ d i ) T o this end let us ﬁx notation: c i : · · · c q q +1 / / c 0 q : · · · c p p +1 / / c 0 p ¯ c i : · · · c 0 q q +1 / / c 00 q : · · · c p p +1 / / c 0 p d i : · · · d q q +1 / / d 0 q : · · · c 0 p p +1 / / c 00 p ¯ d i : · · · d 0 q q +1 / / d 00 q : · · · c 0 p p +1 / / c 00 p The p -comp osition ◦ p : C p +1 ( c p , c 0 p ) × C p +1 ( c 0 p , c 00 p ) → C p +1 ( c p , c 00 p ) is functorial w.r.t. all q -comp ositions, with p < q . Indeed q -comp ositions in the pro duct C p +1 ( c p , c 0 p ) × C p +1 ( c 0 p , c 00 p ) are pro ducts of q -comp ositions in the comp onen ts. Hence the pair ( c i ◦ q ¯ c i , d i ◦ q ¯ d i ) is really a com- p osition ( c i , d i ) ◦ q ( ¯ c i , ¯ d i ) and its image under ( − ) ◦ p ( − ), namely ( c i ◦ q ¯ c i ) ◦ p ( d i ◦ q ¯ d i ), m ust b e equal to the q-comp osition of the images of the p -comp osites c i ◦ p d i and ¯ c i ◦ p ¯ d i , namely ( c i ◦ p d i ) ◦ q (¯ c i ◦ p ¯ d i ). A.2 The gr oup oid c ondition 205 Vic e-versa a globular n -category C univ o cally deﬁnes a n -category C . ( Ide a of a ) pr o of. The degenerate case n = 0 giv es immediately a (0-truncated globular) set. F or n = 1 Deﬁnition A.1 is precisely the deﬁnition of a category as a 1-truncated globular set. So let us supp ose n > 1. W e deﬁne a n -category C in the following wa y . C 0 is the set C 0 of C . F or every pair of elements c 0 , c 0 0 of C 0 , we can consider the ( n − 1)-truncated globular set C ( c 0 , c 0 0 ), where [ C ( c 0 , c 0 0 )] 0 = { c 1 ∈ C 1 s . t . s ( c 1 ) = c 0 , t ( c 1 ) = c 0 0 } and inductiv ely [ C ( c 0 , c 0 0 )] i = { c i +1 ∈ C i +1 s . t . s ( c i +1 ) ∈ [ C ( c 0 , c 0 0 )] i − 1 , t ( c i +1 ) ∈ [ C ( c 0 , c 0 0 )] i − 1 } k -Sources, k -targets, k -starting iden tities and k -comp ositions maps, with k = 1 , . . . n , restrict prop erly to C ( c 0 , c 0 0 ), hence it is a globular ( n − 1)- category . By induction hypothesis hence a ( n − 1)-category C 1 ( c 0 , c 0 0 ). Moreo ver 0-comp osition deﬁnes ( n − 1)-functors ◦ 0 : C ( c 0 , c 0 0 ) × C ( c 0 0 , c 00 0 ) → C ( c 0 , c 00 0 ) for ev ery triple c 0 , c 0 0 , c 00 0 , and 0-starting iden tities deﬁne ( n − 1)-functors u 0 ( c 0 ) : I → C ( c 0 , c 0 ) These data form indeed a n -category . A.2 The group oid condition Our notion of n -group oid corresponds, mo dulo the con versions recalled ab o v e, to the notion of n -gr oup oid of Kapranov and V o ev o dsky in [ KV91 ]. Ac- cording to their deﬁnition a n -group oid is a globular strict- n -category which satisﬁes a so-called gr oup oid-c ondition . This basically sa ys that every equa- tion of the kind cx = c 0 or y c = c 0 is (w eakly) solv able, when the equation mak es sense. F or sak e of completeness this condition