Coherence in monoidal track categories

Coherence in monoid al track categories Yves Guiraud Philippe Malbos INRIA Uni ver sité L yon 1 Institu t Camille Jord an Institu t C amille Jordan guirau d@math.uni v- lyon1.fr malbos@math .uni v-lyo n1.fr Abstract – W e intr oduce homotopi cal methods based on r ewriting on higher -dimensio nal cate gories to pr ov e coh er ence re sults in cate gori es with an alg ebraic structur e. W e exp res s the co her ence pr oblem for (symmetric) monoidal cate gories as an aspher icity pr oblem for a trac k cate gory and we use re writing methods on polygra phs to solve it. The sett ing is exte nded to mor e gener al coher ence pr oblems, seen as 3-dimens ional wor d pr oblems in a trac k cate gory , inclu ding the case of br aided monoid al cate gorie s. Key words – coh ere nce; monoi dal cate gory; highe r -dimensi onal cate gory; r ewritin g; polygra ph. M.S.C. 2000 – 18C10, 18D10, 68Q42. I N T RO D U C T I O N A monoida l cate gory is a cate gory equip ped with a p roduct, associa ti ve up to a natura l isomorphism, and ha ving a disting uished object, which is a unit for the product up to natural isomorp hisms. Associat iv ity and unity satisfy , in turn, a cohere nce condition : all the diagrams b uilt from the corres pondi ng natur al isomorph isms are commutati ve. A cornerston e result for m onoid al categorie s was to redu ce the infinite requir ement “e very diagram commutes ” to a finite require ment “if a spec ifi ed finite set of diag rams commute then e very d iagram commutes”, [10, 14]. W e c all co her ence bas is su ch a fini te set o f d iagrams. A symmetr ic monoi dal cate gory is a monoidal cate gory whose produ ct is commutati ve up to a natu ral isomorph ism, called symmetry . In a symmetri c monoidal category the coherence problem has the same formulat ion as in monoid al categori es, with addit ional coherence diag rams for the symmetry , [10]. In a symmetric mono idal catego ry the symmetry is its o wn in verse. Braided monoid al categorie s are monoida l cate gories co m mutati ve up to an isomorphism which is not its own in verse. The coherence proble m in brai ded cate gories has anoth er formulation: a diagram is commutati ve if and only if its two sides corresp ond to the same braid , [9]. In this paper , we formula te the cohere nce p roblem for monoidal track 2 -cate gories in the homotop ical terms of higher -dimensiona l cate gories, as introd uced by the a uthors in [6]. This formulati on gi ves a way to reduce the coherence problem to a 3 -dimensi onal word prob lem in track cate gories . The construc tion of con ver gent ( i.e. , terminati ng and confluent) presen tations of monoid al track 2 -cat ego ries allo ws us to reduce the problem “e very diagram commutes” to “if the diagr ams induced by critical branchin gs com- mute then e very diagra m commutes”: the confluence diag rams of critical branching s form a coheren ce basis. Let us illus trate this methodology on a si mple example. Coher ence for categories with an associative product Let us co nsider a cate gory C equipp ed with a functo r ⊗ : C × C → C which is associa ti ve up to a n atural isomorph ism, i.e. , there is a natur al isomorp hism α x,y,z : ( x ⊗ y ) ⊗ z − → x ⊗ ( y ⊗ z ) , such that the follo wing diagram commutes in C : ( x ⊗ ( y ⊗ z )) ⊗ t α / / x ⊗ (( y ⊗ z ) ⊗ t ) α & & M M M M M M M M M M (( x ⊗ y ) ⊗ z ) ⊗ t α 8 8 q q q q q q q q q q α + + V V V V V V V V V V V V V V V V V V V c  x ⊗ ( y ⊗ ( z ⊗ t )) ( x ⊗ y ) ⊗ ( z ⊗ t ) α 3 3 h h h h h h h h h h h h h h h h h h h (1) Presentat ion of such cate gories by generat ors an d relatio ns can be achie ved using the notion of polygra ph. This notion of pres entation of higher -dimensional cate gories was intr oduced by Burron i, [3 ], and by Street u nder the terminology of computads, [15, 16]. In this paper , we use Burroni’ s termino logy , as usual in re w riting th eory . An n -polygr aph is a family ( Σ 0 , . . . , Σ n ) , where Σ 0 is a set and, for ev ery 0 ≤ k < n , Σ k + 1 is a family of paralle l k -cells of the free k -cate gory Σ ∗ k ov er Σ k . W e call such a family a cellul ar e xtension of Σ ∗ k . Categ ories w ith an associati ve produ ct can be presented us ing th e notion of po lygrap h as follo ws. Let us cons ider the 3 -polygraph A s 3 with one 0 -cell , one 1 -cell , one 2 -cel l and one 3 -cell: _ % 9 Let As ⊤ 3 be the free trac k 3 -cate gory gener ated by As 3 , i.e. , the free 3 -cate gory ov er As 3 whose 3 -cells are in vertible. The relation (1) satisfied by the assoc iati vity isomorph ism can be presente d by a cellular ext ension As 4 of As ⊤ 3 with one 4 -cell : _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   (2) Let AsCat be the track 3 -cat egor y obt ained as the quot ient of As ⊤ 3 by the cellular extensi on As 4 . The cate gory of (small) categorie s w ith a prod uct, associati ve up to a natural isomorphism, is isomorphic to the cate gory Alg ( AsCat ) of algebr as over the 3 -categor y AsCat . S uch an algebra is a 3 -functor from AsCat to th e monoida l 2 -categor y C at of s mall cate gories, func tors and natura l transformation s, seen as a 3 -categor y w ith only one 0 -cell. The correspond ence associates, to a categ ory ( C , ⊗ , α ) , the algebra A : AsCat → Cat defined by: A ( ) = C , A ( ) = ⊗ , A ( ) = α. A diagram in a AsCat -algebr a A is the image A ( γ ) of a pair γ = ( A, B ) of parallel 3 -cell s in AsCat . This diagram commutes if A ( A ) = A ( B ) hold s in Cat . The coheren ce probl em for AsCat -algebr as can be formulated as “does e very dia gram co mmute in ev ery AsCat -algeb ra”. In this w ay , the co herence proble m is reduced to sho w ing that As 4 forms a homoto py basis of the tr ack 3 -cate gory As ⊤ 3 , i.e. , if 2 ( A, B ) is a pair of parallel 3 -cells of AsCat , then A = B . A 3 -categor y that satisfies this last proper ty is called aspher ical . Provin g asphe ricity is a special case of a word problem i n a trac k 3 -cate gory . In [6], the authors pro ve that for a con v er gent, i.e . , terminat ing and conflue nt, n -poly graph Σ , the critical bra nchi ngs genera te a homotop y basis of the free track n -cate gory Σ ⊤ . In our e xample, the 3 -polyg raph As 3 is con v ergen t and has a unique critical branchi ng, formed by the two differe nt application s of the 3 -cell on the same 2 -cell: j * > V 4 The 4 -cell of (2) forms a confluence dia gram for this critic al branc hing. As a consequenc e, th e cellula r ext ension As 4 is a homotop y basis of the track 3 -cate gory As ⊤ 3 : this prov es the coheren ce result for AsCat -algeb ras. Organisation of the paper In Section 1, we recal l notion s on high er -dimensional track catego ries, presenta tions by polyg raphs and polyg raphic re w riting, includi ng critical bra nching s. W e intro duce the notion of highe r -dimensio nal pro(p) s in Section 1.3. For n ≥ 1 , a (trac k) n -pr o is a (tra ck) n - cate gory with one 0 -cell, suc h tha t i ts un derlyin g 1 -cate gory is the monoid of natural numbers with ad dition. Equi valen tly , for n ≥ 2 , a (track) n -pro is a strict monoida l cate gory (seen as a 2 -cate gory with one 0 - cell), enrich ed in (track) ( n − 2 ) -catego ries and whose underlying monoid of objects is the monoid of natural numbers with the addit ion. A (trac k) n -pr op is a (track) n -pro, whose underly ing monoidal c ateg ory is symmetric. In particular , 2 -pro(p)s coinci de with M ac Lane’ s P R O(P)s, an acrony m for “product (and permutation) cate gories", introduced in [11]. For coherence problems, we c onside r special cases of track 3 -pro(p )s: the track 3 - pros AsCat of cat - ego ries with an associa ti ve prod uct and MonCat of m onoida l catego ries and the track 3 -prop s SymCat of symmetric monoida l categorie s and BrCat of braided monoida l categorie s. An alg ebra over a 3 -pr o(p) P is a strict (symmetric ) monoidal 2 -func tor from P to Cat . Here Cat is considered as a 3 -categ ory with one 0 -cell, cate gories as 1 -cells, functor s as 2 -cell s and natural trans- formatio ns as 3 -cel ls, see Paragra ph 1.3.3. In Propositi on 1.3.5, we relate t he co herence pro blem for algebr as over a 3 -pro(p) P to the aspheri city of P : if the 3 -pro(p ) P is aspheri cal, then e very P -diagram commutes in e very P -alg ebra. Thus, red ucing the coherence