Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)

Journal of Homotopy and R elate d Structu r es , vol. ??(??), ????, pp.1–36 CUBICAL COSP ANS AND HIGHER COBORDISMS (COSP ANS IN ALGEBRAIC TOPOLOGY, I I I) MARC O G RANDIS Abstr a ct After tw o pap ers on weak cubica l ca tegories and c ol lar able cospans, resp ectively , we put things together and co nstruct a we ak cubical catego ry of cubical c ol lar e d cospans of top ological spaces. W e also build a second structure, calle d a quasi cubica l category , formed of arbitrary cubica l co spans conca tenated by homotopy pushouts. This structure, simpler but w eaker, has lax identit ies. It co n tains a similar framework for co bo rdisms of manifolds with cor ners and could therefore be the basis to extend the study of TQFT’s of Part II to hig he r cubical degr ee. In tro duction This is a s e q uel to tw o pap ers, cited a s Part I [ 6 ] a nd Part I I [ 7 ]. A r eference I.2, or I.2.3, relates to Section 2 or Subsectio n 2 .3 of Part I. Similarly for Part II. In Part I we co nstructed a cubical struc tur e of higher cospans C osp ∗ ( X ) o n a category X with pusho uts, and abstra cted from the constr uction the g eneral notion of a w eak cubical categor y . An n -cubical cospan in X is deﬁned a s a functor u : ∧ n → X , where ∧ is the category ∧ : − 1 → 0 ← 1 (the formal c osp an ). (1) These diagrams for m a cubical set, equipp ed with comp ositions u + i v of i - consecutive n -cub es, for i = 1 , ..., n. Suc h cubical comp ositio ns are co mputed by pushouts, and behave ‘catego rically’ in a weak s e nse, up to suitable co mparisons. T o make roo m for the latter, the n -th comp onent of C osp ∗ ( X ) C osp n ( X ) = Cat ( ∧ n , X ) , (2) is no t just the set o f functors u : ∧ n → X (the n -cub es of the s tr ucture), but the c ate gory of such functors a nd their natural transformations f : u → u ′ : ∧ n → X (the n -maps of the structure). The comparisons a re invertible n -maps; but general n - maps ar e als o impo rtant, e.g . to deﬁne limits and co limits (I.4.6, I I.1.3). Thus, a we ak cubic al c ate gory ha s coun tably many we ak (or cubic al ) dir ections i = 1 , 2 , ..., n, ... all of the same so rt, and one st rict (or tr ansversal ) direction, which is g enerally of a diﬀeren t sor t. W ork supp orted b y a research grant of U niv ersit` a di Genov a. Receiv ed , revised . 2000 Mathematics Sub ject Cl assiﬁcation: 18D05, 55U10, 55P05, 57N70 Key words and phrases: spans, cospans, weak double category , cubical sets, we ak cubical category , homotop y pushout, cob ordisms c  ????, Marco Grandis. Permission to copy for priv ate use granted. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 2 T r uncating cub es and transv ers al maps at cubica l degree 1, we get the we ak double c ate gory C osp( X ) = 1 C osp ∗ ( X ) , consisting of cospans and their natural transformatio ns, with one strict composition (the transv ers a l one) and one weak comp osition (by pushouts). T r uncating at cubical degr ee 2, we get a structure re- lated to Morton’s construction for 2-cubical cospans (cf. [ 21 ] and 1.7). Now, the w eak cubical category C osp ∗ ( T op ) of cubical cos pans of top olo gical spaces is not well suited for Algebraic T op olog y . Indeed, the comp osition by o rdinary pushouts is not ho motopically stable, and is no t preserved b y (co)homotopy or (co)homology functors, even in a w eak sense. This is why , in Part I I, working in cubic al de gr e e 1 , we considered c ol lar able c osp ans , forming a weak double c ategory C blc( T op ) ⊂ C o s p( T op ) . Indeed, a push- out o f collar able maps is a homotopy pushout (Thm. I I.2.5 ). Therefor e, cohomoto py functors induce ‘functor s’ fro m collara ble co spans to spans of s ets, pr oviding - by linearisatio n - top olo gical qua n tum ﬁeld theo ries (TQFT) on ma nifolds and their cob ordisms (I I.3). Similarly , (co)homology and homotopy functors ta ke collarable cospans to r elations of a belia n groups or (co )spans of g r oups, yielding other algebr aic inv a riants (II.4 ). Notice that, as motiv ated in I I.1.6, the deﬁnition of a collar able cospan is more general than one might exp ect. Indee d, we wan t to include the degenerate cospan e 1 ( X ) = (id X , id X ) , instead of re placing it with the cylindric al de gener acy E 1 ( X ) , as usually done for co bo rdisms: E 1 ( X ) = ( d − : X → X × [0 , 1] ← X : d + ) , d − ( x ) = ( x, 0) , d + ( x ) = ( x, 1) . (3) The main pr oblem is that such degenerac ie s do not s atisfy an axio m of cubical sets: w e only get E 1 E 1 ( X ) ∼ = E 2 E 1 ( X ) (see 5.2). Therefore, acco rding to our deﬁnition in Part I I, a c ol lar able c osp an de c om- p oses into a sum of a trivial ly c ol lar able p art (a pair of homeomo rphisms) and a 1-c ol lar able p art ; only the seco nd do es admit co lla rs (which a re embedding s of cylinders, with disjo in t images ). Howev er, e 1 ( X ) a nd E 1 ( X ) a r e we akly e quivalent , as deﬁned in I I.2.8 (and here, in 5 .3); homotopy invarianc e of functors on to polo gical cospans is deﬁned with resp ect to this notion. In the pr esent pap er, we combine the pr e vious Parts. The ﬁrst main construction, in Sections 2 -4, is a weak cubical categ o ry C c ∗ ( T op ) of c ol lar e d c osp ans . It extends to un b ounded cubica l degree a weak double ca tegory C c( T op ) , which is a v a riant of C blc( T op ) where collars are assigne d . The sec o nd main framework, in Section 5 , is simpler, but satisﬁes weak er ax ioms on degenera c ies and requires more c o mparisons. W e work now with arbitr ary cospa ns (not suppo sed to ha ve collar s) and r e pla ce: - the ordinary degeneracies e i with the cylindrical ones, E i , - the o r dinary concatenations with the cylindric al ones , c onstructed by means of homotopy pushouts. Notice that the la tter is, in itself, a homotopy-inv ariant op eration, which explains why her e colla rs ar e not neede d. As a pr ice for this simplicity , we obtain a symmetric quasi cubical category C OSP ∗ ( T op ) , where degeneracie s only satisfy the c ubica l relation mentioned a bove up t o isomorph ism , a nd b ehav e a s lax iden tities: the (left and right) unit co mparisons ar e no t inv ertible. An extensive study of motiv ations Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 3 and goals of the homotopical weak ening of iden tities can b e found in J. Ko ck [ 18 ]; see also Joy al-Ko ck [ 15 ] (and other pap ers in prepara tion, by the sa me author s ). This new structure is made precise at the e nd, in Section 7, while in Section 6 we construct the quasi cubica l ca tegory C OB ∗ ( k ) ⊂ C OSP ∗ ( T op ) of k -manifolds and their cubica l cob ordisms, based on the notion o f m anifold s with c orners [ 4, 19, 13 ]. Extending the coho motopy functors to these structures (after I I.3) should y ield higher TQFTs, but this is not dealt with here. The 2-cubical trunca tion of our construction is re lated with the constructio n of Mo rton and Baez [ 2 1, 1 ] (which use cylindrical identit ies a nd pusho ut-concatenation). As disc us sed in 6 .1, we do not endeav our to cons truct a we ak cubical category C ob ∗ ( k ) ⊂ C c ∗ ( T op ) base d on the ﬁrst main construction. It should b e p oss ible , but so heavy tha t the goa l of getting a b etter b e haviour of degeneracies might not justify its c omplication; moreover, c o nsidering the imp ortance of ‘units-up- to- homotopy’ in modelling homotopy t yp es [ 18, 15 ], one ma y question the in terest of this goal. (Notice als o that in Part I I we deﬁned a weak double subcatego ry C ob( k ) ⊂ C blc( T op ) of k - dimensional manifold s and their c ob or disms , based on the fact that ‘cob ordisms are alw ays collar a ble’.) Size problems ca n b e dealt with as in Part I, us ing a hierarch y of tw o universes, U 0 ∈ U . Small ca tegory means U -small. The co nstructions C o sp ∗ ( − ), C c ∗ ( − ), e tc. apply to the small catego ry T op of U 0 -small spaces. Cat is the 2 -categor y of U -small categorie s, to which T op b elongs. Finally , the index α ta k es v alues ± 1 , written ± in supe r scripts, and I X d enotes the standar d cy linder X × [0 , 1] on a s pa ce X . 1. The symmetric weak cubical category of cospans W e begin b y r ecalling cubical c o spans, from Part I; we also introduce symmetr ic quasi cubical sets, whic h will be us ed later to deﬁne quasi c ubica l categorie s . 1.1. Symmetric cubical and quasi cubical sets A cubical set (( A n ) , ( ∂ α i )) , ( e i )) has fac es ( ∂ α i ) and de gener acies ( e i ) ∂ α i : A n ⇄ A n − 1 : e i ( i = 1 , ..., n ; α = ± ) , (4) satisfying the cubical relations: ∂ α i .∂ β j = ∂ β j .∂ α i +1 ( j 6 i ) , (5) e j .e i = e i +1 .e j ( j 6 i ) , (6) ∂ α i .e j = e j .∂ α i − 1 ( j < i ) , or id ( j = i ) , or e j − 1 .∂ α i ( j > i ) . (7) As in I.2.2, a symmetric cubical set is a cubical set which is fur ther equipped with tr ansp ositions s i : A n → A n ( i = 1 , ..., n − 1 ) , (8) satisfying the Mo ore re la tions (see Coxeter-Moser [ 3 ], 6.2; o r Johnson [ 14 ], Section 5, Thm. 3) s i .s i = 1 , s i .s j .s i = s j .s i .s j ( i = j − 1) , s i .s j = s j .s i ( i < j − 1) , (9) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 4 and the follo wing equations: j < i j = i j = i + 1 j > i + 1 ∂ α j .s i = s i − 1 .∂ α j ∂ α i +1 ∂ α i s i .∂ α j s i .e j = e j .s i − 1 e i +1 e i e j .s i . (10) W e will sp eak o f a symm et ric quasi cubic al set when the axio m (6), on pure degeneracies , is omitted, which will b e imp or tan t for ‘cylindr ical degenera cies’ (cf. 5.2). Actually , the presence o f transp ositions makes all faces and degener acies deter - mined by the ones b elonging to a ﬁxed dir ection, e.g. the 1 -directed o nes, ∂ α 1 and e 1 . In fact, from ∂ α i +1 = ∂ α i .s i and e i +1 = s i .e i , w e deduce that: ∂ α i = ∂ α 1 . s ′ i , e i = s i .e 1 ( i = 2 , ..., n ; α = ± ) , (11) where w e are using the in verse ‘p ermutations’: s i = s i − 1 .....s 1 , s ′ i = s 1 .....s i − 1 . (12) 1.2. A reduced presenta tion These relations lead to a mor e economical pr esent atio n of our structures. Prop osition 1. A symmetric quasi cubic al set c an b e e quivalently deﬁne d as a system A = (( A n ) , ( ∂ α 1 ) , ( e 1 ) , ( s i )) , ∂ α 1 : A n ⇄ A n − 1 : e 1 , s 1 : A n +1 → A n +1 ( n > 1) , (13) under the Mo or e rel ations (9) and the axioms: ∂ α 1 .∂ β 1 = ∂ β 1 .∂ α 1 .s 1 , ∂ α 1 .e 1 = id , s i .∂ α 1 = ∂ α 1 .s i +1 , e 1 .s i = s i +1 .e 1 . (14) F or a symmetric cubic al set one adds the axiom: e 1 e 1 = s 1 .e 1 e 1 (symmetry of se c ond-or der de gener acies). (15) Pr o of. Deﬁning the other faces and degenerac ies by (11 ), one can pr ov e the globa l cubical relations. F or instanc e , letting j 6 i , we ha ve: ∂ α i .