is recalled b elow. Deﬁnition A.2 (Kaprano v and V o ev o dsky , Deﬁnition 1.1 [ KV91 ]) . A n - c ate gory C is c al le d a n -gr oup oid if for al l i < k ≤ n the fol lowing c onditions hold A.2 The gr oup oid c ondition 206 (GR 0 i,k , i < k − 1 ) F or e ach a ∈ C i +1 , b ∈ C k u, v ∈ C k − 1 such that s i ( a ) = t i ( u ) = t i ( v ) , u ? i a = s k − 1 ( b ) , v ? i a = t k − 1 ( b ) ther e exist x ∈ C k , φ ∈ C k +1 such that s k ( φ ) = x ? i a, t k ( φ ) = b, s k − 1 ( x ) = u, s k − 1 ( x ) = v . (GR 0 k − 1 ,k ) F or e ach a ∈ C k , b ∈ C k such that t k − 1 ( a ) = t k − 1 ( b ) ther e exist x ∈ C k , φ ∈ C k +1 such that s k ( φ ) = x ? k − 1 a, t k ( φ ) = b. (GR 00 i,k , i < k − 1 ) F or e ach a ∈ C i +1 , b ∈ C k u, v ∈ C k − 1 such that s i ( a ) = t i ( u ) = t i ( v ) , a ? i u = s k − 1 ( b ) , a ? i v = t k − 1 ( b ) ther e exist x ∈ C k , φ ∈ C k +1 such that s k ( φ ) = a ? i x, t k ( φ ) = b, s k − 1 ( x ) = u, s k − 1 ( x ) = v . (GR 00 k − 1 ,k ) F or e ach a ∈ C k , b ∈ C k such that s k − 1 ( a ) = s k − 1 ( b ) ther e exist x ∈ C k , φ ∈ C k +1 such that s k ( φ ) = a ? k − 1 x, t k ( φ ) = b. The notion recalled ab o ve is a generalization of the deﬁnition by Street in [ Str87 ] which included only axioms for inv erses (GR 0 k − 1 ,k ) and (GR 00 k − 1 ,k ). Moreo ver it is a wider generalization of the deﬁnition b y Brown and Higgins in [BH81] in whic h such axioms where taken in a strict form ( φ = id ). Notice that Kaprano v and V o ev o dsky motiv ate the new axioms for they w ould ensure not only the existence of (weak) inv erses. In fact they claim (but not pro ve) that all four axioms together imply the existence of a coherent system of suc h. That our deﬁnition is equiv alent to A.2 ab o ve is a corollary to Simpson accurate analysis dev elop ed in [ Sim98 ], where he uses the T amsamani’s approac h for the treatment of a group oid condition for weak n -categories [T am96]. This is resumed in the follo wing inductive theorem-deﬁnition A.2 The gr oup oid c ondition 207 Theorem A.3 (Simpson, The or em 2.1 [Sim98]) . Fix n < ∞ . I. Group oids Supp ose C is a [ globular ] strict n -c ate gory. The fol lowing thr e e c onditions ar e e quivalent (and in this c ase we say that C is a strict n -gr oup oid). (1) C is a n gr oup oid in the sense of Kapr anov and V o evo dsky (Deﬁnition A.2) ; (2) for al l x, y ∈ C 0 , C ( x, y ) is a strict ( n − 1) -gr oup oid, and for any 1- c el l f : x → y in C , the two [ families of ] morphisms of [ left and right ] c omp ositions with f ar e e quivalenc es of strict ( n − 1) -gr oup oids; (3) for al l x, y ∈ C 0 , C ( x, y ) is a strict ( n − 1) -gr oup oid, and τ ≤ 1 C is a gr oup oid. I I. T runcation If C is a strict n -gr