pr oblem “ev ery dia gram commutes” to “i f s ome diagrams commute then e very diagram commutes” c onsists in constructi ng an algebr aic presenta tion of the 3 -pro(p) pro ving that it is aspheric al. W e sho w th at a con ver gent presen tation gi ves a procedu re to solve the cohere nce proble m. 3 1. Pr eliminaries The monoidal coher ence problem. In Section 2 , we consider th e cas e o f 3 -pros. A con ver gent pre- sentat ion for a 3 -pro P is a pair ( Σ 3 , Σ 4 ) , where Σ 3 is a con ver gent 3 -polygr aph toge ther with a cellu lar ext ension Σ 4 of gene rating conflue nces of Σ 3 . W e ha ve: Theor em 2.1.2. If a trac k 3 -pr o P admits a con ver gent pre sentat ion, then ever y P -dia gra m commutes in eve ry P -algebr a. In Secti on 2.3, we consider the cohe rence probl em for monoid al cate gories. W e prove that the 3 - pro MonCat of monoidal categori es is asph erical, see Paragraph 2.3.4, hence the coherence theo rem for monoida l categorie s, prov ed in [10]. The symmetric monoidal coher ence pro blem. In Secti on 3, for the c oheren ce probl em for symmetric monoida l catego ries, we consider the asphericity problem of algebr aic track 3 -props, i.e. , track 3 -props whose genera ting 2 -cells and 3 -cells ha ve coarity 1 , see Paragraph 3.2.1. In that case, we ha ve a con ver - gent presenta tion o f the symmetry , see [3, 4]. This giv es the follo wing suf fi cient condition for provin g that an al gebrai c track 3 -prop is asp herical , where π ( Γ Σ 3 ) is a cellul ar extens ion generated by the cr itical branch ings that do not dep endent on the symmetry only: Theor em 3.2.4. If a tr ack 3 - pr op P admits an alg ebraic con ver ge nt pr esenta tion ( Σ 3 , Σ 4 ) such t hat Σ 4 is T ietze-e quival ent to π ( Γ Σ 3 ) , then P is aspher ical. In the case of the 3 -prop SymCat of symmetric monoidal categorie s, this result giv es the corre spondi ng cohere nce theorem, see Coro llary 3.3.6. The braided monoidal case and the generalised coher ence proble m. For braided monoi dal cate - gories , we consid er a gener alised version of the coher ence problem: “gi ven a 2 -prop P , decide, for any 3 -sphe re γ of P , whether or not the diag ram A ( γ ) commutes i n eve ry P -alge bra A ”. T o solv e it, we procee d in two step s. First , we prove that cohere nce is prese rved by aspherical quotients, so that we can reduce a 3 -prop to its non-a spheric al part: Theor em 4.3.1. Let P and Q be 3 -pr ops w ith Q asphe rical and Q ⊆ P . Then, for eve ry 3 -sph er e ( A, B ) of P , we have A = B if and only if π ( A ) = π ( B ) . Then, gi ven an algebraic 3 -prop P , we define the initial P -algeb ra P , see Section 4.4, and we prov e: Theor em 4.4.3. Let P be an algeb rai c 3 -pr op and let ( A, B ) be a 3 -spher e of P . T hen we have A = B if and only if P ( A ) = P ( B ) . In th e case o f the 3 -pro p of braided monoida l catego ries, the initial algebra B associates, to ev ery 3 - cell A , a bra id B ( A ) . Hence, the introdu ced methodol ogy reco vers the c oheren ce result o f J oya l and Street, [9]: a diag ram commutes if and only if its two sides are associat ed to the same braid. 1 . P R E L I M I N A R I E S In this sectio n, we recall from [6] notion s a nd result s o n hig her -dimensiona l (track) catego ries, homoto py bases and present ations by poly graphs . 4 1.1. Higher -dimensional categori es and homotopy bases 1.1. H igher -dimensional categories and homotopy bases Let n be a natura l number and let C be an n -cate gory (we alw ays consid er strict, glob ular n -categor ies). W e denot e by C k the set (and the k -categ ory) of k -cells of C . If f is in C k , then s i ( f ) and t i ( f ) respe c- ti vely deno te the i -source and i -tar get of f ; we drop the suffix i when i = k − 1 . The sourc e and tar get maps satisfy the glob ular r elations : s i ◦ s i + 1 = s i ◦ t i + 1 and t i ◦ s i + 1 = t i ◦ t i + 1 . W e respecti vely denote by f : u → v , f : u ⇒ v , f : u ⇛ v or f : u  ? v a 1 -cell, 2 -cell, 3 -cel l or 4 -cell f with source u and tar get v . If ( f, g ) is a pair of i -composable k -cells, that is when t i ( f ) = s i ( g ) , we deno te by f ⋆ i g their i -composi te. The composi tions satisf y the exc hange r elations giv en, for e very i 6 = j and ev ery possible cells f , g , h and k , by: ( f ⋆ i g ) ⋆ j ( h ⋆ i k ) = ( f ⋆ j h ) ⋆ i ( g ⋆ j k ) . If f is a k -cell, w e denote by id f its identity ( k + 1 ) -cell. Wh en id f is compos ed with cells of dimension k + 1 or high er , we simply deno te it by f . A cell is de gener ate w hen it is an identi ty cell. 1.1.1. T rack n -cate gories. In an n -cate gory C , we say that a k -cell f with sou rce u and ta r get v is in vertib le when it admits an in ver se for the highe r -dimensio nal compositio n ⋆ k − 1 defined on it, i.e. , when there exists a (necessarily unique ) k -cell in C , with source v and tar get u in C , denoted by f − and called the in vers e of f , tha t satisfies f ⋆ k − 1 f − = id u and f − ⋆ k − 1 f = id v . A tr ack n - cate gory is an n -categor y whose n -c ells are in vertible . One can a lso d efine track n -cate gories by in ductio n on n , with track 1 -cate gories being gr oupoid s and track ( n + 1 ) -cat ego ries being ca tego ries enrich ed in trac k n -catego ries. 1.1.2. Cellular extensions. Let C be an n -cate gory . A k -spher e of C is a pair γ = ( f, g ) of parallel k - cells of C , i.e. , with s ( f ) = s ( g ) and t ( f ) = t ( g ) . W e call f the sou r ce of γ and g its ta r get . When f = g , the k -sphere γ is de ge ner ate . An n -c ateg ory C is aspherical when eve ry n -sph ere of C is degenera te. A cellula r ext ension o f C is a family Γ of n -sphe res of C . By consid ering all the formal compositio ns of elements of Γ , seen as ( n + 1 ) -cells w ith sour ce and target in C , one bu ilds the fr ee ( n + 1 ) -cate gory gen era ted by Γ o ver C , denoted by C [ Γ ] . The quotient of C by Γ , denote d by C /Γ , is t he n -cate gory one g ets from C by identificati on of t he n - cells s ( γ ) and t ( γ ) for e very element γ of Γ . T wo cellular e xtensio ns Γ 1 and Γ 2 of C are T ietze-equ ivalent if the n -cate gories C /Γ 1 and C /Γ 2 are isomorph ic. The fr ee trac k ( n + 1 ) -cate gory ge nera ted by Γ over C is defined by C ( Γ ) = C [ Γ , Γ − ] / In v ( Γ ) , where Γ − and In v ( Γ ) are the followin g cellular extensio ns of C and C [ Γ , Γ − ] , respect iv ely: Γ − =  γ − : t ( γ ) → s ( γ ) | γ ∈ Γ  and In v ( Γ ) =  γ ⋆ n γ − → id sγ , γ − ⋆ n γ → id tγ | γ ∈ Γ  . 5 1. Pr eliminaries 1.1.3. Homotopy bases. A cellular exte nsion Γ of an n -cate gory C is a homotopy basis when the quotie nt n -catego ry C / Γ is asphe rical, i.e. , when, for e very n -spher e γ of C , th ere exists an ( n + 1 ) -cell from s ( γ ) to t ( γ ) in the track ( n + 1 ) -cat ego ry C ( Γ ) . 1.2. Presentations by polygraphs W e define, by induc tion on n , the notions of n -poly graph, of presented ( n − 1 ) -categ ory a nd of freel y genera ted (track) n -categor y . For a dee per treatment, we refer the reader to [3, 12, 6]. A 1 -polygr aph is a graph Σ = ( Σ 0 , Σ 1 ) . W e denote by Σ ∗ the free 1 -cate gory and by Σ ⊤ the free track 1 -c ategory ( i.e. , groupoid ) it generates. A n ( n + 1 ) -polygr aph is a pair Σ = ( Σ n , Σ n + 1 ) made of an n -polygr aph Σ n and a cellul ar extensio n Σ n + 1 of the free n -cat ego ry Σ ∗ n genera ted by the n - polyg raph Σ n . The n -cate gory pr esent ed by Σ , the fr ee ( n + 1 ) -cate gor y genera ted by Σ and the fre e tra c k ( n + 1 ) -cat e gory gener ated by Σ are respe cti vely denoted by Σ , Σ ∗ and Σ ⊤ and d efi ned as fo llo ws: Σ = Σ ∗ n /Σ n + 1 , Σ ∗ = Σ ∗ n [ Σ n + 1 ] , Σ ⊤ = Σ ∗ n ( Σ n + 1 ) . An n -poly graph yield s a diagram of cellular extensi ons, as gi ven in [3]: Σ 0 Σ ∗ 1 Σ ∗ 2 Σ ∗ 3 Σ 0 Σ 1 f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O Σ 2 f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O Σ 3 f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O ( · · · ) g g N N N N N N N N N N N N N g g N N N N N N N N N N N N N If C is an n -cate gory , a pr esenta tion of C is an ( n + 1 ) -poly graph Σ such that Σ is isomorphic to C . 