∂ β j = ∂ α 1 . ( s 1 .....s i − 1 ) .∂ β 1 . ( s 1 .....s j − 1 ) = ∂ α 1 ∂ β 1 . ( s 2 .....s i ) . s ′ j = ∂ β 1 ∂ α 1 . ( s ′ i +1 s ′ j ) , ∂ β j .∂ α i +1 = ∂ β 1 . ( s 1 .....s j − 1 ) .∂ α 1 . s ′ i +1 = ∂ β 1 ∂ α 1 . ( s 2 .....s j ) . s ′ i +1 = ∂ β 1 ∂ α 1 . ( s 1 s ′ j +1 s ′ i +1 ) . Now, it suﬃces to verify that the t wo op erato rs at the end o f these equalities coin- cide. In fact, in the symmetric gro up S n of per m utations o f the set { 1 , ..., n } , s ′ i is ident iﬁed with the p ermutation: σ i = ( i, 1 , ..., ˆ i, ..., n ) . Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 5 Thu s, a lw ays for j 6 i, the op erators s i +1 s j and s j +1 s i +1 corres p ond to the per m utations: σ i +1 σ j = ( i + 1 , j, 1 , ..., ˆ j , ..., ( i + 1) ˆ , ..., n ) , σ j +1 σ i +1 = ( j, i + 1 , 1 , ..., ˆ j , ..., ( i + 1) ˆ , ..., n ) , and the transpo sition s 1 turns one per m utation into the other. 1.3. A se tting for cospans The mo del of our co nstruction of cubical cospa ns, in P ar t I, is the formal c osp an category ∧ and its Cartesian p ow ers ∧ n ( n > 0) ( − 1 , − 1 ) / /   (0 , − 1)   (1 , − 1) o o   • 1 / / 2   − 1 / / 0 1 o o ( − 1 , 0) / / (0 , 0) (1 , 0) o o ∧ ( − 1 , 1) / / O O (0 , 1) O O (1 , 1) o o O O ∧ 2 (16) (Iden tities and comp osed a rrows are alwa ys understo o d in s uc h diagra ms o f ﬁnite categorie s.) Thus, an n -c osp an in the ca tegory X is a functor u : ∧ n → X . B ut, in order to b e a ble to comp ose them, in direction i = 1 , ..., n, we need (a full choice of ) pushouts in X . More generally , according to the terminolog y of Part I, a pt-c ate gory , or c ate gory with distinguishe d pu s houts , is a ( U -small) ca tegory where some spans ( f , g ) hav e one distinguishe d pushout ( f ′ , g ′ ) (in a s ymmetric w ay , of course) • f / / g   • f ′   x f / / 1   x ′ 1   _ _   _ _   • g / / • x f / / x ′ (17) and w e assume the following unitarity c onstr aints : (i) each square of ident ities is a distinguished pushout, (ii) if the pair ( f , 1) has a distinguished pushout, this is (1 , f ) (as in the rig h t diagram ab ove). A pt-functor F : X → Y is a functor betw een pt-categ ories which strictly pre- serves the distinguished pushouts. W e sp eak o f a ful l (resp. trivial ) choic e , or of a c ate gory X with ful l (r e sp. trivial ) distinguishe d pushouts , when a ll the spans in X (resp. only the spans of iden tities) hav e a distinguished pusho ut. The catego ry pt C at o f pt-categ ories a nd pt-functors is U -complete and U -co com- plete. F or instance, the pr o duct of a family ( X j ) j ∈ J of pt-categorie s indexed on a U -small set is the Cartesian pr o duct X (in Cat ) , equipp ed with thos e (pushout) squares in X whose pro jection in ea c h factor X j is a distinguished pushout. In Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 6 particular, the ter minal ob ject of pt Cat is the terminal catego r y 1 with the unique po ssible c hoice: its only square is a distinguished pushout. Cat em beds in pt Cat , equipping a small category with the trivial c hoice of pullbacks (a pr o cedure which is le ft a djo int to the for getful functor pt Cat → Cat ) . Limits and colimits are pr e served by this embedding. Our co nstruction requires this sort o f double se tting Cat ⊂ pt Cat , with ‘mo dels’ ∧ n having a trivia l choice a nd cubical co spans ∧ n → X liv ing in ca tegories with a full choice (whic h is necessa ry to compo s e them). Notice that the category ∧ n has all pu sho uts; ho wev er, should we use the full choice suggested by diagram (16), a pt-functor ∧ 2 → X would only re ach v ery particular 2-cubical co spans. Notice a lso that, in the absence of the unitar it y con- straint (i) on the choice o f pushouts, the terminal ob ject of pt Cat would still b e the same, but a functor 1 → X co uld only r each an ob ject whose squar e of identities is distinguished. On the o ther hand, condition (ii) just simpliﬁes things , mak ing our units work strictly . 1.4. The structure of the formal cospan The category ∧ has a basic structure of formal symmetric in terv al, with resp ect to the Car tes ian product in Cat (a nd pt Cat ) . This structure co nsists of tw o fac es ( ∂ − , ∂ + ) , a de gener acy ( e ) and a tr ansp osition symmetry ( s ) ∂ α : 1 ⇒ ∧ , e : ∧ → 1 , s : ∧ 2 → ∧ 2 ( α = ± 1) , ∂ α ( ∗ ) = α, s ( t 1 , t 2 ) = ( t 2 , t 1 ) . (18) (A functor with v alues in the ordere d set ∧ n is determined by its v a lue on ob- jects). Comp osition is - fo rmally - mo r e c omplicated. The mo del of binary c omp o- sition is the pt- categor y ∧ 2 display ed below, with one non-trivial distinguished pushout 0 {∗} ∂ + / / ∂ −   ∧ k −   P P n n a 8 8 p p p p p c f f N N N N N − 1 6 6 m m m m m b g g O O O O O 7 7 o o o o o 1 g g O O O O O ∧ k + / / ∧ 2 (19) Now, the commutativ e square at the right ha nd above is not a pushout; in fact, in Cat or pt Cat , the corr esp onding pushout is the subca tegory ∧ (2) lying at the bo ttom of ∧ 2 in the diagram above: − 1 → a ← b → c ← 1 ∧ (2) . (20) But the relev ant fact is that a ca tegory X with ful l distinguis hed pushouts ‘b elieves’ that the squa re ab ov e is (also) a pushout. Explicitly , we have a - so to s ay - p ar a- universal prop erty of ∧ 2 : (a) given t wo cospans u , v : ∧ → X in a categ ory with full distinguished pushouts, with ∂ + 1 u = ∂ − 1 v , there is one pt- functor [ u , v ] : ∧ 2 → X such that [ u , v ] .k − = u and [ u, v ] .k + = v . Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 7 The c onc atenation map k : ∧ → ∧ 2 , (21) is an em b edding, alrea dy displayed ab ove by the labe lling of ob jects in ∧ 2 . As usua l in for mal ho motopy theor y , the functors ( − ) n i = ∧ i − 1 × − × ∧ n − i : pt Cat → pt Cat produce the higher s tructure o f the in terv al, for 1 6 i 6 n and α = ± 1 ∂ α i : ∧ n − 1 → ∧ n , ∂ α i ( t 1 , ..., t n − 1 ) = ( t 1 , ..., α, ..., t n − 1 ) , e i : ∧ n → ∧ n − 1 , e i ( t 1 , ..., t n ) = ( t 1 , ..., ˆ t i , ..., t n ) , s i : ∧ n +1 → ∧ n +1 , s i ( t 1 , ..., t n +1 ) = ( t 1 , ..., t i +1 , t i , ..., t n ) . (22) Moreov er, a c ting on (19) and k , these functor s yie ld the n-dimensional i-c onc ate- nation mo del ∧ ni 2 and the n-dimensional i-c onc atenation map k i : ∧ n → ∧ ni 2 ∧ n − 1 ∂ + i / / ∂ − i   ∧ n k − i   ∧ ni 2 = ∧ i − 1 × ∧ 2 × ∧ n − i , ∧ n k + i / / ∧ ni 2 k i = ∧ i − 1 × k × ∧ n − i : ∧ n → ∧ ni 2 . (23) Again, the squa re ab ov e is not a pushout, but X (having full distinguished pushouts) b elieves it is. 1.5. The sym metric pre-cubical category o f cospans A symmetric pr e-cubic al c ate gory A = (( A n ) , ( ∂ α i ) , ( e i ) , ( s i ) , (+ i )) , (24) is a symmetric cubica l set with co mpos itio ns, s atisfying the c o nsistency axioms (cub.1-2) o f I.1.2 , where the tr ansp ositions s i agree with the comp ositions + i (see I.2.3). Notice that we a re not (yet) assuming that the cubical comp o sitions + i behave in a categor ical wa y or satisfy interchange, in any sense, even weak. F or a c a tegory X with full distinguished pushouts, there is suc h a str ucture A = Cosp ∗ ( X ) . (Below, we will use a diﬀer en t no ta tion for the we ak cubic al c ate- gory C o s p ∗ ( X ) , a richer structur e whose comp onents ar e ca teg ories instead o f sets.) An n -cub e, or n-cubic al c osp an , is a functor u : ∧ n → X ; faces, degeneracies and transp ositions are co mputed according to the formulas (22) fo r the formal interv a l ∧ Cosp n ( X ) = Cat ( ∧ n , X ) = pt Cat ( ∧ n , X ) , ∂ α i ( u ) = u.∂ α i : ∧ n − 1 → X , ∂ α i ( u )( t 1 , ..., t n − 1 ) = u ( t 1 , ..., t i − 1 , α, ..., t n − 1 ) , e i ( u ) = u.e i : ∧ n → X , e i ( u )( t 1 , ..., t n ) = u ( t 1 , ..., ˆ t i , ..., t n ) , s i ( u ) = u.s i : ∧ n +1 → X , s i ( u )( t 1 , ..., t n +1 ) = u ( t 1 , ..., t i +1 , t i , ..., t n +1 ) . (25) The i -comp osition u + i v is computed on the i -concatenation mo del ∧ ni 2 (23), a s u + i v = [ u , v ] .k i : ∧ n → ∧ ni 2 → X ( ∂ + i ( u ) = ∂ − i ( v )) . (26) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 8 A symmet ric cubic al functor F : A → B betw een symmetric pr e -cubical cat- egories is a morphism of symmetric cubical sets whic h preserves all c o mpo sition laws. F or a n ordina ry (i.e., 1-cubical) co s pan u : ∧ → X , we write u = ( u − , u + ) : X − · → X + to sp ecify its cubical faces (notice the dot-ma rked a rrow). 1.6. The weak cubical category o f cospans Now, star ting from a category X with full distinguished pushouts, we ha ve a symmetric we ak cubic al c ate gory C osp ∗ ( X ) (as deﬁned in I.4), which is unitary (under the unitarity constra in t 1.3(i)-(ii) in X ) . It c onsists of the following data. (a) Our previo us Cosp ∗ ( X ) forms the symmetric pre-cubical categor y of cubical ob jects. (b) A tr ansversal n -map f : u → u ′ , is a natura l transformatio ns of n - cubes f : u → u ′ : ∧ n → X , or equiv a len tly a n n - cube in the pt-category X 2 of morphisms o f X (equipped with the coherent choice of distinguis hed pusho uts). T ra nsversal maps form a symmetric pre-cubical category Cosp ∗ ( X 2 ) , with: ∂ α i ( f ) = f .∂ α i : u∂ α i → u ′ ∂ α i : ∧ n − 1 → X ( i 6 n, α = ± 1) , e i ( f ) = f .e i : u.e i → u ′ .e i : ∧ n → X ( i 6 n ) , s i ( f ) = f .s i : u.s i → u ′ .s i : ∧ n +1 → X ( i 6 n ) , f + i g = [ f , g ] .k i : ∧ n → X 2 ( ∂ + i f = ∂ − i g ) . (27) Notice that an n -map should b e vie w ed as (n+1) -dimensiona l, a nd will also be called an (n+1)-c el l . (c) The symmetric pre-cubica l functors of tr ansversal fac es ( ∂ α 0 ) and tr ansversal de gener acy ( e 0 ) simply derive, con trav aria n tly , from the ob vious functors linking the categories 1 and 2 e 0 : Cosp ∗ ( X ) ← − − → ← − Cosp ∗ ( X 2 ) : ∂ α 0 ( ∂ α 0 : 1 − → ← − − → 2 : e 0 , α = ± ) . (28) (d) The comp osition hf : u → u ′′ of tr ansversally conse cutiv e n -maps is the com- po sition o f natural transforma tions. It is categor ical and preser ves the symmetric cubical structure. (e) The cubical c ompo sition laws b ehave catego rically up to suitable compa risons for asso cia tivit y and interc hange, which are in vertible sp e cial tr ansversal maps. (A transversal n -ma p f is said to b e sp e cial if its 2 n vertic es f ( α 1 , ..., α n ) are identities, for α i = ± ; see I.4.1.) These comparisons are explicitly constructed in I.4.4, after a study of the struc- ture of the models ∧ n (see I.3.5, I.3.6). 1.7. T runcation T r uncating everything a t cubical degr ee n, we g et the symmet ric we ak (n+1)- cubic al c ate gory n C osp ∗ ( X ) , which contains the k -cub es and k - maps of C o sp ∗ ( X ) for k 6 n (I.4.5). Indeed, its n -maps are ‘actually’ (n+1)-dimensiona l cells. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 9 In the 1 -truncated cas e 1 C o sp ∗ ( X ) = C osp( X ) there is only one cubical dir ection and no tr ansp osition, and w e drop the term ‘sy mmetric’: a we ak 2-cubic al c ate gory amounts, precise ly , to a weak (or pseudo) double catego ry , as studied in [ 9 ]-[ 12 ]: a structure with a str ic t ‘hor izontal’ comp o sition and a weak ‘vertical’ comp osition, under strict in terchange. The 2-trunca ted str ucture 2 C os p ∗ ( X ) , a symmetric quas i 3-cubical ca tegory , is related to Mo r ton’s constructions [ 21 ]: a ‘double bicategor y ’ of 2- cubical cospans. Lo osely speak ing, and starting with 2 C o sp ∗ ( X ) , one should omit the transp osition and restrict transversal maps to the sp e c ial ones. See also [ 1 ]. 2. Collarable and collared cospans After recalling c ol lar able (ordinar y) cospans, from Part I I, we deﬁne now c ol lar e d c osp ans, a v ariation of the former (already hin ted a t in II.2.2), where collar s are part of the data instead of just existing. W e b egin with the w eak double catego ry of p r e-c ol lar e d cospans p C c( T op ) and single out within the la tter the w eak dou- ble sub categor y C c( T op ) of colla red co s pans. Higher cubical degre e is deferred to Section 4. 2.1. T op olog ical e m b edding s. Let us r ecall that a top olo gic al emb e dding is a n injective map f : X → Y in T op , where the spa c e X has the pre-imag e top olog y . A clos ed injective map is necessarily an embedding, and will b e called a close d emb e dding . Op en embeddings ar e simila rly deﬁned, but not explicitly used here. It is ea sy to see that a pushout of top olog ical embeddings (re sp. clo sed embed- dings) in T op co ns ists of top olog ical em b eddings (re s p. closed embeddings ) and is also a pullback. Decomp osing a space X into a categor ical sum X = P X j (in T op ) amoun ts to giving a partition of the space X into a family of clop ens (closed and op en subspaces). 2.2. Collarable m aps Now, we r ecall from P ar t I I the construction of the w eak double subc a tegory C blc( T op ) ⊂ C osp( T op ) of collar able cospans. Motiv ations for these deﬁnitions hav e b een reca lled in the Intro duction. T o begin with, a c ol lar able map f : X → Y (II.2 .1) is a cont inuous mapping which can b e decomp osed in to a sum of t wo maps, so that: f = f 0 + f 1 : X 0 + X 1 → Y 0 + Y 1 ( c ol lar able de c omp osition ), (29) (i) f 0 : X 0 → Y 0 is a home omorphi sm , also ca lled a trivial ly c ol lar able , or 0-c ol lar able map, (ii) f 1 : X 1 → Y 1 is a 1-c ol lar able ma p, i.e. it admits a c ol lar F whic h extends it F : I X 1 → Y 1 , f 1 = F ( − , 0) : X 1 → Y 1 , (30) to the cylinder I X 1 = X 1 × [0 , 1] and is a close d emb e dding (2.1), with F ( X 1 × [0 , 1[) op en in Y 1 . Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 10 Both conditions ar e only satisﬁed b y id ∅ . Omitting empty comp onents, an irr e- ducible co llarable decomp osition is uniquely determined. Co llarable maps are also closed em b eddings; they are not closed under comp osition in T op (I I .2 .9). 2.3. Collarable cospans Limits a nd colimits in the weak double categor y C o sp( T op ) a re reca lled in I I.1.3. As deﬁned there, a s ub-c osp an of a top ologica l cospa n u = ( u − : X − → X 0 ← X + : u + ) , (31) is a regular subo b ject of u , i.e. an equa liser o f t wo 1-maps u → v in C osp( T op ) . It amounts to assigning three subspaces ( Y − , Y 0 , Y + ) such that Y t ⊂ X t , u α ( Y α ) ⊂ Y 0 ( t ∈ ∧ ; α = ± ) . (32) W e say that the sub-cospan is op en (resp. close d ) in u if so are the three subspaces Y t ⊂ X t . The sub-cospans o f u form a complete lattice, which is a subla ttice of P ( X − ) × P ( X 0 ) × P ( X + ) . Thu s, to give a decomp osition u = P u j int o a sum o f co spans amounts to give a clop en p artition ( u j ) o f u , i.e. a cov er o f u b y disjoint sub-cos pans, close d and op en in u (II.1.3 ). A c ol lar able c osp an (I I.2.2 ) o f top olo gical spaces is a top ologica l cos pan u : X − · → X + which admits a c ol lar able de c omp osition , i.e. can be decomposed in a binary sum, where u = u 0 + u 1 = ( X − 0 + X − 1 → X 0 0 + X 0 1 ← X + 0 + X + 1 ) , u 0 = ( u − 0 : X − 0 → X 0 0 ← X + 0 : u + 0 ) , u 1 = ( u − 1 : X − 1 → X 0 1 ← X + 1 : u + 1 ) , (33) (i) u 0 = ( u − 0 , u + 0 ) is a p air of home omorphisms , also calle d a trivial ly c ol lar able , or 0-c ol lar able cospan, (ii) u 1 = ( u − 1 , u + 1 ) is a 1-c ol lar able c osp an ; by this we mean that it admits a c ol lar c osp an (or, simply , a c ol lar ), i.e. a cos pan ( U − , U + ) formed o f a pa ir o f collar s of its maps having disjoint images . In other words, we have tw o disjoint closed embeddings U = ( U − : I X − 1 → X 0 1 ← I X + 1 : U + ) , u α 1 = U α ( − , 0) : X α 1 → X 0 , (34) where U α ( X α 1 × [0 , 1[) is open in X 0 1 . Both conditions are o nly satisﬁed by the empty cospan e 1 ( ∅ ) . O mitting empt y comp onents, an irr e ducible collarable decompositio n is uniquely determined. The maps u α are also closed em be dding s; in the 1-collarable case, they ha ve dis jo in t images. Cubical faces and degeneracy a re inherited from C osp( T op ) ∂ α 1 u = X α , e 1 ( X ) = (id : X → X ← X : id) ( α = ± 1) . (35 ) W e have proved (Thm. I I.2.4) that c o llarable cos pans are clo s ed under concatena - tion: given a consecutive co llarable co span v : Y − · → Y + (with X + = A = Y − ) , the cospan w = u + 1 v deco mpo ses in a trivial part, computed b y a pushout of homeomorphisms (over the clop en subspace X + 0 ∩ Y − 0 of A on which u and v are b oth 0-colla rable), and a 1 - collarable par t, computed by a 1-c ol lar able pushout (over Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 11 the co mplemen t clop en subs pa ce o f A ) . The latter pusho ut is home omorph ic to a standard homotopy pushout (Thm. I I.2.5 ). T op olo gical spa c es, co llarable cospans and their transversal maps form thus a transversally full weak do uble subca tegory C blc( T op ) ⊂ C osp( T op ) . (36) Notice that a to p olo gical ma p, even if colla rable, ca n not b e viewed as a colla rable cospan, generally: the categ ory T op is tr ansversal ly emb e dde d in C blc( T op ) , sending a map f : X → Y to the same 0-map. 2.4. Pre-collared cospans A pr e-c ol lar e d (top olo gic al) c osp an U = ( u ; U − , U + ) will consist of a co s pan u = ( u − : X − → X 0 ← X + : u + ) o f t op olo gic al emb e ddings equipped with t wo pr e- c ol lars U α ; these are maps which ex tend to the cylinder the co rresp onding maps u α of u I X − U − / / X 0 I X + U + o o U α : I X α → X 0 , X − d − O O u − ; ; v v v v v v v v X + d − O O u + c c H H H H H H H H u α ( x ) = U α ( x, 0) . (37) Notice that these pre-collars need not b e injectiv e, nor have disjoint ima g es, a nd we are not (yet) as suming the existence o f a ‘collar ed decomp ositio n’. The underlying c osp an o f U is | U | = u. F ac e s a nd degenera cies ar e deﬁned as follows, consistently with the pro cedure | − | : ∂ α U = ∂ α u = X α , e 1 ( X ) = ( X → X ← X ; eX : I X → X ← I X : eX ) . (38) Pre-co llared cospans form a 1-cubical set, with topo logical spaces in deg r ee 0. Every co llarable cospa n u = ( u − : X − → X 0 ← X + : u + ) underlie s so me pre- collared cos pan U . In fact, if u is 1 -collara ble, it admits a collar-c o span (2.3); if u is 0 -collara ble, i.e., a pair o f homeomo r phisms, then it admits a trivial pr e-c ol lar , similar to a deg enerate one, with U α = u α .eX α : I X α → X α → X 0 . In the g eneral case, u ha s a collar able decomp ositio n (33 ), and it suﬃce s to take the top olog ical sum of the previous solutions - also b ecause the cylinder functor preser ves s ums . 2.5. The weak double category of pre-collared cospans Let us supp ose we hav e tw o pre -collared cospans, U = ( u ; U α ) as ab ov e (in (37)) and V = ( v ; V − , V + ) , v = ( v − : Y − → Y 0 ← Y + : v + ) , (39) which ar e consecutive: X + = A = Y − . As in the collara ble case (Part I I), to deﬁne their c onc atenation W = U + 1 V , we concatenate their underlying cos pans getting a cospan w = u + 1 v (of top ological embeddings, by 2 .1) and then for m the n ew Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 12 pre-collar s using t wo of the o ld o nes, namely U − and V + Z 0 U + 1 V = ( w ; W α ) , C C { { w = u + 1 v I X − U − / / X 0 h − = = z z z z z z z Y 0 h + a a C C C C C C C I Y + V + o o = ( h − u − , h + u + ) , W − = h − U − , X − d − O O u − < < z z z z z z z z A u + ` ` B B B B B B B B v − > > } } } } } } } } Y + d − O O v + a a D D D D D D D D W + = h + V + , (40) A tr ansversal map of pre-colla red cospans f = ( f − , f 0 , f + ) : U → V , f t : X t → Y t ( t ∈ ∧ ) , (41) is a triple o f maps f t which commute with the pre -collars of U , V , via their cylindr ical extensions F α = I f α : I X α → I Y α . The underlying tr ansversal map | f | : | U | → | V | has the same compone nts f t . F ac es, degener acies and transp ositions of transversal maps are deﬁned in the obvious way , co nsistently with domains a nd co domains. Finally , the comparison for asso ciativity is provided by the homolo gous comparis on fo r the conca tenation of the underlying cos pans in the weak double ca teg ory C o s p( T op ) , which is easily seen to satisfy the condition of consistence with pre-collars. W e have thus deﬁned the we ak double c ate gory p C c( T op ) of pr e-c ol lar e d c osp ans and equipped it with a forgetful do uble functor | − | : p C c( T op ) → C osp( T op ) , (42) which is the identit y o n ob jects and faithful o n tra ns v er s al ma ps . (Concatena tion is strictly preserved, since we a r e using one choice of distinguished pusho uts in T op to concatenate cospans, in both structure s .) 2.6. Collared cospans Let U = ( u ; U α ) b e a pre-collared cospan, as in (37). W e say that U is c ol lar e d if u can be decomp osed into a (uniquely deter mined) binar y sum, where: u = u 0 + u 1 = ( P u − i : P X − i → P X 0 i ← P X + i : P u + i ) ( i = 0 , 1) , (4 3) U α ( I X α 0 ) ⊂ X 0 0 , U α ( I X α 1 ) ⊂ X 0 1 , (44) so tha t the restrictions ( U − i , U + i ) fo r m a pre-collar cospan of u i ; more precis ely , we wan t tha t: (i) u 0 = ( u − 0 , u + 0 ) is a p air of home omorphisms a nd U α 0 = u α 0 .eX α 0 : I X α 0 → X α 0 → X 0 0 ; (ii) the pair ( U − 1 , U + 1 ) is a 1-c ol lar of u 1 = ( u − 1 , u + 1 ) , i.e. it cons is ts of tw o disjo int closed em b eddings (extending the maps u α 1 ) and U α 1 ( X α 1 × [0 , 1[) is op en in X 0 1 . Collared co spans form the transversally full we ak double su b c ate gory C c( T op ) ⊂ p C c( T op ) . The fact that they are closed under c o ncatenation in the latter is prov ed as in Part I I for collarable cospans (Thm. I I.2.4 ). Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 13 W e have thus a comm utative tria ngle of double functors C c( T op )   / / ( ( Q Q Q Q Q Q Q Q Q Q Q p C c( T op ) |−|   C osp( T op ) (45) which ar e transversally faithful (the inclusion is a lso tr ansversally full). 3. Pre-collars in cubical degree 2 W e deﬁne here the weak 3 -cubical categor y 2p C c ∗ ( T op ) of 2-cubica l pre- collared cospans. The collared case will be trea ted dir e c tly in unbounded cubica l deg ree, in the next section. 3.1. Notation for cubical cospans Let t = ( t 1 , ..., t n ) ∈ ∧ n be a multi-index with co ordina tes t i ∈ {− 1 , 0 , 1 } . If t i 6 = 0 , we write t ♯i = ( t 1 , ..., t i − 1 , 0 , t i +1 , ..., t n ) , (46) the po in t of ∧ n obtained b y annihilating the i -th co ordinate. A cubical cospan u : ∧ n → T op will b e written as follo ws u = ( X ( t ) , u ( i, t )) , u ( i, t ) : X ( t ) → X ( t ♯i ) ( i = 1 , ..., n ; t ∈ ∧ n ; t i 6 = 0) , (47) where u ( i, t ) is a map in dir ection i, as exempliﬁed below for n = 1 , 2 X − u (1 − ) / / X 0 X + u (1+) o o (48) X −− u (1 −− ) / / u (2 −− )   X 0 − u 0 −   X + − u (1+ − ) o o u (2+ − )   X − 0 u − 0 / / X 00 X +0 u +0 o o • 1 / / 2   X − + u (1 − +) / / u (2 − +) O O X 0+ u 0+ O O X ++ u (1++) o o u (2++) O O (49) Note, in the latter , the simpliﬁed notatio n of the central arrows: u − 0 = u (1 , − , 0) , and so on. F or a ﬁxed i, ther e are 2 . 3 n − 1 maps u ( i, t ) in dir ection i. Globally , there are 2 n. 3 n − 1 maps in u and n . 3 n − 1 cospans. Of course, when w e sp eak of a c osp an of u we always mean tw o maps with the same co domain and the same dir e ction , i.e. a pair ( u ( i, t )) , with t i = ± 1 (the other c o o rdinates being ﬁxed, as well a s i ). Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 14 An n -cubical cospa n u ca n also b e view ed as a c ospan (in direction 1 ) o f ( n − 1)- cubical cospans (in directions 2 , ..., n ) u : ∧ → T o p ∧ n − 1 , u = ( u − : X − → X 0 ← X + : u + ) , X t = ( X ( t, t ) , u ( i, t, t )) , u ( i, t, t ) : X ( t, t ) → X ( t, t ♯i ) , (50) where i = 1 , ..., n − 1 and t = ( t 2 , ..., t n ) ∈ ∧ n − 1 , t i 6 = 0 . (Other presentations of u ca n be obtained from the tr ansp osed cospans us i , using the tr a nsp o sition symmetries s i : ∧ n → ∧ n , cf. 1.4.) 3.2. Square cos pans and pre-collars Let u : ∧ 2 → T op b e a 2-cubical to polo gical cospan, with the previous notation (in (49)). As considered ab ov e (in degree n ), u will a lso be viewed as a cospa n (in directio n 1) of cospans (in direction 2), as follows u : ∧ → T op ∧ , u = ( u − : X − → X 0 ← X + : u + ) , (51) X t = ( u (2 , t, − ) : X t − → X t 0 ← X t + : u (2 , t, +)) , u α = ( u (1 , α, − ) , u (1 , α, 0) , u (1 , α, +)) ( t ∈ ∧ , α = ± ) . (52) The symmetric presentation of u can b e obtained as a bove from the trans po s ed cospan s 1 ( u ) = us : ∧ 2 → T op , using the transpos itio n symmetry s : ∧ 2 → ∧ 2 (1.4). Let us assume that a ll maps of u are top ologica l embeddings . A family of pr e- c ol lars of u in dir e ction 1 co nsists of six maps U (1 , α, t ) : I X αt → X 0 t ( α = ± , t ∈ ∧ ) , (53) I X −− U (1 −− ) / / I u (2 −− )   X 0 − u 0 −   I X + − U (1+ − ) o o I u (2+ − )   I X − 0 U − 0 / / X 00 I X +0 U +0 o o • 1 / / 2   I X − + U (1 − +) / / I u (2 − +) O O X 0+ u 0+ O O I X ++ U (1++) o o I u (2++) O O (54) so that: (i) ea ch of them is a pre-colla r of the cor resp onding map u (1 , α, t ) , i.e. u (1 , α, t ) = U (1 , α, t )( − , 0) (2.4); (ii) the diagram (54) comm utes and its four squares are pullbac ks. Symmetrically , a family of pr e-c ol lars of u in dir e ction 2 amo un ts to the previous notion for the transpo sed 2-cospan s 1 ( u ) = us. It consists thus of six maps U (2 , t, α ) : I X tα → X t 0 ( α = ± , t ∈ ∧ ) , (55) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 15 which satis fy symmetric conditions. In particular , they form four pullbacks I X −− I u (1 −− ) / / U (2 −− )   I X 0 − U 0 −   I X + − I u (1+ − ) o o U (2+ − )   X − 0 u − 0 / / X 00 X +0 u +0 o o • 1 / / 2   I X − + I u (1 − +) / / U (2 − +) O O I X 0+ U 0+ O O I X ++ I u (1++) o o U (2++) O O (56) A pr e-c ol lar e d top olo gic al 2-c osp an U = ( u ; U (1 , α, t ) , U (2 , t ′ , β )) will be a 2- cospan u : ∧ 2 → T op of top olog ic al embeddings, equipp e d with a fa mily of pr e- c ol lars , i.e. a pa ir of suc h families in b oth directions. Its underlying 2-c osp an is | U | = u . One can note that, bec ause of the pullback condition (ii), a ll the pre-co llars are determined by the c ol lar-cr oss , consisting of the 4 c entr al c ol lars , i.e. those which reach the central ob ject I X 0 − U 0 −   I X − 0 U − 0 / / X 00 I X +0 U +0 o o • 1 / / 2   I X 0+ U 0+ O O (57) 3.3. F aces and d egeneracies The faces of the pr e -collared 2 -cospan U are the fo llowing pre-collar ed 1-cos pans ∂ α 1 U = ( ∂ α 1 u ; U (2 , α, − ) : I X α − → X α 0 ← I X α + : U (2 , α, +)) , ∂ β 2 U = ( ∂ β 2 u ; U (1 , − , β ) : I X − β → X 0 β ← I X + β : U (1 , + , β )) . (58) F or instanc e , the tw o faces ∂ α 1 U can be pictur ed a s follo ws: Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 16 ( ∂ − 1 U ) ( ∂ + 1 U ) I X − + I X ++ X − 0 X +0 I X −− I X + − X 00 I X 0+ I X 0 − U ( − +) U 0+ U (++) U ( −− ) U 0 − U (+ − ) / / o o / / o o / / o o O O O O O O       (59) The dege neracies e 1 U , e 2 U of a pr e - collared 1-co span U = ( u ; U α ) are deﬁned as follows e 1 U = ( e 1 u ; eX t , U β ) , e 2 U = ( e 2 u ; U α , eX t ) . (60) In particular , e 1 U has pre-colla rs for ming the following tw o diagr a ms of pullbacks I X − e / / I u −   X − u −   I X − e o o I u −   I X − id / / U −   I X − U −   I X − id o o U −   I X 0 e / / X 0 I X 0 e o o X 0 id / / X 0 X 0 id o o • 1 / / 2   I X + e / / I u + O O X + u + O O I X + e o o I u + O O I X + id / / U + O O I X + U + O O I X + id o o U + O O (61) 3.4. Concatenating pre-col lared 2-cos pans Theorem 1. Pr e-c ol lar e d 2-c osp ans have wel l-deﬁne d c onc atenation in b oth dir e c- tions, c onsistently with their underlying 2-c osp ans. Pr o of. It suﬃces to prov e t wo p oints: (A) F a milies of pre-collars in dir ection 1 c a n be concatena ted in dir e ction 1, (B) F amilies o f pr e-collars in direction 2 ca n b e conca tenated in directio n 1 . Poin t (A) is proved as in Thm. I I.2.4 , working in T op ∧ instead of T op (cf. (51)). W e only have to check that the new s quares built by pushouts are indeed pullbacks. This is done in the following lemma, 3.5. F or p oint (B), let us a ssume that u, v are equipp ed with pre-co llars in direction 2, and prove tha t their co nca tenation w = u + 1 v can b e ca nonically so equipp ed. Below, ‘in direction 2’ is understo o d most of the time. Consider the cospan ( h ; H − , H + ) = ∂ + 1 u = ∂ − 1 v along which the concatena tion is computed. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 17 The 1-faces of w (whose ma ps are in directio n 2 ) b elong to u or v w (2 , − , α ) = u (2 , − , α ) , w (2 , + , α ) = v (2 , + , α ) , (62) and are already equipped with pre-c o llars ( U (2 , − , α )) a nd ( V (2 , + , α )) . F urther more, we have a central co s pan ( w 0 − , w 0+ ) = ( w (2 , 0 , − ) , w (2 , 0 , +)) which is computed in the left diagram below, via thr ee pushouts Y 0 − y − / / v 0 −       Z 0 − w 0 −   I Y 0 − I y − / / V 0 −       I Z 0 − W 0 −   A − u − / / v − ; ; w w w w h −   X 0 − x − : : u u u u u 0 −   I A − I u − / / I v − 9 9 t t t t H −   I X 0 − I x − 9 9 r r r r U 0 −   Y 00 y 0 / / _ _ _ _ _ Z 00 Y 00 y 0 / / _ _ _ _ _ _ _ Z 00 A 0 u 0 / / v 0 ; ; w w X 00 x 0 : : u u u u A 0 u 0 / / v 0 9 9 t t X 00 x 0 9 9 r r r r Y 0+ y + / / _ _ _ _ _ v 0+ O O     Z 0+ w 0+ O O I Y 0+ I y + / / _ _ _ _ _ _ V 0+ O O     I Z 0+ W 0+ O O A + u + / / v + ; ; w w h + O O X 0+ u 0+ O O x + : : u u u u I A + I u + / / I v + 9 9 t t H + O O I X 0+ U 0+ O O I x + 9 9 r r r r (63) Here, the notation of some maps is simpliﬁed: u α = u (1 , α, +) , v α = v (1 , α, − ) , h α = u (2 , + , α ) = v (2 , − , α ) , and similarly for their collars, denoted by the corresp onding capital letters. Now, for this last co span ( w 0 − , w 0+ ) . we hav e to construct pre-colla rs, co ns is- ten tly with the previous c o llars ( U (2 , − , − ) , U (2 , − , + )) and ( V (2 , + , − ) , V (2 , + , +)) . The cylinder functor I : T op → T op preserves pushouts, a s a left adjo in t. There- fore, the righ t diagram ab ov e a llows us to deﬁne - co ns isten tly - the vertical arrows ( W 0 − , W 0+ ) , on the basis of the consistent co llars a ppea ring in the other vertical arrows. They are also topo lo gical em b eddings, and the ne w squares built in the right diagr am ab ov e are pullbacks, aga in by 3.5. 3.5. A di agrammatic lem ma W e end this section with a dia grammatic pr o per t y , which has already b een used in the pro o f of the previous theorem. (In that cas e, the b ottom square of the diagram below is a pushout of embedding s , whic h is also a pullbac k.) Lemma 1 . L et us supp ose we have, in T op , a c ommutative cubic al diagr am of top olo gic al emb e ddings: Y / /       Z   A / / ; ; w w w w   X ; ; v v v v   Y ′ / / _ _ _ _ _ _ Z ′ A ′ / / ; ; x x X ′ ; ; w w w w (64) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 18 If t he fr ont and b ott om squar es a r e pul lb acks, and the top squar e is a pushout, then also the b ack squ ar e is a pul lb ack. Pr o of. Since our maps a r e top olog ical embeddings , we can forg e t top ologies and work in the category o f se ts . The pa s ting of the fro n t a nd b ottom squares is a pullback. T he r efore, the pasting of the top and back squar es is also a pullback (the same). W e hav e thu s a commutativ e diagram of sets A / /   Y / /   Y ′   _ _   X / / Z / / Z ′ (65) where the left square is a pusho ut and the o uter rectang le a pullback. Knowing that all maps a re injective, it is quite easy to chec k, on elements, that the right square is also a pullbac k. 4. Collared cubical cospans W e extend the previous constructions to unbounded cubical degr e e, constructing the symmetric weak cubical ca teg ory of pre- collared cubical cos pa ns p C c ∗ ( T op ) and its w eak cubical sub category C c ∗ ( T op ) of collared cubical cospans. 