oup oid, then deﬁne τ ≤ k C to b e the strict k -c ate gory whose i -c el ls ar e those of C for i < k , and whose k -c el ls ar e the e quivalenc e classes of k -c el ls of C under the e quivalenc e r elation that two ar e e quivalent if ther e is a ( k + 1) -c el l joining them. The fact that this is an e quivalenc e r elation is a statement ab out ( n − k ) -gr oup oids. The set τ ≤ 0 C wil l also b e denote d π 0 C . The trunc ation is again a k -gr oup oid and for n -gr oup oids C the trunc ation c oincide with the op er ation deﬁne d in [ KV91 ]. I I I. Equiv alence A morphism F : C → D of strict n -gr oup oids is said to b e an e quivalenc e if the fol lowing e quivalent c onditions ar e satisﬁe d: (a) (this is the deﬁnition in [ KV91 ]) F induc es an isomorphism π 0 C → π 0 D , and for every obje ct c ∈ C F induc es isomorphisms π i ( C , c ) → π i ( D , F ( c )) , wher e these homotopy gr oups ar e deﬁne d in [KV91]; (b) F induc es a surje ction π 0 C → π 0 D and for every p air of obje cts x, y ∈ C F induc es an e quivalenc e of ( n − 1) -gr oup oids C ( x, y ) → D ( F ( x ) , F ( y )) ; (c) if u, v ar e i-c el ls in C sharing sour c e and tar get, and if r : F ( u ) i +1 / / F ( v ) is a ( i + 1) -c el l in D , ther e exists an ( i + 1) -c el l t : u i +1 / / v of C and a ( i + 2) -c el l F ( t ) i +2 / / r in D (this includes the limiting c ases i = − 1 wher e u and v ar e not sp e ciﬁe d, and i = n − 1 , n wher e “ ( n + 1) -c el l” me ans e quality and “ ( n + 2) -c el ls” ar e not sp e ciﬁe d). Finally w e can compare these conditions with our group oid condition. W e ha v e already discussed ab out the corresp ondence b etw een globular and enric hed versions of n -category . Hence w e can fo cus on characterizing conditions. F ormally our Deﬁnition 4.11 amoun ts precisely to condition I. (2) abov e. What is to b e chec k ed is then the notion of equiv alence. This is done by Pr op osition 4.8 that is the inductive version of I I I. (c). A.3 System of adjoint inverses 208 The notion of truncation is not directly inv olved, nevertheless it can b e reco vered by successive applications of the sesqui-functor π 0 . A.3 System of adjoin t inv erses In order to b e more precise ab out the c hoices of in verses, only for this section, w e give a sharp er deﬁnition of equiv alence that take in to account directions of cells. Deﬁnition A.4. L et n-c ate gory morphism F : C → D b e given. • F is c al le d equiv alence of n-categories if it satisﬁes the fol lowing pr op erties: n = 0 F is an isomorphism. n > 0 1. for every obje ct d 0 of D , ther e exists an obje ct c 0 of C and a 1-c el l d 1 : d 0 → F c 0 such that for every d 0 0 in C , the morphism d 1 ◦ − : D 1 ( d 0 , d 0 0 ) → D 1 ( F c 0 , d 0 0 ) is an e quivalenc e of ( n − 1) -c ate gories, and the morphism − ◦ d 1 : D 1 ( d 0 0 , F c 0 ) → D 1 ( d 0 0 , d 0 ) is a c o-e quivalenc e of ( n − 1) -c ate gories. 