1.2.1. Poly graphic rewriti ng. Let Σ be an n -polygr aph. W e say that an ( n − 1 ) -cell u of Σ r educes to some ( n − 1 ) -cell v in Σ when there exists a non- deg enerate n -cell from u to v in Σ ∗ . A r eduction sequen ce of Σ is a countable famil y ( u i ) i ∈ I of ( n − 1 ) -cel ls of Σ such that each u i reduce s to the follo- wing u i + 1 . W e say that Σ termina tes when it has no infinit e reduction sequence. A bra nch ing of Σ is a non- ordered pair ( f, g ) of n -cells of Σ ∗ with the same source, called the sour ce of ( f, g ) . A branch ing ( f, g ) is conflu ent when there exists a pair ( f ′ , g ′ ) of n -cells of Σ ∗ with the same tar get and such that ( f, f ′ ) and ( g, g ′ ) are composable , as in the follo wing diagram, called a confluen ce dia gra m for the br anching ( f, g ) : f ′   f 2 2 g , , g ′ G G W e say that the n -poly graph Σ is confl uent when e very bra nching of Σ is confluent. Finally , the n -poly graph Σ is c on ver gen t when it terminates an d it is conflue nt. Follo wing [17], finite and con ver gent re writing systems, such as con ver gent polyg raphs, giv e an algorith mic way , the normal form algor ithm , to solve the word probl em for the algebra ic struct ure they pres ent: see [2] for presen - tation s of monoids by word (or string) rewritin g systems, [1] for presentat ions of equation al theories by term rewritin g systems and [6] for presentat ions of n -cate gories by polygra phs. Here, w e are interest ed in con ver gent polyg raphs becau se the y gi ve a way to compu te homotopy bases. 6 1.2. P r esentatio ns by polygra phs 1.2.2. Critical branchings. Here, we giv e the informal idea underlyin g the notion of critica l branch- ings. W e refer the reader to other works for a fuller trea tment of the subject: [2] for word rewrit ing systems; [1] for term re writing systems; [6 ], where the autho rs giv e a general theory of branching s in n - polyg raphs and a thoroug h study of critical branc hings of 3 -polygraph s; [7], where the authors describe resolu tions of small cate gories based on the critical branch ings (and gener alisatio ns) of presenta tions by con v er gent 2 -polygrap hs. Branchin gs in an n -polygraph Σ occur w hen an ( n − 1 ) -cell u of Σ ∗ contai ns the sources of two n -cells ϕ and ψ of Σ . When thos e source s are disjoint in u , the branch ing is confluent, such as in the follo wing simple case, w ith 0 ≤ i ≤ n − 1 : t ( ϕ ) ⋆ i s ( ψ ) t ( ϕ ) ⋆ i ψ " " u = s ( ϕ ) ⋆ i s ( ψ ) ϕ ⋆ i s ( ψ ) 5 5 s ( ϕ ) ⋆ i ψ ) ) t ( ϕ ) ⋆ i t ( ψ ) s ( ϕ ) ⋆ i t ( ψ ) ϕ ⋆ i t ( ψ ) < < Note that, in this example, both composites are equal to the n -cell ϕ ⋆ i ψ , due to the exchang e rela tion between ⋆ i and ⋆ n in Σ ∗ . Otherwise, when th e sources of ϕ and ψ overla p in u , in such a way that u is a minimal ( n − 1 ) - cell such that this ov erlapping occurs, we ha ve a critical branching . F or ex ample, in a 2 -polygra ph, w e can ha ve two dif ferent shape s of critical bran chings : / / / / F F / / ϕ E Y ψ   / /   / / B B / / ϕ E Y ψ   Here, we are intere sted in 3 -pol ygraph s e xclusi vely , for which we hav e giv en a complete classificatio n of critical branchi ngs, see [6]. The y are or ganised in three families, cov ering eight differe nt topolog ical configura tions of the o verl apping , tha t we will enco unter here in differe nt ex amples. In the case of the 3 -poly graph As 3 , as we hav e seen in the introducti on, we ha ve exac tly one critical bran ching, whose source is an ov erlapping of two copies of the source of the single 3 -cell : j * > V 4 7 1. Pr eliminaries The critica l branchings are essential in the study of con ver gence because , under the hypoth esis of termi- nation , their confluence ensure s the confluenc e of ev ery b ranchi ng. T his results relies on th e fun damental theore m of re writing theory , namely Newman’ s lemma, see [13], and on another result that depends on the type of re writing system we consi der . The case of n -polygr aphs is examin ed in [6]. Also, critic al branchings of con ver gent n -poly graphs giv e an algorith mic way to b uild homotopy bases of trac k n -cate gories. Indeed, for a gi ven con v ergent n -polygra ph Σ , we de fine a basis of g enerat- ing conflu ences of Σ as a cel lular extensio n of the free n -cate gory Σ ∗ made of one ( n + 1 ) -cell f ′     f 2 2 g , , g ′ F F for each c ritical bran ching ( f, g ) of Σ , where f ′ and g ′ are arbi trarily chosen n -ce lls of Σ ∗ with th e same tar get and such that ( f, f ′ ) and ( g, g ′ ) are composa ble. T hen we ha ve the follo wing result : 1.2.3. Theor em ([6]). Let Σ be a con ver gent n -polygr aph. Then every basi s of ge nera ting confluen ces of Σ is a homotopy basis of the trac k n -cate gory Σ ⊤ . 1.2.4. Example. The 3 -p olygra ph As 3 , se en in the intro duction, has one 0 -cell, one 1 - cell, one 2 -cell and one 3 -cell _ % 9 This 3 -pol ygraph terminates, see [6 ] or the proof of Proposit ion 2.3.3. It has exactl y one critical branch- ing, which is confluent : _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   As a conseque nce, fi lling this 3 -spher e with the 4 -cell , as abov e, yields a homotopy basis A s 4 of the track 3 -catego ry As ⊤ 3 . In other terms, any two parall el 3 -cells f and g of As ⊤ 3 are identified in the quotie nt track 3 -ca tego ry As ⊤ 3 / As 4 . 1.3. H igher -dimensional pr o(p)s 1.3.1. Higher -dimensional monoids. For n ≥ 1 , a (trac k) n -monoid is a (track) n -catego ry with exa ctly one 0 -cell, see [3] . In parti cular , a 1 -monoid is a monoid , a track 1 -monoid is a gro up, a 2 - monoid is a strict mon oidal cat ego ry , a track 2 -mono id is a strict mon oidal gr oupoid . More gen erally , for n ≥ 2 , a (track) n -monoid is a strict m onoida l cate gory enrich ed in (track) ( n − 2 ) -cate gories. When the corresp ondin g (enr iched) monoidal catego ry is symmetri c, we say that an n -monoid is symmetric . 8 1.3. H igher -dimensional pro(p)s 1.3.2. Higher -dimensional pro (p)s. For n ≥ 1 , a (tr ack) n -pr o is a (track) n -monoid whose under ly- ing ( 1 -)mono id is the monoid N of natural numbers with the addition . A (trac k) n -pr op is a symmetric (track) n -pr o. In pa rticula r , 2 -pro(p)s coincide with M ac Lane ’ s PR O(P)s, see [11]. Let us note tha t we could consider a more general definition of n -pro (p)s by repla cing the monoid N , which is the free monoid on one gener ator , by any free monoi d. Here, we are intereste d in track 3 -pr o(p)s and, more preci sely , in four main examples : the track 3 - pros AsCat of ca tego ries with an a ssociat i ve product (s ee the intr oducti on) and MonCat of mon oidal cate gories (see 2.3.1) and the track 3 -prop s SymCat of symmetric monoid al cate gories ( see 3.3.1) and BrCat of braid ed monoid al cate gories (see 4.1.2). W e will sho w , for each one of those 3 -pro(p )s, how to us e homotop y bases b uilt from con ver gent present ations in order to prove a coherence theorem for the corres pondin g algebras, a notion we introduce now . 1.3.3. Algebras o ver 3 -pro (p)s. W e see the (lar ge) monoidal 2 -cate gory Cat of (small) catego ries, functo rs and natural transformat ions as a (large) 3 -monoi d w ith cate gories as 1 -cel ls, functors as 2 - cells, natura l tran sformatio ns as 3 -cells, carte sian product as 0 -compo sition, composition of functors as 1 -compo sition, ve rtical compositio n of natu ral trans formation s as 2 -composi tion. If P is a 3 -pro (resp. 3 -prop), a P -algeb ra is a 3 -functor from P to Cat (resp. whose corre spondi ng strict monoida l 2 -func tor preserv es the symmetry). If A and B are P -al gebras, a morphism of P -alg ebra s fr om A to B is a natura l tran sformatio n from A to B , i.e. , a pair ( F , Φ ) where F : A ( 1 ) → B ( 1 ) is a functo r and Φ is a map se nding e very 2 -cell f : m ⇒ n in P to a natura l isomorphism with t he fol lo w ing shape, where C = A ( 1 ) and D = B ( 1 ) , D m B ( f )   Φ f   C m F m 3 3 A ( f ) , , D n C n F n @ @ such that the follo wing relations hold: • for ev ery 2 -cells f : m ⇒ n and g : p ⇒ q of P , we ha ve Φ f ⋆ 0 g = Φ f × Φ g : D m + p B ( f ⋆ 0 g ) # # Φ f ⋆ 0 g   D m × D p B ( f ) × B ( g ) % % Φ f × Φ g   C m + p F m + p 3 3 A ( f ⋆ 0 g ) + + D n + q = C m × C p F m × F p 3 3 A ( f ) × A ( g ) , , D n × D q C n + q F n + q ; ; C n × C q F n × F q 9 9 • for ev ery 2 -cells f : m ⇒ n and g : n ⇒ p in P , we ha ve Φ f ⋆ 1 g = ( Φ f ⋆ 1 B ( g )) ⋆ 2 ( A ( f ) ⋆ 1 Φ g ) : D m B ( f ⋆ 1 g )   Φ f ⋆ 1 g   C m F m 3 3 A ( f ⋆ 1 g ) , , D p C p F p @ @ = D m B ( f )   Φ f   C m F m 3 3 A ( f ) , , D n B ( g )   Φ g   C n A ( g ) , , F n q q 8 8 q q D p C p F p @ @ 9 2. Coher ence in monoidal cat egorie s • for ev ery 3 -cell A : f ⇛ g : m ⇒ n in P , we ha ve Φ f ⋆ 2 ( A ( A ) ⋆ 1 F n ) = ( F m ⋆ 1 B ( A )) ⋆ 2 Φ g : D m B ( f )   Φ f   C m F m 3 3 A ( f ) & & M M M M M M M A ( g ) ; ; A ( A )     {      D n C n F n @ @ = D m B ( f )   B ( g ) & & M M M M M M Φ g   B ( A )     |      C m F m 3 3 A ( g ) , , D n C n F n @ @ The P -algebr as and their morphi sms form a categor y , denoted by A lg ( P ) . 