4.1. Pre-collared cubical cospans W e extend now the previous deﬁnitions (2 .4, 3.2) to higher cubical degree , using the notation for cubical cospans in tro duced in 3.1. A pr e-c ol lar e d n-cu bic al c osp an U = ( u ; U ( i , t )) consists of an n -cubical cospan u = | U | of top ologica l embeddings , equipp ed with a family o f 2 n. 3 n − 1 pr e-c ol lars U ( i, t ) of its ma ps u = | U | = ( X ( t ) , u ( i, t )) , U ( i, t ) : I X ( t ) → X ( t ♯i ) , u ( i, t )( x ) = U ( i, t )( x, 0) ( i = 1 , ..., n, t = ( t 1 , ..., t n ) ∈ ∧ n , t i 6 = 0) . (66) Moreov er, the fo llowing s quares m ust (commute a nd) b e pul lb acks I X ( t ) I u ( i, t ) / / U ( j, t )   I X ( t ♯i ) U ( j, t ♯i )   _ _ _ _   ( i 6 = j, t i 6 = 0 , t j 6 = 0) . X ( t ♯i ) u ( j, t ♯j ) / / X ( t ♯i♯j ) (67) F ac es a nd degener acies a re deﬁned as follows, for ming a sy mmetric cubica l set (consistently with the forg etful pro cedure | − | ) , where i ♯j is i if i < j, and i + 1 if i > j ∂ α j U = ( ∂ α j u, U ( i ♯j , ∂ α j t )) ( i = 1 , ..., n − 1; t ∈ ∧ n − 1 ) , e j ( U ) = ( e j u, e j ( U )( i, t )) , e j ( U )( i, t ) = U ( i, e j t ) ( i 6 = j ) , e i ( U )( i, t ) = eX ( t ) : I X ( t ) → X ( t ) , (68) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 19 ( ∂ α j t , e j t are deﬁned in (2 2).) 4.2. A weak cubical category W e can now form the symmetric weak cubical categ ory p C c ∗ ( T op ) o f pr e-c ol lar e d cubic al c osp ans . The op eratio n of i -conca tenation of pre-colla red n -cubical cos pans is deﬁned as in 3.4 for n = 2: p oint (A) and (B) sp ecify , re spec tively , how to construct the new collars in direction i and j 6 = i. A tr ansversal map of pre-colla red n -cubical cospans , also called an ( n + 1)-c e ll, f : U → V , U = ( u, U ( j, t )) , V = ( v , V ( j, t )) , ( 69 ) is a transversal map | f | : | U | → | V | o f the underly ing n -cubical cospa ns w hich commutes with the collars , via the cylindrical extensions F ( t ) = I f ( t ) : I X ( t ) → I Y ( t ) ( t ∈ ∧ n ) . (70) F ac es, degener acies and transp ositions of transversal maps are deﬁned in the obvious way , consistent ly with domains and co domains. The compar isons for asso- ciativity and interc hange derive from the comparis ons of C osp ∗ ( T op ). W e have a lso deﬁned a c ubica l forgetful functor | − | : p C c ∗ ( T op ) → C o sp ∗ ( T op ) , (71) which is tra nsversal ly faithful : given t wo transversal maps f , g : U → V , the c o ndition | f | = | g | implies f = g . 4.3. Cubical col lared cospans Let U = ( u , U ( i , t )) b e a pr e-collared n -cubical cospans. W e say that U is c ol lar e d in dir e ction i (= 1 , ..., n ) if the under lying cubica l cospan u can be decomp osed into a binar y sum, coher ent ly with its pre-colla rs u ( i, t ) = u 0 ( i, t ) + u 1 ( i, t ) , u j ( i, t ) : X j ( t ) → X j ( t ♯i ) ( j = 0 , 1; t i 6 = 0) , (72) U ( i, t )( I X 0 ( t )) ⊂ X 0 ( t ♯i ) , U ( i, t )( I X 1 ( t )) ⊂ X 1 ( t ♯i ) , (73) so that the restrictions U 0 ( i, t ) : I X 0 ( t ) → X 0 ( t ♯i ) , U 1 ( i, t ) : I X 1 ( t ) → X 1 ( t ♯i ) , (74) yield a collared decompo sition of each or dinary cospan ( u ( i, t ); t i = ± ) . By 2.6, this means that: (i) each cospa n ( u 0 ( i, t ); t i = ± ) is a p air of home omorphisms with trivial collar s U 0 ( i, t ) = u 0 ( i, t ) .eX 0 ( t ) : I X 0 ( t ) → X 0 ( t ) → X 0 ( t ♯i ); (ii) the pair ( U 1 ( i, t ); t i = ± ) is a 1-c ol lar of ( u 1 ( i, t ); t i = ± ) , i.e. it co nsists o f tw o disjoint closed embeddings (extending the maps u 1 ( i, t )) and U 1 ( i, t )( X 1 ( t ) × [0 , 1[) is open in X 1 ( t ♯i ). W e say that U is c ol lar e d if it is collared in ev ery direction. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 20 4.4. Concatenating coll ared cubical cospans Theorem 2. Col lar e d cubic al c o-sp ans ar e stable under c onc atenation in al l dir e c- tions, in p C c ∗ ( T op ) . Pr o of. Since tr a nspo sitions p ermute directions, it suﬃces to cons ide r a c o ncatena- tion W = U + 1 V in direction 1 and prov e that: (A) The new cospans in dire ction 1 pro duced b y 1-concatenation a re co lla red in direction 1, (B) The same are co llared in dir ection 2. Poin t (A) is proved a s in Thm. I I.2.4, working in T op ∧ n − 1 instead of T op . F or point (B), w e use the no tation of 3.4, and we only write t he indic es in dir e c- tions 1, 2 (which amoun ts to working in T op ∧ n − 2 ) . Co llared will mean: co llared in direction 2. Consider the cospa n H = ( h ; H − , H + ) = ∂ + 1 U = ∂ − 1 V along which the concate- nation is computed, with collar-pair H − : I A − → A 0 ← I A + : H + , H = H 0 + H 1 . (75) (B ′ ) Fir st, let us c o nsider the case h 6 = e 1 ( ∅ ), so t hat in the co llared decomp osi- tion h = h 0 + h 1 bo th comp onents are non t riv ial. Accordingly , we can split the concatenation w int o the sum w = w 0 + w 1 = ( u 0 + 1 v 0 ) + ( u 1 + 1 v 1 ) , (76) of the concatenations of the 0- a nd 1-collar ed parts (in directio n 2). Plainly , the ﬁrst component is 0-co llared. W e can f or get it , assuming that u , v are 1- c ollared and prov e that w = u + 1 v is also. W e pro ceed now as in the pro of of 3.4 , p oint (B). The 1-faces of w (see (62)) are already 1-colla red, and we cons tr uct the new pr e-collars ( W 0 − , W 0+ ) as in (63). W e end po in t (B ′ ) proving that the latter are indeed 1-collar s. Some length y calculations are needed to show that they ar e disjoint; essentially , this dep ends not o nly on the fact that the old collar s are disjoint, but also on the pullback-h yp othesis (67). First, s ince the to p and b ottom sq uares are pushouts Im( W 0 α ) = Im( W 0 α .I x α ) ∪ I m ( W 0 α .I y α ) = x 0 (Im U 0 α ) ∪ y 0 (Im V 0 α ) . (77) W e hav e thus to cons ider four in terse ctions; b y symmetry , it is s uﬃcien t to prove that: x 0 (Im U 0 − ) ∩ x 0 (Im U 0+ ) = ∅ , x 0 (Im U 0 − ) ∩ y 0 (Im V 0+ ) = ∅ . (78) The ﬁrst fact is obvious, b ecaus e x 0 is injective a nd the collar s U 0 α are disjoin t. F or the seco nd, we w ill use the fact that the squar e of ( u 0 , v 0 , x 0 , y 0 ) is a pullback (as a pushout of em b eddings) and the square ar ound H − is a lso (by hypothesis ). Therefore: x 0 (Im U 0 − ) ∩ y 0 (Im V 0+ ) ⊂ x 0 (Im U 0 − ) ∩ Im y 0 = x 0 (Im U 0 − ∩ Im u 0 ) = x 0 (Im( u 0 H − )) = Im( x 0 u 0 H − ) , Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 21 and symmetrically x 0 (Im U 0 − ) ∩ y 0 (Im V 0+ ) ⊂ Im( y 0 v 0 H + ) . W e co nc lude noting that x 0 u 0 = y 0 v 0 , and that the collars H α are disjoin t. (B ′′ ) Finally , let us ex a mine the degenera te case h = e 1 ( ∅ ) . If, in dir e ction 2 , the cospans u and v ar e b oth 0 -collared or b oth 1-c o llared, we come back to the previo us argument. But here one of them can b e 0-colla red, say u , a nd the o ther 1- collared. Then the cospan ( w 0 − , w 0+ ) is the sum Y 0 − y − / / v 0 −       Z 0 − w 0 −   ∅ / / ; ; x x x x   X 0 − x − 9 9 s s s s u 0 −   Y 00 y 0 / / _ _ _ _ _ _ _ Z 00 ∅ / / ; ; x x X 00 x 0 9 9 s s s s Y 0+ y + / / _ _ _ _ _ _ v 0+ O O     Z 0+ w 0+ O O ∅ / / ; ; x x O O X 0+ u 0+ O O x + 9 9 s s s s (79) of a 0-collar ed comp onent ( u 0 − , u 0+ ) and a 1-c ollared comp onent ( v 0 − , v 0+ ) , which means that it is collared. 4.5. The structure of collared cubical cospans W e ca n no w deﬁne the tr ansversal ly ful l weak cubical subca teg ory of c ol lar e d cubic al c osp ans C c ∗ ( T op ) ⊂ p C c ∗ ( T op ) . (80) Its n -c ubes are the co llared n -cospa ns , as deﬁned ab ov e. Its transversal maps ar e all the natural transformations f : u → u ′ : ∧ n → T op b etw een collared n - cospans. These data are pla inly close d under faces and deg eneracies. They are also clo sed under concatenation in an y cubical dir ection, as proved in 4.4. The comparisons of the weak structure are inherited fro m p C c ∗ ( T op ) , since they are (inv ertible sp ecial) transversal maps - and we are taking all of them betw een colla red cub es. 5. Cylindrical degeneracies and cylindrical concatenation W e deﬁne a new framework, C OSP ∗ ( T op ) , fr om which we will abs tract the notion of a symmetric quasi cubical category (Sectio n 7). 5.1. Comments W e come back to arbitrar y topo logical cospans a nd b egin a diﬀerent constr uction, which do es not use colla rs but cylindric al degener a cies and cylindric al concatena- tions (by ho motopy pushouts). Notice tha t the latter grants, by itself, a homotopy- inv a riant co ncatenation. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 22 With res pect to the symmetric weak cubical categor y C osp ∗ ( T op ) , the new struc- ture C O SP ∗ ( T op ) diﬀers with res pect to dege ne r acies, concatenatio ns and c o mpar- isons, and satisﬁes weak er a xioms: degeneracies b ehave now in a weak er way a nd - for ins ta nce - ar e just lax ident ities. W e get thus a symmetric quasi cubical cate- gory , as deﬁned in Section 7. (The imp o rtance of weak units in homo topy theory is discussed in [ 18 ] and references therein.) With resp ect to C c ∗ ( T op ) , the new fra mew or k is simpler a nd can eas ily be restricted to manifolds with fac es and their cob or disms (see the next section). This is actually the main r eason for using here cy lindrical deg eneracies instead of the ordinary ones - which would also give lax iden tities, with respect to cylindrical concatenation. 5.2. Cylindrical dege neracies Let us come back to the symmetric weak cubical categor y C osp ∗ ( T op ) , recalled in Section 1. After the or dinary degener acy e 1 ( X ) = (id X : X → I X ← X : id X ) of a space X , we also ha ve a cylindric al de gener acy : E 1 ( X ) = ( d − : X → I X ← X : d + ) , d − ( x ) = ( x, 0) , d + ( x ) = ( x, 1) . (81) One degree up, a cospan u = ( u − : X − → X 0 ← X + : u + ) has t wo cylindr ical degeneracies , E 1 ( u ) and E 2 ( u ) = E 1 ( u ) .s (with ∂ α i E i ( u ) = u ) X − d − X − / / u −   I X − I u −   X − d + X − o o u −   X 0 u − / / d − X −   X 0 d − X 0   X + u + o o d − X +   X 0 d − X 0 / / I X 0 X 0 d + X 0 o o I X − I u − / / I X 0 I X + I u + o o • 1 / / 2   X + d − X + / / u + O O I X + I u + O O X + d + X + o o u + O O X − u − / / d + X − O O X 0 d + X 0 O O X + u + o o d + X + O O E 1 ( u ) E 2 ( u ) = E 1 ( u ) .s. (82) While ordina ry degenera cies satisfy the cubical relation e 1 e 1 = e 2 e 1 (1.1), the 2-cospa ns E 1 E 1 ( X ) and E 2 E 1 ( X ) = E 1 E 1 ( X ) .s are diﬀerent, since E 1 E 1 ( X ) is not symmetric X d − / / d −   I X I d −   X d + o o d −   I X d − I / / I 2 X I X d + I o o • 1 / / 2   X d − / / d + O O I X I d + O O X d + o o d + O O (83) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 23 How ever, the tra nspo sition symmetry s : I 2 → I 2 induces an in v ertible s pecia l compariso n σ 1 X : E 1 E 1 ( X ) → E 2 E 1 ( X ) ( E 2 E 1 = s 1 E 1 E 1 ) , (84) which repla ces the or dinary c ubical relations for degenera c ie s. In general, a n n -cospan u = ( X ( t ) , u ( j, t )) (with the notation of 3 .1) has an i -directed cylindric al degenerac y (for i = 1 , ..., n ) , which is an ( n + 1 )-cospan: E i ( u ) = ( X ( e i t ) , E i ( u )( j, t )) , ( j = 1 , ..., n + 1; t ∈ ∧ n +1 ) , E i ( u )( j, t ) = u ( j ♯i , e i t ) ( j 6 = i ) , E i ( u )( i, t ) = d α X ( e i t ) : X ( e i t ) → I X ( e i t ) ( α = t i 6 = 0) , (85) where e i : ∧ n +1 → ∧ n omits the i -th co ordinate (see (22)) a nd w e let: j ♯i = j ( f or j 6 i ) , j ♯i = j − 1 ( f or j > i ) . (86) Finally , we hav e an invertible sp ecial symmetry c omp arison for cylindric al de gen- er acies . F o r every n -c ub e u, we have a n inv ertible sp ecial ( n + 2 )-map σ 1 ( u ) , which is natural on n -maps a nd has the follo wing faces σ 1 u : E 1 E 1 ( u ) → E 2 E 1 ( u ) ( symmetry 1-c omp arison ), ∂ α 1 σ 1 ( u ) = ∂ α 2 σ 1 ( u ) = id( E 1 u ) , ∂ α j +2 σ 1 ( u ) = σ 1 ( ∂ α j u ) . (87) Via transp ositions, σ 1 generates all the other symmetry compar isons E j .E i → E i +1 .E j ( j 6 i ). 