2. for every p air c 0 , c 0 0 in C , F c 0 ,c 0 0 1 : C 1 ( c 0 , c 0 0 ) → D 1 ( F c 0 , F c 0 0 ) is an e quivalenc e of ( n − 1) -c ate gories. • F is c al le d co-equiv alence of n-categories if it satisﬁes the fol lowing pr op- erties: n = 0 F is an isomorphism. n > 0 A.3 System of adjoint inverses 209 1. for every obje ct d 0 of D , ther e exists an obje ct c 0 of C and a 1-c el l d 1 : F c 0 → d 0 such that for every d 0 0 in C , the morphism d 1 ◦ − : D 1 ( d 0 , d 0 0 ) → D 1 ( F c 0 , d 0 0 ) is a c o-e quivalenc e of ( n − 1) -c ate gories, and the morphism − ◦ d 1 : D 1 ( d 0 0 , F c 0 ) → D 1 ( d 0 0 , d 0 ) is an e quivalenc e of ( n − 1) -c ate gories. 2. for every p air c 0 , c 0 0 in C , F c 0 ,c 0 0 1 : C 1 ( c 0 , c 0 0 ) → D 1 ( F c 0 , F c 0 0 ) is a c o-e quivalenc e of ( n − 1) -c ate gories. This suggests to impro ve Deﬁnition 4.2 : Deﬁnition A.5. A 1-c el l c 1 : c 0 → c 0 0 of a n-c ate gory C is said to b e w eakly in vertible , or simply an equiv alence , if, for every obje ct ¯ c 0 of C , the morphism c 1 ◦ − : C 1 ( c 0 0 , ¯ c 0 ) → C 1 ( c 0 , ¯ c 0 ) is an e quivalenc e of ( n − 1) -c ate gories, and the morphism − ◦ c 1 : C 1 (¯ c 0 , c 0 ) → C 1 (¯ c 0 , c 0 0 ) is a c o-e quivalenc e of ( n − 1) -c ate gories. The dual deﬁnition for a we akly c o-invertible 1-cell. When a 1-cell is weakly inv ertible, then it has indeed left and righ t w eak- in verses. In fact for c 1 : c 0 → c 0 0 , c 1 ◦ − : C 1 ( c 0 0 , c 0 ) → C 1 ( c 0 , c 0 ) b eing an equiv alence implies that for the 1-cell 1 c 0 : c 0 → c 0 there exists a pair ( c ∗ 1 , i 2 : 1 c 1 ∼ + 3 c 1 ◦ c ∗ 1 ) , similarly for − ◦ c 1 : C 1 ( c 0 0 , c 0 ) → C 1 ( c 0 0 , c 0 0 ) b eing a co-equiv alence implies there exists a pair ( c † 1 , e 2 : c † 1 ◦ c 1 ∼ + 3 1 c 0 1 ) . A.3 System of adjoint inverses 210 Left and right inv erses are indeed equiv alent: following a classical group- theoretical argumen t, the 1-comp osition c † 1 = c † 1 ◦ 1 c 1 c † 1 ◦ i 2 + 3 c † 1 ◦ 1 c 1 ◦ c ∗ 1 e 2 ◦ c ∗ 1 + 3 1 c 0 1 ◦ c ∗ 1 = c ∗ 1 witnesses the equiv alence. Let us supp ose we hav e chosen a system of in verse in a n -group oid C , i.e. for ev ery k -cell c k w e can exhibit a weak inv erse c ∗ k and equiv alences i c k : 1 + 3 c k ◦ k − 1 c ∗ k , e c k : c ∗ k ◦ k − 1 c k + 3 1 Then it is alw ays p ossible to get a system of adjoin t in v erses, according to the follo wing Deﬁnition/Prop osition A.6. By induction over n . In 0 Gp d and in 1 Gp d inverses ar e unique. So let us supp ose n > 1 . L et us supp ose further we have chosen