1.3.4. Coher ence pr oblem f or algebras o ver a 3 -pro (p). Let P be a 3 -pro(p) and let A be a P -a lgebra. A P -diagr am in A is the image A ( γ ) o f a 3 -sp here γ in P . A P -di agram A ( γ ) in A commutes if th e relatio n A ( s ( γ )) = A ( t ( γ )) is satisfied in C at . T he coher ence problem for algebras over a 3 -p ro(p) is: C O H E R E N C E P RO B L E M : Given a 3 -pr o(p ) P , does eve ry P -diagr am commute in ever y P -algebr a? As a conseq uence of the definition of an aspheri cal 3 -pro(p), we ha ve the follo wing sufficien t condit ion for gi ving a posit i ve answer to the coheren ce problem: 1.3.5. Pro position. If P is an aspherical 3 -pr o(p) then every P -diag ram commutes in ever y P -algebr a. 1.3.6. Example. Let AsCat be the track 3 -pro defined as the follo wing quotie nt: As = As ⊤ 3  As 4 = ( , , ) ⊤  . The cat egor y A lg ( AsCat ) is isomo rphic to the cate gory of (small) asso ciati ve categori es, the correspon - dence between an associ ati ve cate gory ( C , ⊗ , α ) and a 3 -fun ctor A : AsCat → Cat being gi ven by A ( ) = C , A ( ) = ⊗ , A ( ) = α. This corre sponde nce is w ell-define d since the coherence diagram satisfied by associati ve categorie s cor - respon ds to the 4 -cel l . W e hav e seen that A s 4 = { } is a homotop y bas is of As ⊤ 3 , so that AsCat is an asph erical track 3 -pro. As a consequen ce, in e very associati ve cate gory C , e very AsCat - diagra m is commutati ve. This fact can be info rmally restat ed as: ev ery diag ram b uilt in C from the functo r ⊗ and the natur al transformation α is commutati ve. 2 . C O H E R E N C E I N M O N O I DA L C A T E G O R I E S 2.1. Coher ence in algebras over track 3 -pr os 2.1.1. Pres entations of track 3 -pro s. Let P be a track 3 -pro. A pr esenta tion of P is a pair ( Σ 3 , Σ 4 ) , where Σ 3 is a 3 -pol ygraph and Σ 4 is a cellu lar ex tensio n of the free track 3 -cat ego ry Σ ⊤ 3 such that P ≃ Σ ⊤ 3 /Σ 4 . 10 2.2. Identities among r elations f or presentat ions of trac k 3 -pros Note that, in that case, the 3 -polyg raph Σ 3 has exa ctly one 0 -cell and one 1 -cell. A pre sentation of a track 3 -pro yie lds a d iagram which is similar to the on e corres pondin g to the induct i ve construc tion of a 4 -polyg raph, see S ection 1.2: { 0 } N Σ 2 Σ ⊤ 3 { 0 } { 1 } f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O Σ ∗ 2 f f L L L L L L L L L L L L L L f f L L L L L L L L L L L L L L O O O O Σ 3 f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O Σ 4 f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M A pre sentation ( Σ 3 , Σ 4 ) of P is con ver gent when Σ 3 is a con ver gent 3 -poly graph and Σ 4 is a cel lular ext ension of generati ng confluence s of Σ 3 . By d efinition, P is a n a spheri cal 3 -pro if and o nly if, for e very presentation ( Σ 3 , Σ 4 ) of P , the cellul ar ext ension Σ 4 is a homoto py basis. The latter cond ition is satisfied by any con ver gent present ation of P , yieldi ng the follo wing sufficien t con dition for gi ving a positi ve answer to the coh erence pro blem for P -algebra s: 2.1.2. Theor em. If a trac k 3 -pr o P admits a con ver gen t pr esentatio n then ev ery P -dia gram commutes in ev ery P -algebr a. 2.2. I dentities among relations f or pr esentations of track 3 -pros This sectio n is based on notions and results from [8], that we briefly recall first. 2.2.1. Contexts and n atural systems. Let C be an n -categor y . A cont e xt of C is an ( n + 1 ) -cell C of some free ( n + 1 ) -cate gory C [ x ] , where x is an n -sph ere of C , such that C contain s exactly one occurr ence of x . If f is an n -cell of C which is paralle l to x , we denot e by C [ f ] the n -cell of C obtained by repla cing x with f in C and, if D is a contex t which is parallel to x , we denote by C ◦ D the conte xt of C obtain ed by repl acing x with D in C . A whisker of C is a contex t of C that contain s only ( n − 1 ) -cells of C , apart from the n -sph ere x . Note that whisk ers of C a re in bijec ti ve corre spondence with conte xts of the ( n − 1 ) -catego ry C n − 1 underl ying C . The contexts of C form a catego ry whose objects are the n -cells of C and whose m orphis ms from f to g are the conte xts C of C such that C [ f ] = g . A natur al system on C is a functor from the cate gory of conte xts of C to the cate gory of abelia n groups . 2.2.2. Abelian tr ack n -ca tegory . Let T be a track n - cate gory . An n -c ell f of T i s c losed when its source and its tar get are equal; this common ( n − 1 ) -cell is the base cell of f . For eve ry ( n − 1 ) -cell u of T , the n -cell s of T , equi pped w ith the compos ition ⋆ n − 1 , form a group , w hich is deno ted by A ut T u . W e say that a track n -categor y T is abelian when eve ry group A ut T u is abeli an, i.e. , when, for ev ery closed n -ce lls f and g with same base cell, the rel ation f ⋆ n − 1 g = g ⋆ n − 1 f is satisfied. Note that, for an abelia n track n -cate gory T , the assignment o f e ach ( n − 1 ) -cel l u of T to t he ab elian gro up Aut T u ext ends to a natural system Aut T on the ( n − 1 ) -catego ry T n − 1 underl ying T . W e denote by T ab the abelia nised trac k n -cat e gory of T , defined as the quotient of T by the cellular ext ension made of one ( n + 1 ) -cell from f ⋆ n − 1 g to g ⋆ n − 1 f for e very pair ( f, g ) of closed n -cel ls of T with the same base cell. 11 2. Coher ence in monoidal cat egorie s 2.2.3. Identities among r elations f or n -polygraphs. Let Σ be an n -polygraph . W e denote by u the image of an ( n − 1 ) -cell u of Σ ⊤ ab by the cano nical projection to the ( n − 1 ) -catego ry Σ present ed by Σ . W e define the natura l system on Σ of identi ties among r elations of Σ , denoted by Π ( Σ ) , as follo ws. For an y ( n − 1 ) -cell u in Σ , the abelian group Π ( Σ ) u is defined as the group with one generator ⌊ f ⌋ for e very n -ce ll f : v → v of Σ ⊤ ab with v = u , subjected to the followin g relations: i) ⌊ f ⋆ n − 1 g ⌋ = ⌊ f ⌋ + ⌊ g ⌋ , for eve ry n -cells f, g : v → v of Σ ⊤ ab with v = u ; ii) ⌊ f ⋆ n − 1 g ⌋ = ⌊ g ⋆ n − 1 f ⌋ , for ev ery n -cells f : v → w and g : w → v of Σ ⊤ ab with v = w = u . For any conte xt C of Σ from u to v , the morphism of groups Π ( Σ ) C from Π ( Σ ) u to Π ( Σ ) v is gi ven, on a generato r ⌊ f ⌋ , by Π ( Σ ) C ( ⌊ f ⌋ ) = ⌊ b C [ f ] ⌋ , w here b C is any whisk er of Σ ⊤ ab that represent s the contex t C of Σ . In [8], the authors prov e that the natural system Π ( Σ ) is well-defined and, in particul ar , that its v alues on cont exts do not depend on the chosen representati ves. Moreo ver , the functor Π ( Σ ) is the uniqu e natural s ystem o n Σ , up to iso m orphis m, such that there exi sts an isomorphis m of natu ral systems on Σ ∗ n − 1 Φ : [ Π ( Σ ) − → Aut Σ ⊤ ab , where [ Π ( Σ ) is defined, on an ( n − 1 ) -cell u of Σ ⊤ ab , by [ Π ( Σ ) u = Π ( Σ ) u . The isomorp hism Φ is giv en, for an ( n − 1 ) -cel l u of Σ ⊤ ab and a closed n -cell f of Π ( Σ ) with base v such that u = v , by Φ ( ⌊ f ⌋ ) = g ⋆ n − 1 f ⋆ n − 1 g − , where g : u → v is any n -ce ll of Σ ⊤ ab . Let Γ be a cellul ar ext ension of Σ ⊤ . For each γ in Γ , w e den ote by e γ the follo wing n -cell of Σ ⊤ : e γ = s ( γ ) ⋆ n − 1 t ( γ ) − . W e define e Γ = { e γ, γ ∈ Γ } . When Γ is a homotop y basis, then the se t ⌊ e Γ ⌋ is a g enerati ng set of th e natura l system Π ( Σ ) , i.e. , e very element a of an y Π ( Σ ) u can be written a = n X i = 1 ε i C i ⌊ b i ⌋ , where each b i is a element o f e Γ , each C i is a context of Σ and each ε i is ± 1 . The p roof relies on an equi val ence between the facts that Γ is a homotopy basis and that ev ery n -cell f of Σ ⊤ can be written f =  g 1 ⋆ n − 1 C 1 [ e γ ε 1 1 ] ⋆ n − 1 g − 1  ⋆ n − 1 · · · ⋆ n − 1  g k ⋆ n − 1 C k [ e γ ε k k ] ⋆ n − 1 g − k  , where each γ i is in Γ , each ε i is ± 1 , e ach C i is a whisk er of Σ ⊤ and eac h g i is an n -cell of Σ ⊤ . W e r efer the reader to [8] for the proof . H ere, in the special case of prese ntations of track 3 -pros, we get: 2.2.4. Pro position. If P is an asp herical tra c k 3 -pr o then , for every pr esentation ( Σ 3 , Σ 4 ) of P , the natur al system Π ( Σ 3 ) on the 2 -pr o Σ 3 is gener ated by the set ⌊ e Σ 4 ⌋ . 