5.3. W eak equiv alences and homotopy inv ariance Extending a prev ious deﬁnition on ordinar y topolo gical cospans (I I .2 .8), we say that a transversal n -map f : u → v betw een cubical top olog ical cospans is a we ak e quivalenc e if it is sp ecial (1.6) and all its comp o nen ts are homotopy eq uiv alences (in T op ) . Thu s, for every cubical cospan u = ( X ( t ) , u ( j, t )) , we hav e an obvious w eak equiv a lence p i ( u ) : E i ( u ) → u , (88) consisting of the sp ecial transversal map whose non- ide ntit y compo nen ts are the pro jections I X ( e i t ) → X ( e i t ). In c ubical degree 1 , p 1 : E 1 ( X ) → e 1 ( X ) has com- po nen ts (1 , eX , 1) X d − / / I X e   X d + o o X id / / X X id o o (89) W e hav e alr eady seen that such ma ps can not b e cons ide r ed as ‘homotopy equiv- alences’ in C osp( T op ) , b ecause there are no transversal maps backw ards (II.1.6). W e say that t wo n -cubical topo logical co spans u , v ar e we akly e quivalent if there exists a ﬁnite sequence of w eak equiv alences co nnecting them: u → u 1 ← u 2 → Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 24 ... → u n = v . Then, they m ust hav e the same vertices: u ( α 1 , ..., α n ) = v ( α 1 , ..., α n ) , for α i = ± . A weak double functor F : C osp ∗ ( T op ) → A , with v alues in a n arbitra r y weak cubical category , will be said to b e homotopy invariant if: (i) it sends weak equiv alences f : u → v betw een top ologica l co spans to in vertible (spe c ial) cells of A , and ther efore weakly equiv alent n -cubical cospans to isomorphic n -cub es of A . 5.4. Review o f homo top y pushouts The ne w concatenatio ns will use a fundamental notion of ho motopy theory , in- tro duced b y Mather [ 20 ] (and also used in Part II ). Let f : A → X , g : A → Y fo r m a span in T op . The standar d homotopy pus hout fr om f to g is a four-tuple ( P ; u, v ; λ ) as in the left diagr am b elow, where λ : u f → v g : A → P is a homotop y satisfying the follo wing univ ersal proper t y (as for c o- c omma s qu ar es o f c a tegories), w hich determines the solutio n up to home omorphism A g / / d +   Y v   A g / / f   Y v   A d − / / f   I A λ A A A A A A A A X u / / λ 2 2 P X u / / P (90) (a) for every homotop y λ ′ : u ′ f → v ′ g : A → W, ther e is precis e ly o ne map h : P → W suc h that u ′ = h u , v ′ = hv , λ ′ = hλ . (W riting h λ w e are using the obvious whisker c omp osition of homo to pies and maps.) In T op , the so lution a lw ays exists and can b e cons tr ucted a s the o r dinary colimit of the r ight-hand diag r am a bove. T his construction is based on the c ylinder I A = A × [0 , 1] and its faces d − , d + : A → I A, d − ( a ) = ( a, 0 ) , d + ( a ) = ( a, 1) ( a ∈ A ) . (91) Therefore, the space P is a pas ting of X a nd Y with the cylinder I A, and c a n b e realised a s a quotient of their top ological sum, under the equiv alence relation which gives the following identiﬁcations: P = ( X + I A + Y ) / ∼ , [ f ( a )] = [ a, 0] , [ g ( a )] = [ a, 1] ( a ∈ A ) . (92) The term ‘standard homotop y pushout’ will generally r efer to this particula r construction. Notice that, if f and g are top olog ical embeddings, the spaces X and Y are embedded in P . As a crucial fea ture, this construction alwa ys has s trong prop erties of homo top y inv a riance (e.g ., see [ 5 ], Section 3), which an o rdinary pushout need no t have. Notice also that the cylinder I A is itself the standard ho motopy pushout from id A to id A. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 25 5.5. Cylindrical concatenation By deﬁnition, the cylindric al i-c onc atenation u ⊗ i v of i -consecutive n -cospa ns is computed on the i -concatenation mode l ∧ ni 2 (of (23)), as u ⊗ i v = [[ u , v ]] .k i : ∧ n → ∧ ni 2 → T op ( ∂ + i ( u ) = ∂ − i ( v )) , (93) where [[ u, v ]] : ∧ ni 2 → T op sends all distinguished pus houts into standa r d homo top y pushouts (and, obviously , restricts to u and v o n k α i : ∧ n → ∧ ni ). (One could further for malise this by in tro ducing the categor y hpt Cat of h- c ate gories with distinguished homoto py pushouts, where an h-categor y - or a c at- e gory with homotopies - is a ca tegory enriched on reﬂexive gra phs with a suitable monoidal structure, as deﬁned in [ 5 ].) 5.6. Comparisons for id en tities, asso ci ativit y and int ercha nge The new structure C OSP ∗ ( T op ) comes with v ario us c omparison maps, whic h make it a symmetric quasi cu bic al c ate gory , accor ding to a deﬁnition which can be found in the last section. (a) First, there are lax compariso ns for identit ies, w hich ar e gene r ally not inv ertible (but w eak equiv alences) λ i u : E i ( ∂ − i u ) ⊗ i u → u, ρ i u : u ⊗ i E i ( ∂ + i u ) → u. (94) They are deﬁned in the ob vious wa y: for λ i u, one collapses to its basis the two cylinders on ∂ − i u whic h we ha ve pasted with u (i.e., the o ne appear ing in E i ( ∂ − i u ) and the one pro duced b y the cylindrical concatenation ⊗ i ). Notice that E i ( X ) ⊗ i E i ( X ) ∼ = E i ( X ) , but λE i ( X ) and ρE i ( X ) are d iﬀerent and not inv ertible: they collapse diﬀerent par ts of the resulting cylinder on X . On the o ther hand, ordinary degeneracies work even worse (with ho motopy pus houts): e i ( X ) ⊗ i e i ( X ) ∼ = E i ( X ) , which is only w eakly equiv alen t to e i ( X ) . (b) Second, the cubical relation for pure degenera cies (which, in the presence of transp ositions, ca n b e reduced to the identit y e 1 .e 1 = e 2 .e 1 ) do es not ho ld. It is replaced with an in vertible symmet ry c omp arison , deﬁned in 5.2 σ 1 u : E 1 E 1 ( u ) → E 2 E 1 ( u ) , ∂ α 1 σ 1 ( u ) = ∂ α 2 σ 1 ( u ) = id( E 1 u ) , ∂ α j +2 σ 1 ( u ) = σ 1 ( ∂ α j u ) , (95) which, via transp ositio ns, gener ates a ll the o ther ones , E j .E i → E i +1 .E j ( j 6 i ). (c) Asso cia tivit y of cylindrical concatenations works up to iso mo rphism (as in the weak cubical case): κ i ( u, v , w ) : u ⊗ i ( v ⊗ i w ) → ( u ⊗ i v ) ⊗ i w. (96) This is expressed b y the following computation (for ordina ry cospans ): [ X 0 + I A + [ Y 0 + I B + Z 0]] ∼ = [ X 0 + I A + Y 0 + I B + Z 0 ] ∼ = [[ X 0 + I A + Y 0 ] + I B + Z 0 ] , (97) where the brack ets [ ... ] stand for a quotient mo dulo the adequate equiv alence rela- tions. (d) Middle-four interc hange also works up to iso morphism χ 1 ( x, y , z , u ) : ( x ⊗ 1 y ) ⊗ 2 ( z ⊗ 1 u ) → ( x ⊗ 2 z ) ⊗ 1 ( y ⊗ 2 u ) : ∧ 2 → T op . (98) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 26 Indeed, the following diagram of pastings sho ws that b o th quaterna ry op era tions ab ov e ar e isomorphic to a symmetr ic one, denoted as ⊗ 12 ( x, y , z , u ) • x • E 1 a • y • • x • • y • • x • E 1 a • y • • • E 2 h • • • E 2 c • E 2 h ′ • E 2 d • • E 2 c • w • E 2 d • • z • E 1 b • u • • z • • u • • z • E 1 b • u • • • • • • • • • • • • • ( x ⊗ 1 y ) ⊗ 2 ( z ⊗ 1 u ) ( x ⊗ 2 z ) ⊗ 1 ( y ⊗ 2 u ) ⊗ 12 ( x, y , z , u ) . (99) Above, we hav e written: a = ∂ + 1 x = ∂ − 1 y , b = ∂ + 1 z = ∂ − 1 u, h = ∂ + 2 x ⊗ 1 ∂ + 2 y , c = ∂ + 2 x = ∂ − 2 z , d = ∂ + 2 y = ∂ − 2 u, h ′ = ∂ + 1 x ⊗ 1 ∂ + 1 z , w = E 1 E 1 ( v ) ∼ = E 2 E 1 ( v ) , (100) where v = ∂ + 1 ∂ + 2 x is the ( n − 2)-cos pan commo n to the four given items x, y , z , u. The symmetric prop erty of the op eration ⊗ 12 is: ⊗ 12 ( x, y , z , u ) .s 1 = ⊗ 12 ( xs 1 , z s 1 , y s 1 , us 1 ) . (101) (e) Finally , w e hav e an invertible nul lary inter change c omp arison , for 1-consecutive n -cub es x, y ι 1 ( x, y ) : E 1 ( x ) ⊗ 2 E 1 ( y ) → E 1 ( x ⊗ 1 y ) . (102) It c a n b e constructed using the isomo rphic cons truction of E 1 ( x ⊗ 1 y ) display ed below, and the symmetry isomorphism (95) (again, we write a = ∂ + 1 x = ∂ − 1 y ) • x E 1 x • x • x E 1 x • x • E 1 a E 2 E 1 a • E 1 a • E 1 a E 1 E 1 a • E 1 a • 1 / / 2   • y E 1 x • y • y E 1 y • y • • • • E 1 ( x ) ⊗ 2 E 1 ( y ) E 1 ( x ⊗ 1 y ) (103) 5.7. Cylindrical coll ared degene racies W e end this section b y r emarking that a sort of ‘c y lindrical degener acies’ also exist for collared cospans, in C c ∗ ( T op ) . Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 27 Beginning at cubica l degr ee 0, every space X ha s a cylindric al c ol lar e d de gener acy (not to be confused with the cylindrica l degeneracy E 1 ( X ) co nsidered a bove) E 1 ( X ) = ( X , I X , X ; E − : I X → I X ← I X : E + ) , E − ( x, t ) = ( x, t/ 3 ) , E + ( x ) = ( x, 1 − t/ 3) . (104) Then, every co lla red cos pa n U = ( X − , X 0 , X + ; U − : I X − → X 0 ← I X + : U + ) has tw o cylindr ical colla red degenerac ies E 1 ( U ), E 2 ( U ) = E 1 ( U ) .s, determined b y the following colla rs I 2 X − E − I X − / / I U −   I X − I U −   I 2 X − E + I X − o o I U −   I 2 X − I U − / / E − I X −   I X 0 E − X 0   I 2 X + I U + o o E − I X +   I X 0 E − X 0 / / I X 0 I X 0 E + X 0 o o I X − I U − / / I X 0 I X + I U + o o • 1 / / 2   I 2 X + E − I X + / / I U + O O I X + I U + O O I 2 X + E + I X + o o I U + O O I 2 X − I U − / / E + I X − O O I X 0 E + X 0 O O I 2 X + I U + o o E + I X + O O E 1 ( U ) E 2 ( U ) = E 1 ( U ) .s. (105) Within manifolds and cob ordism, E 1 ( X ) is g enerally used as the degenerate cob ordism on the manifold X ( cf. [ 21 ]). But again the cubica l r elation e 1 e 1 = e 2 e 1 is not satisﬁed: E 1 E 1 ( X ) a nd E 2 E 1 ( X ) are diﬀerent (a nd isomorphic): I 2 X E − I / / I E −   I 2 X I E −   I 2 X E + I o o I E −   I 2 X I E − / / E − I   I 2 X E − I   I 2 X I E + o o E − I   I 2 X E − I / / I 2 X I 2 X E + I o o I 2 X I E − / / I 2 X I 2 X I E + o o • 1 / / 2   I 2 X E − I / / I E + O O I 2 X I E + O O I 2 X E + I o o I E + O O I 2 X I E − / / E + I O O I 2 X E + I O O I 2 X I E + o o E + I O O E 1 E 1 ( X ) E 2 E 1 ( X ) . (106) 6. Cob ordisms W e obta in here a quasi cubical catego ry of k -manifolds and cubical cob ordisms, based on the notion of a diﬀerentiable manifold with fa ces [ 4, 13, 19 ]. 