systems of adjoint inverses in the hom- ( n − 1) -gr oup oids. Then the fol lowing pr o c e dur e gives a system of adjoint inverses in C . F or every four-tuple ( c 1 , c ∗ 1 , i c 1 , e c 1 ) one deﬁnes a new four-tuple ( c 1 , c ∗ 1 , i 0 c 1 , e 0 c 1 ) wher e e 0 c 1 = e c 1 and i 0 c 1 is given by the 1-c omp osition (0-c omp osition is jux- tap osition) 1 c 0 i c 1 + 3 c 1 c ∗ 1 c 1 e ∗ c 1 c ∗ 1 + 3 c 1 c ∗ 1 c 1 c ∗ 1 i ∗ c 1 c 1 c ∗ 1 + 3 c 1 c ∗ 1 Then for ev ery 1-cell c 1 : c 0 → c 0 0 one has the triangular equiv alences: c 1 i c 1 c 1 + 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? c 1 c ∗ 1 c 1 c 1 e c 1   c 1  w                       c ∗ 1 c ∗ 1 i c 1 + 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? c ∗ 1 c 1 c ∗ 1 e c 1 c ∗ 1   c ∗ 1  7 G                      Pr o of. It suﬃces to paste 3-cells in the following tw o diagrams, where hexagons comm ute quite trivially , rectangles are identities. Concerning triangles, induction hypothesis provide adjoint inv erses for higher dimen- sional cells used there. A.3 System of adjoint inverses 211 c 1 i c 1 + 3 c 1 c ∗ 1 c 1 c 1 e ∗ c ∗ 1 c 1 + 3 c 1 c ∗ 1 c 1 c ∗ 1 c 1 i ∗ c 1 c ∗ 1 c 1 + 3 c 1 c ∗ 1 c 1 c 1 e + 3 c 1 c 1 i c 1 + 3 c 1 c ∗ 1 c 1 c 1 e ∗ c ∗ 1 c 1 + 3 c 1 c ∗ 1 c 1 c ∗ 1 c 1 c 1 c ∗ 1 c 1 e + 3 c 1 c ∗ 1 c 1 i ∗ c 1 + 3 c 1 c 1 i c 1 + 3 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c 1 c ∗ 1 c 1 c 1 e + 3 Q Q Q Q Q Q Q Q Q Q Q Q Q c 1 c 1 e ∗ + 3 c 1 c ∗ 1 c 1 i ∗ c 1 + 3 m m m m m m m m m m m m m c 1 m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m c 1 c ∗ 1 c 1 c 1 c 1 η ∗ e    η ∗ i c 1    c ∗ 1 c ∗ 1 i + 3 p p p p p p p p p p p • c ∗ 1 c 1 e ∗ c ∗ 1 + 3 p p p p p p p p p p p • c ∗ 1 c 1 c ∗ 1 i ∗ + 3 p p p p p p p p p p p c ∗ 1 c 1 c ∗ 1 ec ∗ 1 + 3 p p p p p p p p p O O O O O O O O O c ∗ 1 O O O O O O O O O O O • c ∗ 1 i + 3 • c ∗ 1 c 1 e ∗ c ∗ 1 + 3 • c ∗ 1 i ∗ c 1 c ∗ 1 + 3 • c ∗ 1 i ∗ + 3 • c ∗ 1 i + 3 • ec ∗ 1 + 3 • • c ∗ 1 i + 3 P P P P P P P P P P P • c ∗ 1 c 1 e ∗ c ∗ 1 + 3 P P P P P P P P P P P • c ∗ 1 c 1 c ∗ 1 i ∗ + 3 P P P P P P P P P P P • c ∗ 1 i ∗ + 3 P P P P P P P P P P P • c ∗ 1 i + 3 • ec ∗ 1 + 3 n n n n n n n n n n n • n n n n n n n n n n n • c ∗ 1 i + 3 • c ∗ 1 c 1 e ∗ c ∗ 1 + 3 • c ∗ 1 c 1 c ∗ 1 i ∗ + 3 • ec ∗ 1 + 3 • • c ∗ 1 i + 3 • c ∗ 1 c 1 e ∗ c ∗ 1 + 3 • ec ∗ 1 c 1 c ∗ 1 + 3 • c ∗ 1 i ∗ 1 + 3 • • c ∗ 1 i + 3 N N N N N N N N N N N N N N N N N N N N N N N N • ec ∗ 1 + 3 O O O O O O O O O • e ∗ c ∗ 1 + 3 • c ∗ 1 i ∗ 1 + 3 p p p p p p p p p • p p p p p p p p p p p p p p p p p p p p p p p p c ∗ 1 c 1 c ∗ 1 c ∗ 1 c ∗ 1 ε i  J T c ∗ 1 ε ∗ i  J T η e c ∗ 1  J T c ∗ 1 η i  J T Bibliograph y [B ´ en67] Jean B ´ enab ou. 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