12 2.3. Application: coher ence f or monoidal categories Pr oof. Since P is aspherical , then Σ 4 is a homotopy basis of the track 3 -cate gory Σ ⊤ 3 and, thus, of the abelia nised track 3 -ca tego ry ( Σ ⊤ 3 ) ab . Hence, any closed 3 -cell A in ( Σ ⊤ 3 ) ab can be written A =  A 1 ⋆ 2 C 1 [ e ω ε 1 1 ] ⋆ 2 A − 1  ⋆ 2 · · · ⋆ 2  A k ⋆ 2 C k [ e ω ε k k ] ⋆ 2 g − k  , where each ω i is in Σ 4 , each ε i is ± 1 , each C i is a whisk er of ( Σ ⊤ 3 ) ab and each A i is a 3 -cel l of ( Σ ⊤ 3 ) ab . Hence, any g enerato r ⌊ A ⌋ of Π ( Σ 3 ) f , for f a 2 -cell of Σ 3 , can be written ⌊ A ⌋ = k X i = 1 ⌊ A i ⋆ 2 C i [ e ω ε i i ] ⋆ 2 A − i ⌋ = k X i = 1 ε i C i ⌊ e ω i ⌋ . Thus, the elements of ⌊ e Σ 4 ⌋ form a genera ting set for Π ( Σ 3 ) . 2.2.5. Cor ollary . Let P be an aspher ical trac k 3 -pr o and let ( Σ 3 , Σ 4 ) be a pr esenta tion of P . If Σ 4 is finite , then the natur al system Π ( Σ 3 ) is finit ely gener ated. 2.3. Application: coherence f or monoidal categories W e recall that a mon oidal cate gory is a cate gory C , equipped with two functors ⊗ : C × C → C and e : ∗ → C , and three natural isomorphi sms α x,y,z : ( x ⊗ y ) ⊗ z → x ⊗ ( y ⊗ z ) , λ x : e ⊗ x → x, ρ x : x ⊗ e → x, such that the follo wing two diagrams commute in C : ( x ⊗ ( y ⊗ z )) ⊗ t α / / x ⊗ (( y ⊗ z ) ⊗ t ) α " " F F F F F F F F (( x ⊗ y ) ⊗ z ) ⊗ t α < < x x x x x x x x α ) ) R R R R R R R R R R R R R R c  x ⊗ ( y ⊗ ( z ⊗ t )) ( x ⊗ y ) ⊗ ( z ⊗ t ) α 5 5 l l l l l l l l l l l l l l x ⊗ ( e ⊗ y ) λ " " E E E E E E E E E ( x ⊗ e ) ⊗ y α 9 9 s s s s s s s s s ρ / / x ⊗ y c  A mono idal functor from C to D is a tri ple ( F , φ, ι ) made of a fu nctor F : C → D and two natu ral natural isomorph isms φ x,y : Fx ⊗ Fy → F ( x ⊗ y ) and ι : e → F ( e ) such that the follo wing diagrams commute in D : Fx ⊗ ( Fy ⊗ Fz ) 1 ⊗ φ / / Fx ⊗ F ( y ⊗ z ) φ ' ' O O O O O O O O O O O ( Fx ⊗ Fy ) ⊗ Fz α 7 7 n n n n n n n n n n n n φ ⊗ 1 ' ' P P P P P P P P P P P P c  F ( x ⊗ ( y ⊗ z )) F ( x ⊗ y ) ⊗ Fz φ / / F (( x ⊗ y ) ⊗ z ) Fα 7 7 o o o o o o o o o o o Fx ⊗ e ρ / / 1 ⊗ ι   c  Fx Fx ⊗ Fe φ / / F ( x ⊗ e ) Fρ O O e ⊗ Fx λ / / ι ⊗ 1   c  Fx Fe ⊗ Fx φ / / F ( e ⊗ x ) Fλ O O 13 2. Coher ence in monoidal cat egorie s 2.3.1. The 3 -pro of monoidal categories . Let M onCat be the 3 -pro presented by ( Mon 3 , Mon 4 ) , where Mon 3 is the 3 -poly graph with two 2 -cell s , and three 3 -cells _ % 9 _ % 9 _ % 9 and Mon 4 is the cellul ar extensio n of Mon ⊤ 3 made of the follo wing two 4 -cells: _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   9  & 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9  6 J                      _ % 9   2.3.2. Lemma. The cate gory of small monoidal cate gor ies and monoidal functor s is isomorp hic to the cate gor y Alg ( MonCat ) . Pr oof. For a monoid al categ ory ( C , ⊗ , e, α , λ, ρ ) , the corresp onding MonCat -algeb ra A is giv en by: A ( ) = C , A ( ) = ⊗ , A ( ) = e, A ( ) = α, A ( ) = λ, A ( ) = ρ. (3) The two commutati ve diag rams satisfied by mon oidal categori es corres pond to the MonCat -diagrams A ( ) and A ( ) . If ( F , φ, ι ) is a monoidal fu nctor , the correspond ing morphism Ψ of M onCat - algebr as is: Ψ = F , Ψ = φ, Ψ = ι. 2.3.3. Pro position ([6]). The cellul ar exten sion Mon 4 of the fr ee trac k 3 -cate gory Mon ⊤ 3 is a ho m otopy basis. Pr oof. First, we check that the 3 -polygraph M on 3 terminate s. W e reca ll the proof from [5], see also [6]. W e consider the 2 -functor X from Mon ∗ 2 to the category of order ed sets and monoto ne maps, seen as a 2 -cate gory with one 0 -cell : X ( ) = N \ { 0 } , X ( )( i, j ) = i + j, X ( ) = 1. Then, we consid er the follo wing assignment of 2 -cells of Mon 2 : ∂ ( )( i, j ) = i, ∂ ( ) = 0. This assignment extend s, in a unique way , to a deri vatio n of Mon ∗ 2 with value s in X , i.e. , a map ∂ that sends each 2 -cell f : m ⇒ n of Mon ∗ 2 to a monoto ne map ∂ ( f ) : N m → N that satisfies the follo wing relatio ns: ∂ ( f ⋆ 0 g )( i 1 , . . . , i m + n ) = ∂ ( f )( i 1 , . . . , i m ) + ∂ ( g )( i m + 1 , . . . , i m + n ) and ∂ ( f ⋆ 1 g )( i 1 , . . . , i m ) = ∂ ( f )( i 1 , . . . , i m ) + ∂ ( g ) ◦ X ( f )( i 1 , . . . , i m ) . 14 2.3. Application: coher ence f or monoidal categories W e check that, for ev ery 3 -cell α of Mon 3 , we ha ve X ( s ( α )) ≥ X ( t ( α )) and ∂ ( s ( α )) > ∂ ( t ( α )) , where m onoton e maps are compare d pointwis e. This implies that, for eve ry non-de generate 3 -cell A of Mon ∗ 3 , we ha ve ∂ ( s ( A )) > ∂ ( t ( A )) . Since ∂ takes its v alues in N , the 3 -polygra ph Mon 3 terminate s. For confluen ce, we stud y the critical branch ings of M on 3 : i t has fiv e critical branchi ngs and each of them is confluent . This yields a cellular ex tension Γ of Mon ⊤ 3 with five 4 -cells, the one s of Mon 4 plus the follo wing three 4 -c ells: <  ' < < < < < < < < < < < < < < < < < < < < < < < < < < < ~ 5 I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ % 9 ω 1   <  ' < < < < < < < < < < < < < < < < < < < < < < < < ~ 5 I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ % 9 ω 2   8  %  9 M ω 3   Hence Γ is a homotop y basis of Mon ⊤ 3 . T o pro ve that M on 4 is a homotop y basis , we sho w that, for each 4 -cell ω i , we ha ve s ( ω i ) = t ( ω i ) in MonCat . For ω 1 , we define the 4 -ce ll γ of Mon ⊤ 3 ( Mon 4 ) by the follo wing relation, w here we ab usiv ely denot e 3 -cells by the gener ating 3 -cell of Mon 3 the y contain: _ % 9    = 5  $    g ) = _ % 9 \ # 7 _ % 9 = _ e y = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 5 I ~ ~ ~ ~ ~ ~ ~ ~ ~ v 1 E γ As a conseq uence of this constr uction , we hav e s ( γ ) = t ( γ ) in M onCat . T hen we b uild the follo wing diagra m, proving that s ( ω 1 ) = t ( ω 1 ) also hold s:  }     3  # | | | | | | | | | | s  | | | | | | F F F F F F F  , _ e y q q q q q q q q q q q q q q q q q q q . B q q q q q q q q q q q q q q q O O O O O O O O O O O O O O O O O O O ] q O O O O O O O O O O O O O O O _ e y = = = For the 4 -cell ω 2 , one proce eds in a similar way , startin g with the 4 -cell . 15 3. Coher ence in symmetri c m onoidal cate gories Finally , let us consider the cas e of the 4 -cell ω 3 . W e define th e 4 -cell δ of Mon ⊤ 3 ( Mon 4 ) by the follo wing relation: A  * _ % 9 } 4 H N  0 p . B _ % 9 ⋆ 1 ω 2 δ = As a conseq uence, we ha ve s ( δ ) = t ( δ ) in MonCat . Hence, we also ha ve equality s ( δ ) ⋆ 2 = t ( δ ) ⋆ 2 . in MonCat , which relates the source and tar get of the follo wing diagram: .  b & : _ % 9 \ $ 8 H  . v 0 D  > R = = This gi ves s ( ω 3 ) = t ( ω 3 ) in MonCat , thus conclud ing the proof. W e can deduce, from this result an d Proposit ion 2.2.4, tha t the follo wing two elements for m a genera ting set fo r the natural s ystem of identi ties among rela tions Π ( Mon 3 ) on the 2 -pro Mon = Mon 3 of mon oids: ⌊ ^ ⌋ = ⌊ ⋆ 2 ⋆ 2 ⋆ 2   − ⋆ 2   − ⌋ ⌊ e ⌋ = ⌊ ⋆ 2 ⋆ 2   − ⌋ From Propositi on 2.3.3, we ha ve: 2.3.4. Cor ollary (Coherence theor em f or monoidal categories, [10]). The 3 -pr o MonCat is aspheri- cal. 3 . C O H E R E N C E I N S Y M M E T R I C M O N O I D A L C A T E G O R I E S 3.1. Presentations of track 3 -pr ops W e recall from [4] the follo wing characte risatio n of 2 -pro ps, deriv ed from a similar result for algebraic theori es [3]. 