6.1. Goals and proble ms W e construct no w the quasi cubical sub categor y C OB ∗ ( k ) ⊂ C OSP ∗ ( T op ) of k - manifolds a nd cubical cob ordisms, with cylindrica l degenerac ie s and concatenation. The 2-cubical truncation 2 C O B ∗ ( k ) of o ur construction is related with the con- struction of Mo rton a nd Baez [ 21, 1 ], which works with assigned colla r s, cy lindrical Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 28 degeneracies and concatenation by pushout. But notice that, her e, k denotes the top ological dimension of the ob jects, while pa per s dealing with a 2 -cubical o r 2- globular structure g enerally refer to the dimension k + 2 of the highest cob ordisms which a ppea r in the str ucture itself. In the unbounded cubical (or glo bular) cas e , there is no upper b ound for such dimensions. Likely , working with ass igned collar s, one can a lso construct a we ak cubical cate- gory C ob ∗ ( k ) ⊂ C c ∗ ( T op ) . How ever, the technical as pects of this construction seem to b e so heavy , that one wonders whether such complication would be justiﬁed by the adv a n tage of obtaining a less weak struc tur e (satisfying all cubical axioms and having all compar is ons inv ertible). 6.2. Manifolds with corners W e b egin b y recalling some basic deﬁnitions. A diﬀer en tiable manifold with c or- ners [ 4, 1 9 ] is a second-countable Hausdor ﬀ space X which admits a diﬀeren tiable atlas of c harts ϕ i : U i → R n + = [0 , ∞ [ n . (107) (These charts ar e homeo morphisms from o pen s ubspaces of X onto op en subspaces of the euclidea n se ctor R n + , and the changes of charts ϕ i ϕ − 1 j are diﬀeomorphisms, in the obvious sense.) Every p oint x ∈ X has a well-deﬁned index c ( x ) betw een 0 and n, which coun ts the num b er of null co o rdinates of ϕ i ( x ) , for whichev er c har t ϕ i deﬁned at x. Th us, c ( x ) = 0 means that X is lo ca lly euclidea n at the po in t x. By deﬁnition, a c onne cte d fac e of X is the closur e of a connected co mponent of the subset of p oint s of index 1; ∂ X is the union o f a ll co nnected fa ces. A manifold with fac es [ 13, 19 ] is a manifold with corners where ev ery p oint x b elongs to precis e ly c ( x ) connected faces. Every manifold with b oundary is a manifold with faces, where the highest p os s ible index is 1. A compact cub e is also a manifold with face s: it has s ix co nnected faces , vertices have index 3, the other edge-p oints hav e index 2 and the r e maining face-p oints index 1. A fac e o f a manifold with faces is a unio n of disjoint connected faces, and is still a manifold with face s ; for instance, a compact cube has three non-connec ted faces. Finally , a manifold with n (distinguishe d) fac es X = ( X ; ∂ 1 X , ..., ∂ n X ) ([ 13, 19 ], where it is called an h n i -manifold ) is a manifold with faces equipp ed with a n indexed family of n faces which cov er ∂ X and such that ∂ i X ∩ ∂ j X is alwa ys a face of ∂ i X (for i 6 = j ) . Pla inly , n is at most equa l to the num be r of connected faces of X ; if it is less, the structure w e are considering is not determined b y X . A morphism f : X → Y o f such manifolds, with v alues in Y = ( Y ; ∂ 1 Y , ..., ∂ m Y ) , will be a c ontinuou s mapping whic h sends eac h face o f X into some face of Y . (The pa per s referr ed to ab ov e consider diﬀer entiable maps; the present c hoice will simplify the relations with top ological cospans.) The categorical sum is : X + Y = ( X + Y ; ∂ 1 X , ..., ∂ n X , ∂ 1 Y , ..., ∂ m Y ) . (108) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 29 6.3. Cubical cos pans of manifolds On this ba sis, it is easy to deﬁne the tra nsversally full quasi cubical sub categor y C OB ∗ ( k ) ⊂ C OSP ∗ ( T op ) of k -manifolds. A cubic al c osp an of k -manifolds u : ∧ n → T op is a top ologica l cospa n u = ( X ( t ) , u ( i, t )) , determined b y a manifold with faces X = ( X ; ( ∂ α i X )) , as made explic it b elow; X ha s dimension k + n a nd 2 n distinguished faces, whic h ar e pairwise disjoint: ∂ − i X ∩ ∂ + i X = ∅ . Namely , the central spac e X (0 , ... 0) of u is X itself, all the o ther spaces are int ers ections of the assigned faces (and faces as well) and all maps of u are inclusions X ( t ) = ∂ t 1 1 X ∩ .... ∩ ∂ t n n X ( t = ( t 1 , ..., t n ) ∈ ∧ n ) , u ( i, t ) : X ( t ) ⊂ X ( t ♯i ) ( t i 6 = 0 ) , (109) where we let ∂ 0 i X = X , for all i. Thus, the central cos pan in each directio n i is g iven by the inclusion in X of its tw o i -faces ∂ − i X → X ← ∂ + i X , (110) Notice that such a par ticular top olo gic al c osp an u : ∧ n → T op fully deter mines X and the family ( ∂ α i X ) of its fac e s . Mor eov er, all the sq ua res in u along tw o a rbitrary directions are pullbacks. By deﬁnition, a tr ansversal map f : u → v of such cubical cospans is an arbitrar y natural transformation f : u → v : ∧ n → T op ; as a consequence, it has an un- derlying morphism f : X → Y of manifolds with dis tinguished fa c es (a contin uous mapping). Plainly C O B ∗ ( k ) is c losed in C OSP ∗ ( T op ) under faces , (cylindrica l) dege nera- cies, transp ositions and concatenations. Being tr a nsversally full, it automatically contains the comparis o ns for identities, as so ciativity and (cubical) in terchanges; recall that the comparisons of iden tities are lax . Since manifolds with faces hav e collars ([ 19 ], Lemma 2.1.6), w e alwa ys ha ve E i ( u ) ⊗ i u ∼ = u. But this isomo r phism dep ends on the choice of co lla rs for u, and is not natura l (in the present s tr ucture). On the other ha nd, the natura l compariso n inherited from C OSP ∗ ( T op ) ‘colla pses cylinder s’ and is no t in vertible. 7. Symmetric quasi cubical categories W e make precise the deﬁnition of a symmetr ic quasi cubical ca teg ory , extending the notion of a symmetric we ak cubical category . (The latter, introduced in P ar t I, is recalled here in Section 1). 7.1. Symmetric quasi pre-cubical categories A symmetric quasi pre-cubical category A = (( A n ) , ( ∂ α i ) , ( e i ) , ( s i ) , (+ i )) , (111) is a symmetric quasi cubical s et (1.1, 1.2), equipped with the following additional op erations. Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 30 F or 1 6 i 6 n , the i - c onc atenation x + i y (or i -comp osition) o f tw o n - cub es x, y is deﬁned when x, y are i - c onse cut ive , i.e. ∂ + i ( x ) = ∂ − i ( y ) , and sa tisﬁes the following ‘geometrical’ int era ctions with faces a nd tr ansp ositions ∂ − i ( x + i y ) = ∂ − i ( x ) , ∂ + i ( x + i y ) = ∂ + i ( y ) , ∂ α j ( x + i y ) = ∂ α j ( x ) + i − 1 ∂ α j ( y ) ( j < i ) , = ∂ α j ( x ) + i ∂ α j ( y ) ( j > i ) , (112) s i − 1 ( x + i y ) = s i − 1 ( x ) + i − 1 s i − 1 ( y ) , s i ( x + i y ) = s i ( x ) + i +1 s i ( y ) , s j ( x + i y ) = s j ( x ) + i s j ( y ) ( j 6 = i − 1 , i ) . (113) Again, we are not (yet) ass uming categorical or int erchange laws for the i - comp ositions. Our structure is a symmetric pr e-cubical categor y (as deﬁned in I.3.4) if its degener acies sa tis fy the cubical relatio ns (6) (or (15)) a nd agree with conca te- nations: e j ( x + i y ) = e j ( x ) + i +1 e j ( y ) ( j 6 i 6 n ) , = e j ( x ) + i e j ( y ) ( i < j 6 n + 1 ) . (114) The presence of transp ositions allo ws us to reduce condition (114) to: e 1 ( x + 1 y ) = e 1 ( x ) + 2 e 1 ( y ) . (115) 7.2. In tro ducing transv ersal maps. As in I.4.1, we in tro duce now a richer structure, having n -dimensiona l maps in a new dir e ction 0, whic h can be viewed as s trict or ‘tr ansversal’ in opp osition with the pre v ious weak o r ‘cubical’ directions. The compariso ns fo r units, asso ciativity and in terchange will b e maps of this kind. Let us start with considering a gener al c ate gory obje ct A within the categor y of symmetric quasi pre-cubical categor ies and their functors A 0 e 0 / / A 1 o o ∂ α 0 o o A 2 c 0 o o (116) W e have thus: (qcub.1) A symmetric qua si pre-cubical categor y A 0 = (( A n ) , ( ∂ α i ) , ( e i ) , ( s i ) , (+ i )) , whose en tries ar e called n-cub es , or n-dimensional obje cts of A . (qcub.2) A symmetric quas i pre-cubical c a tegory A 1 = (( M n ) , ( ∂ α i ) , ( e i ) , ( s i ) , (+ i )) , whose en tries ar e called n-maps or n-dimensional m aps of A . (qcub.3) Symmetric cubical functors ∂ α 0 and e 0 , called 0- fac es and 0- de gener acy , with ∂ α 0 .e 0 = id . Typically , a n n -map will b e written as f : x → x ′ , where ∂ − 0 f = x, ∂ + 0 f = x ′ are n -cub es. Every n -dimensio nal ob ject x has an identity e 0 ( x ) : x → x. No te tha t ∂ α 0 and e 0 preserve cubical faces ( ∂ α i , with i > 0) , cubical degener acies ( e i ) , tra nspo - sitions ( s i ) a nd cubical c o ncatenations (+ i ) . In par ticular, given t wo i -consecutive n -maps f , g , their 0-faces are also i -consecutive and w e hav e: f + i g : x + i y → x ′ + i y ′ ( f : x → x ′ , g : y → y ′ ; ∂ + i f = ∂ − i g ) . (117) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 31 (qcub.4) A comp osition law c 0 which as s igns to tw o 0-consecutive n -maps f : x → x ′ , h : x ′ → x ′′ an n -map hf : x → x ′′ (also written h.f ) . This compo sition law is (strictly) categorical, and forms a category A n = ( A n , M n , ∂ α 0 , e 0 , c 0 ) . It is also consistent with the symmetric quasi pre-cubical structure, in the following sense ∂ α i ( hf ) = ( ∂ α i h ) . ( ∂ α i f ) , e i ( hf ) = ( e i h )( e i f ) , s i ( hf ) = ( s i h )( s i f ) , ( h + i k ) . ( f + i g ) = h f + i k g , (118) • ∂ − i f / / x   • ∂ − i h / /   • x ′′   f / / h / / • / / y   • / /   • y ′′   • 0 / / i   g / / k / / • ∂ + i g / / • ∂ + i k / / • The last condition, repres en ted in the diagra m ab ove, is the (strict) middle- four interc hange be tw een the strict c o mpo sition c 0 and any weak one . A n n -map f : x → x ′ is said to be sp e cial if its 2 n vertices are identities ∂ α f : ∂ α x → ∂ α x ′ ∂ α = ∂ α 1 1 ∂ α 2 2 ...∂ α n n ( α i = ± ) . (119) In degree 0, this just means an iden tity . 