16 3.1. Pre sentations of track 3 -pr ops 3.1.1. Pro position. A 2 -pr o P is a 2 -pr op if and only if it conta ins a 2 -ce ll τ : 2 ⇒ 2 , r epr esented by , suc h that the following r elations hold: • The symmetry r elation τ ⋆ 1 τ = id 2 , = (4) • The Y ang- Baxter r elation ( τ ⋆ 0 1 ) ⋆ 1 ( 1 ⋆ 0 τ ) ⋆ 1 ( τ ⋆ 0 1 ) = ( 1 ⋆ 0 τ ) ⋆ 1 ( τ ⋆ 0 1 ) ⋆ 1 ( 1 ⋆ 0 τ ) , = (5) • F or every 2 -c ell f : m ⇒ n of P , the left and righ t naturali ty r elations for f , ( f ⋆ 0 1 ) ⋆ 1 τ n,1 = τ m,1 ⋆ 1 ( 1 ⋆ 0 f ) and ( 1 ⋆ 0 f ) ⋆ 1 τ 1,n = τ 1,m ⋆ 1 ( f ⋆ 0 1 ) , with the inductive ly defined notatio ns τ 0,1 = τ 1,0 = id 1 , τ n + 1,1 = ( n ⋆ 0 τ ) ⋆ 1 ( τ n,1 ⋆ 0 1 ) an d τ 1,n + 1 = ( τ ⋆ 0 n ) ⋆ 1 ( 1 ⋆ 0 τ 1,n ) . Gra phicall y , we r epr esent f by , an y τ n,1 by and any τ 1,n by , so that the nat ura lity r elation s for f ar e = and = (6) 3.1.2. The 2 -pro p o f permutatio ns. T he initial 2 -prop is the 2 -prop of permutation s, denoted by P erm , whose 2 -cells from n to n are the permutation s of { 1, . . . , n } and with no 2 -cell from m to n if m 6 = n . The 2 -prop Perm is prese nted by the 3 -pol ygraph with one 2 -cell and two 3 -cells , correspo nding to the symmetry relatio n (4) and the Y ang-Baxte r relation (5): ⇛ and ⇛ There ex ists an isomorphism between the categ ory of small categor ies and functors and the cate gory Alg ( Pe rm ) . The corresp ondence between a categ ory C and a Per m -algebr a A : Perm → C at is giv en by A ( 1 ) = C and A ( ) = T C , C , where T C , C is the endof unctor of C × C sending ( x, y ) to ( y, x ) . 3.1.3. Pres entations of 2 -pr ops. Let Σ be a 2 -pol ygraph with one 0 -c ell an d one 1 -cell. W e denote by SΣ the 3 -poly graph obtai ned from Σ by adjoining a 2 -cell : 2 ⇒ 2 and the follo wing 3 -cells: • The symmetry 3 -cell and the Y ang-Baxter 3 -cell, as in the 2 -prop Perm . • T wo 3 -cell s for e very 2 -cell f = of Σ , corresp onding to the natur ality relations for f : ⇛ and ⇛ 17 3. Coher ence in symmetri c m onoidal cate gories The fr ee 2 -pr op gener ated by Σ is the 2 -categ ory , denoted by Σ S , presente d by the 3 -po lygrap h SΣ . Let P be a 2 -prop. A pr esentat ion of P is a pair ( Σ 2 , Σ 3 ) , made of a 2 -poly graph Σ 2 with one 0 -cell and one 1 -cell and a cellula r extension Σ 3 of the free 2 -pro p Σ S 2 , such that P ≃ Σ S 2 /Σ 3 . 3.1.4. Pro position. A 3 -p r o P is a 3 -p r op if and o nly if it con tains a 2 -cell τ : 2 ⇒ 2 su ch that the followin g rel ations hold: • The symmetry r elation (4) and the Y ang-Baxte r rel ation (5) . • The natu ral ity r elations (6) for every 2 -cell of P . • F or every 3 -c ell A : f ⇛ g : m ⇒ n , the left and right natur ality r elations for A : ( A ⋆ 0 1 ) ⋆ 1 τ n,1 = τ m,1 ⋆ 1 ( 1 ⋆ 0 A ) and ( 1 ⋆ 0 A ) ⋆ 1 τ 1,n = τ 1,m ⋆ 1 ( A ⋆ 0 1 ) . Graph ically , we r epr esent f by and g by , so that the natur ality r elations for A ar e A A A A A A A A A A A A A A A } 4 H } } } } } } } } } } } } } } } } } } A A A A A A A A A A A A A A = A } 4 H } } } } } } } } } } } } } } } } } } A A A A A A A A A A A A A A A } 4 H } } } } } } } } } } } } } } } } } } A A A A A A A A A A A A A A = A } 4 H } } } } } } } } } } } } } } } } } } (7) Pr oof. This is an immediate exten sion of Proposition 3.1.1. 3.1.5. Pres entations of track 3 -props. L et Σ be a presen tation of a 2 -prop . W e denot e by SΣ the 4 - polyg raph obtain ed from the 3 -polyg raph SΣ 2 by adjoini ng the 3 -cells of Σ 3 and a cellular extensio n Σ 4 made of the follo wing two 4 -cel ls for each 3 -cell A of Σ 3 , correspond ing t o the natu rality relati ons (7) for A : A  * A A A A A A A A A A A A A A A A A A   A } 4 H } } } } } } } } } } } } } } } } } } A  * A A A A A A A A A A A A A A A A A A A } 4 H } } } } } } } } } } } } } } } } } } A  * A A A A A A A A A A A A A A A A A A   A } 4 H } } } } } } } } } } } } } } } } } } A  * A A A A A A A A A A A A A A A A A A A } 4 H } } } } } } } } } } } } } } } } } } 18 3.2. C on ver gent pres entations of algebraic track 3 -props and asphericit y The fr ee trac k 3 -pr op genera ted by Σ is the track 3 -categ ory , denoted by Σ S , gi ven by: Σ S = Σ S 2 ( Σ 3 ) /Σ 4 . Let P be a track 3 -prop. A pr esentation of P is a pair ( Σ 3 , Σ 4 ) , where Σ 3 is a presenta tion of a 2 -pro p and Σ 4 is a cellu lar ex tensio n of the free track 3 -pr op Σ S 3 , such that P ≃ Σ S 3 /Σ 4 . T o summari ze, a pre sentati on of P yields a diagra m which is s imilar to the o ne corres ponding to the induct i ve construc tion of a 4 -polyg raph, see S ection 1.2: { 0 } N Σ S 2 Σ S 3 { 0 } { 1 } f f M M M M M M M M M M M M M f f M M M M M M M M M M M M M O O O O Σ 2 f f L L L L L L L L L L L L L L f f L L L L L L L L L L L L L L O O O O Σ 3 f f L L L L L L L L L L L L L f f L L L L L L L L L L L L L O O O O Σ 4 f f L L L L L L L L L L L L L f f L L L L L L L L L L L L L 3.2. Co n ver gent pre sentations of algebraic track 3 -prop s and a sphericity 3.2.1. Con ver gent pre sentations of algebraic tra ck 3 -pr ops. A presentation Σ of a t rack 3 -pr op is con ver ge nt when the 3 -polygraph SΣ is con ver gent. A prese ntation Σ of a 2 -pro p (resp. track 3 -prop) is alg ebra ic when eve ry 2 -c ell (r esp. e very 2 - cell a nd e very 3 -cell ) of Σ has 1 -tar get equal to the gene rating 1 -cell 1 . A track 3 -prop is alg ebraic w hen it admits an algebr aic prese ntation. 3.2.2. Classificatio n of cri tical b ranchings. Let Σ be an algeb raic presenta tion of a 2 -prop P . W e recall from [4, 5] that the critical branch ings of the 3 -pol ygraph S Σ are classi fied as follo ws: 1. Fi ve cri tical branchin gs generate d by the symmetry and Y ang-Baxter 3 -cells, whose sources are: 2. For ev ery 2 - cell ϕ = o f Σ , fiv e critica l branch ings, genera ted, on the one hand, by the n aturali ty 3 -cells for ϕ and, on the other hand, by the symmetry and Y ang-Baxter 3 -cells: 3. For e very pair ( ϕ, ψ ) of 2 -cells of Σ , one critical branching gener ated by the left naturalit y 3 -cell of ϕ = and the right natur ality 3 -cell of ψ = : 19 3. Coher ence in symmetri c m onoidal cate gories 4. For e very algeb raic 3 -cell α : f ⇛ g of Σ , two critical branch ings gen erated by α and the natural ity 3 -cells for f = : 5. The other critica l branchings, called the pr oper critical br anc hings of Σ . All of the critical branchin gs of the first three famili es are confluent and their confluence diag rams are sent to commutati ve diagrams by the canonica l project ion π : SΣ ⊤ → Σ S . The critic al branchings of the fourth f amily are co nfluent and the ir confluence diag rams are sent to 3 - sphere s w hich are t he boun daries of natur ality 4 -cell s. A basis of pr oper conflue nces o f Σ is a cellula r ex tension of the free track 3 -cate gory Σ ⊤ that conta ins, for each prope r critical branchi ng b of Σ , one 4 -cell ω b : A  ? B , where the 3 -sph ere ( A, B ) is a confluen ce diagram for b . W e as sume that, when Σ is a con v ergent 3 -polygraph , we hav e chosen a basis of prop er confluences, w hich we denote by Γ Σ . 3.2.3. Lemma. Let Σ be an algeb rai c con ver gen t pr esentatio n of a 2 -pr op P . Then the imag e π ( Γ Σ ) of the cellula r extens ion Γ Σ thr ough the canonica l pr ojecti on π : S Σ ⊤ → Σ S is a homotopy basis of P . 3.2.4. Theor em. If a trac k 3 -pr op P admits an algebr aic con ver ge nt pr esentation ( Σ 3 , Σ 4 ) suc h that Σ 4 is T ietze -equiva lent to π ( Γ Σ 3 ) , then P is asph erical . 3.3. Application to symmetric monoidal categories A symmetric monoi dal ca te gory is a mono idal c ateg ory ( C , ⊗ , e, α, λ, ρ ) equipped with a natural isomor - phism τ x,y : x ⊗ y − → y ⊗ x, called the symmetry and such that the follo wing two diagrams commute in C : y ⊗ x τ ! ! B B B B B B B B x ⊗ y τ = = | | | | | | | | x ⊗ y c  x ⊗ ( y ⊗ z ) τ / / ( y ⊗ z ) ⊗ x α & & L L L L L L L L L L ( x ⊗ y ) ⊗ z α 8 8 r r r r r r r r r r τ & & L L L L L L L L L L c  y ⊗ ( z ⊗ x ) ( y ⊗ x ) ⊗ z α / / y ⊗ ( x ⊗ z ) τ 8 8 r r r r r r r r r r (8) A symmetric monoidal functor from C to D is a monoida l functo r ( F , φ, ι ) such that the follo wing dia- gram commutes in D : Fx ⊗ Fy τ / / φ   c  Fy ⊗ Fx φ   F ( x ⊗ y ) Fτ / / F ( y ⊗ x ) (9) 20 3.3. Application to symmetric monoidal catego ries 3.3.1. The track 3 -pr op of symmetric monoidal cate gories . Let SymCat be th e track 3 -pro p pre- sented by Sym gi ven as follo w s: • Sym 2 is the 2 -poly graph Mon 2 , contain ing two 2 -ce lls and . • Sym 3 is the cell ular e xtensi on of the fre e 2 -prop S ym S 2 genera ted by S ym 2 contai ning the t hree 3 -cells of Mon 3 _ % 9 _ % 9 _ % 9 plus the follo wing extra 3 -cell: _ % 9 • Sym 4 is the cellular extensio n of the free 3 -prop Sym S 3 genera ted by Sym 3 contai ning the two 4 -cells of Mon 4 _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   .  ! x 1 E _ % 9   plus the follo wing two extra 4 -cells: -  x 1 E   _ % 9 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 x 1 E x x x x x x x x x x x x x x x x x x x x x x x x 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 _ % 9 z 2 F 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   3.3.2. Lemma. The ca te gory of s mall sy mmetric monoida l cate gor ies and sy mmetric mon oidal functo rs is isomorp hic to the ca te gory Alg ( SymCat ) . Pr oof. Giv en a symmetric monoid al cate gory ( C , ⊗ , e, α, λ, ρ, τ ) , the corres ponde nce with a SymCat - algebr a A is giv en by (3 ) for the monoid al under lying structure and by A ( ) = τ for the symmetry . The two c omm utati ve diagrams of a monoidal cate gory correspond to A ( ) and A ( ) and the commutati ve diagrams (8) corresp ond to A ( ) and A ( ) . 21 3. Coher ence in symmetri c m onoidal cate gories The cor respon dence of a symmetric monoidal fun ctor ( F , φ, ι ) with a m orphis m Ψ between the asso - ciated Sym -algebra s is gi ven by: Ψ = F , Ψ = φ, Ψ = ι. The relatio n (9) corr espond s to the properti es of the morphi sm Ψ . 3.3.3. A con ver gent pr esentation of SymCat. W e define Sym ′ as the presentatio n Sym of SymCat , ext ended with one 3 -cell _ % 9 and the follo wing 4 -cel l: E  , E E E E E E E E E E E E E E E E E E ω   y 2 F y y y y y y y y y y y y y y y y y y C  + C C C C C C C C C C C C C C C C C C { 3 G { { { { { { { { { { { { { { { { { { 3.3.4. Lemma. The trac k 3 -pr op SymCat is pr esente d by Sym ′ . Pr oof. The 4 -cell ω induces the relatio n =   − ⋆ 2 ⋆ 2 in the quotient tr ack 3 -prop ( Sym ′ 3 ) S / Sym ′ 4 . As a consequenc e, it is isomorphic to th e quotient tra ck 3 -prop SymCat = ( Sym 3 ) S / Sym 4 . 3.3.5. Pro position. The 3 -poly grap h S ( Sym ′ 3 ) is con ver gen t and the cellula r ext ension Sym ′ 4 is T ietze- equiva lent to π ( Γ S ( Sym ′ 3 ) ) . Pr oof. The con ver gence of the 3 -polygraph S ( Sym ′ 3 ) is pro ved in [5]. The image through the canoni cal projec tion π : S ( Sym ′ 3 ) ⊤ → ( Sym ′ 3 ) S of the cellular extens ion Γ S ( Sym ′ 3 ) has ten 4 -cells. Inde ed, i t contains the images of four 4 -cells of Sym 4 _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   .  ! x 1 E _ % 9   22 3.3. Application to symmetric monoidal catego ries and -  x 1 E   _ % 9 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 x 1 E x x x x x x x x x x x x x x x x x x x x x x x x 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 _ % 9 z 2 F 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   plus the ext ra 4 -cell ω of Sym ′ 4 E  , E E E E E E E E E E E E E E E E E E ω   y 2 F y y y y y y y y y y y y y y y y y y C  + C C C C C C C C C C C C C C C C C C { 3 G { { { { { { { { { { { { { { { { { { and, finally , the follo w ing five 4 -cells: N N N N N N N N N N N N N N N N N N N N N N N N p . B 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 : : : : : : : :  8 L             ω 1   <  ' < < < < < < < < < < < < < < < < < < < < < ω 2   } 4 H } } } } } } } } } } } } } } } } } } A A A A A A A A A A A A A A  7 K                      <  ' < < < < < < < < < < < < < < < < < < < < < ω 3   } 4 H } } } } } } } } } } } } } } } } } } A A A A A A A A A A A A A A  7 K                      and Q  2 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 Q Q Q 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 m m m m = = = = = = = = = = _ % 9  6 J                ω 4   N  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 N N N N N N N N N N N N p . B 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 p p p p p p p p p = = = = = = = = = = _ % 9  6 J                ω 5   In order to sho w that Sym ′ 4 is Ti etze-eq ui valen t to π ( Γ S ( Sym ′ 3 ) ) , we check that, for each one of the fiv e 4 - cells ω i , we hav e the rela tion s ( ω i ) = t ( ω i ) in the qu otient track 3 -p rop SymCat . The proje ction sends ω 1 to one of the natural ity relation s for . For each one of the other 4 -cells ω i , with 2 ≤ i ≤ 5 , we consid er a 4 -cell W i of the tra ck 4 -pro p S ym ⊤ 3 ( Sym 4 ) , b uilt as an inst ance of the 4 -c ell compose d with 2 -cell s: W 2 = , W 3 = , W 4 = , W 5 = . 23 4. Coher ence for braided monoidal categ ories On the one hand, by definiti on, the boundary of W i satisfies the relati on s ( W i ) = t ( W i ) in the quotien t track 3 -prop SymCat . On th e other hand, we progressi vely fi ll the boundary of W i , as in the case of the track 3 -pro p Mo nCat , w ith 4 -cel ls of Sym 4 , plus exchang e and natura lity re lations , until reaching the bound ary of the 4 -c ell ω i (or of ω − i ), thus yieldi ng the result . 3.3.6. Cor ollary (Coher ence theor em for symmetric monoidal categori es, [10]). The trac k 3 -pr op SymCat is asphe rical. 4 . C O H E R E N C E F O R B R A I D E D M O N O I D A L C A T E G O R I E S 4.1. G eneralised coher ence problem A braid ed monoidal cate gory is a monoidal cate gory ( C , ⊗ , e, α , λ, ρ ) equipped w ith a natural isomor - phism β x,y : x ⊗ y − → y ⊗ x, called the brai ding and such that the follo wing diagr ams commute in C : x ⊗ ( y ⊗ z ) β / / ( y ⊗ z ) ⊗ x α & & L L L L L L L L L L ( x ⊗ y ) ⊗ z α 8 8 r r r r r r r r r r β & & L L L L L L L L L L c  y ⊗ ( z ⊗ x ) ( y ⊗ x ) ⊗ z α / / y ⊗ ( x ⊗ z ) β 8 8 r r r r r r r r r r and x ⊗ ( y ⊗ z ) β − / / ( y ⊗ z ) ⊗ x α & & L L L L L L L L L L ( x ⊗ y ) ⊗ z α 8 8 r r r r r r r r r r β − & & L L L L L L L L L L c  y ⊗ ( z ⊗ x ) ( y ⊗ x ) ⊗ z α / / y ⊗ ( x ⊗ z ) β − 8 8 r r r r r r r r r r A braided monoidal functo r from C to D is a monoidal funct or ( F , φ, ι ) such that the following diagram commutes in D : Fx ⊗ Fy β / / φ   c  Fy ⊗ Fx φ   F ( x ⊗ y ) Fβ / / F ( y ⊗ x ) 24 4.1. Generalise d coherence problem 4.1.1. Generalised coher ence th eor em. Contrary to the case of monoidal and symmetric monoidal cate gories , we do not ha ve that ev ery diagram commutes in a braide d monoidal cate gory . For examp le, the morphisms β x,y and β − y,x , from x ⊗ y to y ⊗ x , hav e no reason to be equa l. In f act, they are equal if and only if β is a symmetry , hence if and only if all diagrams commute. As a cons equenc e, the cohe rence prob lem for braided monoidal catego ries requires a generalised ver sion of the coherenc e proble m we ha ve consider ed so far . T H E G E N E R A L I S E D C O H E R E N C E P RO B L EM : Given a trac k 3 -pr op P , decide , for any 3 -spher e γ of P , w hether or not the diagr am A ( γ ) commutes in ever y P -algebr a A . Hence, a solution for the generalise d cohere nce p roblem is a decision proced ure for the equality of 3 - cells of P . For the coherence problems consi dered so far , this decision procedure answers yes for e very 3 -sphe re. W e consid er methods to study the generalis ed coherence theorem of 3 -props and we illustrate those methods on the track 3 -pro p of braid ed monoidal categor ies. 4.1.2. The track 3 -prop of b raided monoidal categor ies. Let BrCat be the track 3 -prop with the presen tation B r defined as follo ws: • The 2 -polyg raph B r 2 is Mon 2 , conta ining the two 2 - cells an d . • The cellula r exte nsion Br 3 of Br S 2 has the same four 3 -cel ls as Sym 3 : _ % 9 _ % 9 _ % 9 _ % 9 • The cellula r exte nsion Br 4 of Br S 3 has four 4 -cells , the tw o 4 -cells of Mon 4 _ % 9 E  , E E E E E E E E E E E E y 2 F y y y y y y y y y y y y S  3 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 S S S S S S S S S S S S S 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 k k k k k k k k k k k k k k k   .  ! x 1 E _ % 9   plus the follo wing two 4 -cells: _ % 9 J  . J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J t 0 D 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 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 H H _ % 9 v 0 D 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 v v v v v 1   25 4. Coher ence for braided monoidal categ ories and _ % 9   − J  . J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J   − t 0 D 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 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 H H   − _ % 9 v 0 D 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 v v v v v 2   The corres ponden ce between symmetric monoidal cate gories and SymCat -algebras can be extend ed to braide d m onoida l cate gories : 4.1.3. Lemma. The cate gory of small braide d monoi dal c ate gorie s and braided monoidal fu nctor s is isomorph ic to the cate gory A lg ( BrCat ) . 