7.3. Comparisons Extending I.4.2, we can now de ﬁne a symmetric quasi cu bic al c ate gory A as a category ob ject within the catego ry of symmetric quasi pre-cubica l categories and symmetric cubical functors, whic h is further equipp ed with spe cial transversal maps, playing the role of compar isons for units, s y mmetry , ass o ciativity and cubical int erchange, a s follows. (W e only assig n the compariso ns in direction 1; all the others can be obtained with transpo sitions. Notice also that the unit comparisons are not assumed to be inv ertible.) (qcub.5) F or every n -cub e x, we have sp ecial n -ma ps λ 1 x and ρ 1 x, which are natur al on n -maps and have the following faces (for n > 0 ) λ 1 x : ( e 1 ∂ − 1 x ) + 1 x → x ( left-unit 1-c omp arison ), ∂ α 1 λ 1 x = e 0 ∂ α 1 x, ∂ α j λ 1 x = λ 1 ∂ α j x (1 < j 6 n ) , (120) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 32 • ∂ − 1 x e 0 ∂ − 1 x • ∂ + j x • ∂ − 1 x x • ∂ + j x • x x x x x x x x x x x x x x x x x x e 1 ∂ − 1 x • x x x x x x x x x x x x x x x x x x λ 1 ∂ + j x • x x x x x x x x x x x x x x x x x x λ 1 ∂ − j x • j / / 1   0 = = { { { { { { { { { • ∂ − j x x • • • ∂ − j x • e 0 ∂ + 1 • • ∂ + 1 x • x x x x x x x x x x x x x x x x x x • ∂ + 1 x x x x x x x x x x x x x x x x x x x • u u u u u u u u u u u u u u u u u u ρ 1 x : x + 1 ( e 1 ∂ + 1 x ) → x, ( right-unit 1-c omp arison ), ∂ α 1 ρ 1 x = e 0 ∂ α 1 x, ∂ α j ρ 1 x = ρ 1 ∂ α j x (1 < j 6 n ) , (121) • ∂ − 1 x e 0 ∂ − 1 x • ∂ + j x • ∂ − 1 x x • ∂ + j x • ∂ − j 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 x x x x x ρ 1 ∂ + j x • ∂ − j x x x x x x x x x x x x x x x x x x x ρ 1 ∂ − j x • j / / 1   0 = = { { { { { { { { { • e 1 ∂ + 1 x • • • • e 0 ∂ + 1 • • ∂ + 1 x • x x x x x x x x x x x x x x x x x x • ∂ + 1 x x x x x x x x x x x x x x x x x x x • u u u u u u u u u u u u u u u u u u (Notice that the 0-direction of ρ 1 x is reversed, with r espe c t to I.4.2 - where ρ 1 x is in vertible a nd its dir ection is iness en tial.) (qcub.6) F o r ev ery n -cub e x, w e hav e an in vertible special ( n + 2)-map σ 1 x, whic h is natural on n -maps a nd has the follo wing faces (for n > 0) σ 1 x : e 1 e 1 ( x ) → e 2 e 1 ( x ) ( symmetry 1-c omp arison ), ∂ α 1 σ 1 ( x ) = ∂ α 2 σ 1 ( x ) = id( e 1 x ) , ∂ α j +2 σ 1 ( x ) = σ 1 ( ∂ α j x ) . (122) (qcub.7) F or three 1- consecutive n -cubes x, y , z , we hav e an invertible sp ecia l n -map κ 1 ( x, y , z ) , which is natura l on n -maps and has the follo wing faces κ 1 ( x, y , z ) : x + 1 ( y + 1 z ) → ( x + 1 y ) + 1 z ( asso ciativity 1-c omp arison ), ∂ − 1 κ 1 ( x, y , z ) = e 0 ∂ − 1 x, ∂ + 1 κ 1 ( x, y , z ) = e 0 ∂ + 1 z , ∂ α j κ 1 ( x, y , z ) = κ 1 ( ∂ α j x, ∂ α j y , ∂ α j z ) (1 < j 6 n ) , (123) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 33 • ∂ − 1 x e 0 ∂ − 1 x • ∂ + j x • ∂ − 1 x x + 1 y • ∂ + j x • ∂ − j x t t t t t t t t t t t t t t t t x • t t t t t t t t t t t t t t t t κ 1 ∂ + j • ∂ + j y • t t t t t t t t t t t t t t t t κ 1 ∂ − j • • ∂ + j y • ∂ + j z • z • ∂ + j z • j / / 1   0 ; ; w w w w w w w w • ∂ − j y y + 1 z • • • ∂ − j z • • • • e 0 ∂ + 1 • • ∂ + 1 x • t t t t t t t t t t t t t t t t • ∂ + 1 x 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 t t t t t t (w cub.8) Given four n -cub es x, y , z , u which make the conca tenations b elow well- formed, we hav e an invertible special n - map χ 1 ( x, y , z , u ), which is na tur al on n - maps and has the follo wing faces (partially display ed below) χ 1 ( x, y , z , u ) : ( x + 1 y ) + 2 ( z + 1 u ) → ( x + 2 z ) + 1 ( y + 2 u ) ( inter change 1-c omp arison ), ∂ − 1 χ 1 ( x, y , z , u ) = e 0 ( ∂ − 1 x + 2 ∂ − 1 z ) , ∂ + 1 χ 1 ( x, y , z , u ) = e 0 ( ∂ + 1 y + 2 ∂ + 1 u ) , ∂ − 2 χ 1 ( x, y , z , u ) = e 0 ( ∂ − 2 x + 1 ∂ − 2 y ) , ∂ − 2 χ 1 ( x, y , z , u ) = e 0 ( ∂ − 2 x + 1 ∂ − 2 y ) , ∂ α j χ 1 ( x, y , z , u ) = χ 1 ( ∂ α j x, ∂ α j y , ∂ α j z , ∂ α j u ) (2 < j 6 n ) , (124) • ∂ − 2 x e 0 • ∂ − 2 y • ∂ + 1 y • ∂ − 2 x x • ∂ − 2 y y • ∂ + 1 y • ∂ − 1 x p p p p p p p p p p p p p p p p p p x + 1 y • • p p p p p p p p p p p p p p p p p p e 0 • ∂ + 1 u • ∂ − 1 x p p p p p p p p p p p p p p p p p p e 0 • + 2 z • + 2 u • ∂ + 1 u • 1 / / 2   0 C C       • ∂ − 1 z z + 1 u • • • • ∂ − 1 z • e 0 • • • ∂ + 2 z • ∂ + 2 u • p p p p p p p p p p p p p p p p p p • ∂ + 2 z p p p p p p p p p p p p p p p p p p • ∂ + 2 u • p p p p p p p p p p p p p p p p p p Moreov er, the nullary interc hange e 1 ( x ) + 2 e 1 ( y ) = e 1 ( x + 1 y ) (of the we ak cubical case (115)) is replaced with an invertible special ( n + 1 )-map ι 1 ( x, y ) , which is deﬁned when x, y ar e 1-consecutive, is natur a l on n -maps and has the following faces (partially display ed below) ι 1 ( x, y ) : e 1 ( x ) + 2 e 1 ( y ) → e 1 ( x + 1 y ) ( nul lary inter change ), ∂ − 1 ι 1 ( x, y ) = ∂ + 1 ι 1 ( x, y ) = e 0 ( x + 1 y ) , ∂ − j +1 ι 1 ( x, y ) = e 0 ( e 1 ∂ − j x ) , ∂ + j +1 ι 1 ( x, y ) = e 0 ( e 1 ∂ + j y ) , (125) Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 34 • e 1 ∂ − j x e 0 • x + 1 y • e 1 ∂ − j x e 1 ( x + 1 y ) • x + 1 y • x v v v v v v v v v v v v v v e 1 ( x ) • v v v v v v v v v v v v v v e 0 • x v v v v v v v v v v v v v v e 0 • 1 / / j +1   0 < < y y y y y y y • y e 1 ( y ) • • • y • e 0 • • e 1 ∂ + j y • v v v v v v v v v v v v v v • e 1 ∂ + j y v v v v v v v v v v v v v v • v v v v v v v v v v v v v v (qcub.9) Finally , these compa risons m ust satisfy some conditions o f coherence (see 7.4). W e say that A is unitary if the compariso ns λ, ρ ar e identities. 7.4. Coherence Extending I.4.3, the cohe r ence axiom (qcub.9) means that the following diagrams of transv er s al maps commute (assuming that a ll the cubical comp ositions mak e sense): (i) c oher enc e p entagon for κ = κ 1 (writing + = + 1 ): x + ( y + ( z + u )) 1+ κ q q d d d d d d d d κ ) ) R R R R R R R R R R R x + (( y + z ) + u ) κ   ( x + y ) + ( z + u ) κ u u l l l l l l l l l l l ( x + ( y + z )) + u κ +1 - - Z Z Z Z Z Z Z Z (( x + y ) + z ) + u (126) (ii) c oher enc e hexagon for χ = χ 1 and κ = κ 1 (alwa ys writing + = + 1 ): ( x + ( y + z )) + 2 ( x ′ + ( y ′ + z ′ )) κ + 2 κ / / χ   (( x + y ) + z ) + 2 (( x ′ + y ′ ) + z ′ ) χ   ( x + 2 x ′ ) + (( y + z ) + 2 ( y ′ + z ′ )) 1+ χ   (( x + y ) + 2 ( x ′ + y ′ )) + ( z + 2 z ′ ) χ +1   ( x + 2 x ′ ) + (( y + 2 y ′ ) + ( z + 2 z ′ )) κ / / (( x + 2 x ′ ) + ( y + 2 y ′ )) + ( z + 2 z ′ ) (127) (iii) c oher enc e c onditions of the inter changes χ = χ 1 , ι = ι 1 with the u nit c omp ar- isons λ = λ 1 and ρ = ρ 1 (writing + = + 1 , e = e 1 and ∂ α = ∂ α 1 ): Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 35 ( e∂ − x + x ) + 2 ( e∂ − y + y ) λ + 2 λ ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q χ   ( x + e ∂ + x ) + 2 ( y + e∂ + y ) ρ + 2 ρ v v m m m m m m m m m m m m m χ   ( e∂ − x + 2 e∂ − y ) + ( x + 2 y ) ι +1   x + 2 y ( x + 2 y ) + ( e∂ + x + 2 e∂ + y ) 1+ ι   ( e ( ∂ − x + ∂ − y ) + ( x + 2 y ) λ 6 6 m m m m m m m m m m m m m ( x + 2 y ) + e ( ∂ + x + 2 ∂ + y ) ρ h h Q Q Q Q Q Q Q Q Q Q Q Q Q (128) 7.5. Comments The symmetric quasi cubical ca tegory A is a symmetric we ak cubical catego ry if the unit-compariso ns λ, ρ are inv ertible, the sy mmetric co mpa rison σ 1 is an identit y and a further coherence axiom ho lds: (iv) c oher enc e triangle for λ 1 , ρ 1 , κ 1 : x + 1 ( e 1 ∂ − 1 y + 1 y ) κ / / 1+ λ ) ) R R R R R R R R R ( x + 1 e 1 ∂ + 1 x ) + 1 y ρ +1 u u l l l l l l l l l x + 1 y (129) Notice that the latter do es no t hold in C OSP ∗ ( T op ) nor in C OB ∗ ( k ) , whe r e the maps 1 + λ and ( ρ + 1) κ collapse diﬀerent cylinder s. In P ar t I, also the n ullary int erchange ι 1 was ta ken to b e an identit y - which works in the situations studied there; the present more general deﬁnition seems to be preferable, from a formal po in t of view. References [1] J. Baez, This W eek ’s Finds in Mathema tica l Physics (W eek 242). http:// math.u cr.edu /home/baez/week242.html [2] R. Brown and P .J. Higgins, T ensor pro ducts and homotopies for ω -group oids and crosse d complexes, J. Pure Appl. Algebra 47 (1 987), 1-33. [3] H .S.M. Coxeter and W.O.J. Moser, Gener ators a nd relatio ns for discrete groups, Springer, Berlin 1957. [4] A . Doua dy , V ari´ et´ es ` a b or ds ang uleux et voisinages tubulair es, S´ eminaire Henri Cartan 14 (1961-6 2), Exp. 1, p. 1-11. http:// www.nu mdam.org/ item?i d=SHC 1961-1 962 14 A1 0 [5] M. Grandis, Homotopical a lgebra in homotopical categ o ries, Appl. Categ. Structures 2 (1994), 351-40 6. [6] M. Gr andis, Hig her cospa ns a nd weak cubical ca tegories (Cospa ns in Algebraic T o polo gy , I), Theory Appl. Categ. 18 (2007), No. 1 2, 321 -347. ht tp://w ww. tac.mt a.ca/t ac/volumes/18/12/18-12.dvi { ps, pdf } Journal of Homotopy and R elate d Stru ctur es, vol. ??(??), ???? 36 [7] M. Gra ndis , Co llared c ospans, cohomotopy and TQFT (Cospans in Algebraic T o polo gy , I I ), Theo r y Appl. Ca teg. 18 (20 07), No. 19, 602- 6 30. http://w ww. tac.mt a.ca/t ac/volumes/18/19/18-19.dvi { ps, pdf } [8] M. Grandis a nd L. Mauri, Cubical s ets and their site, Theo ry Appl. Ca teg. 11 (2003), No. 8, 185-2 11. htt p://ww w.tac.mta.ca/tac/volumes/11/8/11-08. dvi { ps, pdf } [9] M. Grandis and R. Par ´ e , Limits in double categor ies, Cah. T o po l. G´ eom. Diﬀ´ er. Cat´ eg . 40 (1999 ), 1 62-220 . [10] M. Grandis a nd R. Par´ e, Adjoint for double catego ries, Cah. T op ol. G ´ eo m. Diﬀ ´ er . Ca t ´ eg. 4 5 (2004), 193-240 . [11] M. Gra ndis and R. Par ´ e, Kan extensions in do uble catego r ies (On weak double categorie s, Part I I I), Theor y Appl. Ca teg. 20 (2 008), No . 8 , 152-1 8 5. [12] M. Grandis and R. Par´ e, La x Ka n extensio ns for double ca tegories (On weak double categories, Part IV), Cah. T op ol. G´ eom. Diﬀ´ e r . Cat´ e g. 48 (2 007), 16 3 - 199. [13] K. J¨ a nic h, On the classiﬁcation of O ( n )-manifolds, Math. Annalen 1 76 (1968), 53-7 6. [14] D.L. Johnson, T opics in the theory of presen tation of groups, Cam bridge Univ. Press, Cambridge 19 80. [15] A. Joy al and J. Ko ck, W eak units and homoto p y 3- type s , Street F estschriﬀ: Categorie s in algebra, geometry and ma thema tical ph ysics, Contemp. Ma th 431 (2007), 257-2 76. [16] D.M. Kan, A bstra ct homotop y I, Proc . Nat. Aca d. Sci. U.S.A. 41 (19 55), 1092- 1096. [17] D.M. Kan, Abstract homoto p y . I I, Pr o c . Nat. Acad. Sci. U.S.A. 42 (1956 ), 255-2 58. [18] J. Ko ck, W eak iden tit y arrows in hig he r categories , IMRP In t. Math. Res. Pap. 20 0 6, 6 9 163, 1 - 54. a rXiv:math/050 7116 v3 [19] G. Laures , On cob ordism of manifolds with corners, T rans. Amer. Math. So c. 352 (2000), 5667 - 5688. [20] M. Mather, P ull-backs in ho motopy theo r y , Can. J. Math. 28 (19 76), 22 5-263. [21] J. Morto n, A double bicatego ry of cob ordisms with c o rners, Preprint. Av ail- able at arXiv: math.CT/0611 930 . Marco Grandis grandi s@dima .unige.it Dipartimento di Matematica Univ er s it` a di Geno v a Via Do decaneso 35 16146 -Genov a, Italy

Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)

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