4.2. Pr eservation of coher ence by equivalences 4.2.1. Equivale nce of track 3 -pr ops. Let P and Q be trac k 3 -props. A morphism of trac k 3 -pr ops from P to Q is a 3 -functor F : P → Q which is the identity on 1 -cells, i.e. , F ( n ) = n for ev ery 1 -cel l n in N . If F , G : P → Q are two morphisms of track 3 -pro ps, a natur al trans formation fr om F to G is a family α of 3 -cells of Q α f : F ( f ) ⇛ G ( f ) inde xed by the 2 -ce lls of P and such that, for ev ery 3 -cell A : f ⇛ g of P , the fol lo w ing diagram commutes in Q : F ( f ) α f _ % 9 F ( A )    c  G ( f ) G ( A )    F ( g ) α g _ % 9 G ( g ) . If F : P → Q is a mor phism of track 3 -pr ops, a quasi-i n vers e for F is a morphism of track 3 -props G : Q → P such that there exist natur al isomorphisms GF ≃ id P and FG ≃ id Q . An equiva lence between P and Q is a morphis m of track 3 -props F : P → Q that admits a quasi- in ver se. 4.2.2. Pro position. Let F : P → Q be an equivalen ce between trac k 3 -pr ops P and Q and let ( A, B ) be a 3 -sph er e of P . Then A = B if and only if F ( A ) = F ( B ) . Pr oof. Let ( A, B ) : f ⇛ g be a 3 -spher e of P such that F ( A ) = F ( B ) . W e denote by G : Q → P a quasi-in ve rse of F and by α the natura l isomorph ism from GF to id P . W e ha ve, by definitio n of α , 26 4.3. P r eser vati on of coher ence b y aspherical quotient commutati ve diagrams in P : GF ( f ) α f _ % 9 GF ( A )    c  f A    GF ( g ) α g _ % 9 g GF ( f ) α f _ % 9 GF ( B )    c  f B    GF ( g ) α g _ % 9 g By hypoth esis, w e ha ve GF ( A ) = GF ( B ) . Thus: A = α − f ⋆ 2 GF ( A ) ⋆ 2 α g = α − f ⋆ 2 GF ( B ) ⋆ 2 α g = B. 4.3. Preserv a tion of coher ence by as pherical quotient If P and Q are trac k 3 -prop s with Q ⊆ P , we denot e by P / Q the quot ient of P b y the 3 -cells of Q and by π : P → P / Q the canonical projection . 4.3.1. Theor em. Let P and Q be tra c k 3 -pr ops w ith Q asp herica l and Q ⊆ P . Then, for e very 3 -sphe r e ( A, B ) of P , we have A = B if and only if π ( A ) = π ( B ) . Pr oof. Let ( f, g ) be a 2 -sph ere of P . Since Q is aspherica l, the 3 -cells of P from f to g are in biject iv e corres ponden ce with the 3 -cells of P / Q from π ( f ) to π ( g ) . By Corollary 2.3.4, the track 3 -pro(p ) M onCat is asphe rical, so that we hav e: 4.3.2. Cor ollary . Let ( A, B ) be a 3 -spher e of BrCat . Then we have A = B in BrCat if and only if we have π ( A ) = π ( B ) in BrCat / MonCat . 4.4. The i nitial algebra of an algebraic 2 -pr op 4.4.1. Algebraic cells. Let P be an algebraic 2 -prop, with an algebraic presen tation Σ . A 2 -cell f of P is pur ely algebr aic when it is algebra ic, i.e . , it has tar get 1 , and it is the image of a 2 -cell of Σ ∗ 2 by the canon ical projection Σ ∗ 2 → P , i.e. , it contains no 2 -cell . If f : n ⇒ 1 is an alg ebraic 2 -cell of P , th en the nat urality relatio ns (6) satisfied in P imply that f can be decompos ed, in a uniq ue way , as f = σ f ⋆ 1 b f, where b f : n ⇒ 1 is a purely alg ebraic 2 -cell of P and σ f : n ⇒ n is the image o f a 2 -cell of Per m by the canonica l inclus ion P erm → P , i.e. , a 2 -cell of P written with only . If we identify σ f with the corresp ondin g permutati on of { 1, . . . , n } , w e ha ve, for ev ery P -algeb ra A , the follo wing relati on, for e ver y family ( x 1 , . . . , x n ) of obje cts of the cate gory A ( 1 ) : A ( f )( x 1 , . . . , x n ) = A ( b f )( x σ f ( 1 ) , . . . , x σ f ( n ) ) . The 2 -cell b f can be iden tified with the equi v alence class of f modulo the congruen ce gener ated by σ ⋆ 1 f ≈ τ ⋆ 1 f 27 4. Coher ence for braided monoidal categ ories for any permutat ions σ and τ . Similarl y , we deno te by b A the equi v alence clas s of an algebraic 3 -cell A of P modulo the congr uence gene rated by σ ⋆ 1 A ≈ τ ⋆ 1 A for any p ermutatio ns σ and τ . 4.4.2. Initial algebras. Let P be an algeb raic track 3 -prop. The in itial P -alg ebr a is the P -a lgebra P defined as follo ws. The categor y P ( 1 ) is gi ven by: • Its objec ts are the purely algebraic 2 -cells of P , i.e. , the equi vale nce clas ses b f , for f any algebraic 2 -cell of P . • Its morp hisms are the equ i v alence classes b A fo r A an y algebr aic 3 -cell of P . For such a 3 -c ell A : f ⇛ g : n ⇒ 1 , the corr espond ing morphism b A of P has source b f and tar get b g . • The composit e of A : f ⇛ g and B : h ⇛ k , with b g = b h , is defined by A · B = ( σ − g ⋆ 1 A ) ⋆ 2 ( σ − h ⋆ 1 B ) . • The identit y of a f : n ⇒ 1 is b id f . If f : n ⇒ 1 is an algebra ic 2 -cell of P , then the functo r P ( f ) : P ( n ) ⇒ P ( 1 ) is defined by P ( f ) ( x 1 , . . . , x n ) = ( x 1 ⋆ 0 · · · ⋆ 0 x n ) ⋆ 1 f. Note that, using the natural ity relat ions for 2 -cells of P , we hav e: P ( f ) ( x 1 , . . . , x n ) ≈ ( x σ f ( 1 ) ⋆ 0 · · · ⋆ 0 x σ f ( n ) ) ⋆ 1 b f. If A : f ⇛ g : n ⇒ A is an al gebraic 3 -cell of P , then the component at ( x 1 , . . . , x n ) of the natural transfo rmation P ( A ) is giv en by P ( A ) ( x 1 ,...,x n ) = ( x 1 ⋆ 0 · · · ⋆ 0 x n ) ⋆ 1 A with sourc e ( x 1 ⋆ 0 · · · ⋆ 0 x n ) ⋆ 1 f ≈ ( x σ f ( 1 ) ⋆ 0 · · · ⋆ 0 x σ f ( n ) ) ⋆ 1 b f and tar get ( x 1 ⋆ 0 · · · ⋆ 0 x n ) ⋆ 1 g ≈ ( x σ g ( 1 ) ⋆ 0 · · · ⋆ 0 x σ g ( n ) ) ⋆ 1 b g. 4.4.3. Theor em. Let P be an algebr aic trac k 3 -pr op and let ( A, B ) be a 3 -spher e of P . Then A = B if and only if P ( A ) = P ( B ) . Pr oof. Let us assume that A, B : f ⇛ g : m ⇒ n are s uch that P ( A ) = P ( B ) . T hen we ha ve, by definitio n of P , for e very algebr aic 2 -cel ls x 1 , . . . , x m of P : ( x 1 ⋆ 0 · · · ⋆ 0 x m ) ⋆ 1 A ≈ ( x 1 ⋆ 0 · · · ⋆ 0 x m ) ⋆ 1 B. In particul ar , we take id 1 for each x i to get A ≈ B . Since A and B hav e the same source and the same tar get, we must ha ve A = B . 28 4.5. The coher ence theor em f or b raided monoidal categ ories 4.5. The coher ence theorem f o r braided monoidal categories 4.5.1. The 2 -pro of braids. W e define the 2 -pro of braids as the 2 -pr o denote d by Brd and presented by the 2 -pol ygraph with two 2 - cells and an d the follo w ing three 3 -cells ⇛ ⇛ ⇛ In parti cular , those 3 -cells also gener ate the fiv e follo wing equalities in Brd : = = = and = = The oppos ite 2 -pro Brd o is the 2 -pro Brd with compositio n ⋆ 1 re versed. 4.5.2. Pro position. The u nderly ing cate gory B ( 1 ) of the initial a lgeb ra B of BrCat / MonCat is iso- morphic to the 2 -pr o Brd o . Pr oof. W e note that, in the quotient track 3 -prop BrCat / MonCat , there is exact ly one purely algebraic 2 -cell for each natural number n . In particular , for n = 0 and n = 1 , thos e are id 0 and id 1 , respecti vely , for n = 2 , that is and, for n ≥ 3 , that is the equiv alence class of any algebra ic 2 -cell of B rCat that contai ns ex actly ( n − 1 ) copies of . Thus, the unde rlying ca teg ory B ( 1 ) of t he initial B rCat / MonCat -algeb ra B has the natu ral nu m bers as objec ts. M oreo ver , it is equi pped with a structure of 2 -pro by the product ⊗ defined by m ⊗ n = m + n and A ⊗ B = ( A ⋆ 0 B ) ⋆ 1 Graphica lly , if A = and B = , this produ ct is written: ⊗ = . Let us define a morphi sm Φ : Brd o → B ( 1 ) of 2 -pro s. On generating 2 -cells, we define Φ ( ) = and Φ ( ) = − Let us pro ve th at this ind uces a morphism of 2 -p ros by ch ecking that thi s is comp atible with the g enerat- ing 3 -cells of Brd . For the first 3 -cell, we hav e: Φ   = Φ ( ) · Φ ( ) ≈ − ⋆ 2 = = Φ ( id 2 ) . 29 4. Coher ence for braided monoidal categ ories W e pro ve, in a similar way , the relatio n Φ   ≈ Φ ( id 2 ) . For the last 3 -cell, we compute , on the one hand: Φ   = ( Φ ( ) ⊗ 1 ) · ( 1 ⊗ Φ ( )) · ( Φ ( ) ⊗ 1 ) ≈ ⋆ 2 ⋆ 2 = ⋆ 2 = W e hav e used the relation induced by the 4 -cel l 1 for the third equality and the ex change relation between ⋆ 1 and ⋆ 2 for the last equa lity . On the other han d, using the same properties, w e get: Φ   = ( 1 ⊗ Φ ( )) · ( Φ ( ) ⊗ 1 ) · ( 1 ⊗ Φ ( )) ≈ ⋆ 2 ⋆ 2 = ⋆ 2 = Con ver sely , let us define a morphism Ψ : B ( 1 ) → B rd o of 2 -pros. U sing the exchan ge relation between ⋆ 1 and ⋆ 2 , one can write any alge braic 3 -cell A of BrCat / MonCat as a composite A = A 1 ⋆ 2 · · · ⋆ 2 A k , (10) where each A i is an algebr aic 3 -cel l of BrCat / MonCat that conta ins exactly one genera ting 3 -cell, i.e. , exa ctly one copy of either or − . Moreov er , this decompos ition is unique up to the in verse relation s and the exch ange relati ons between ⋆ 0 and ⋆ 2 and between ⋆ 1 and ⋆ 2 . The 4 -cells 1 and 2 genera te the followin g relations in BrCat / MonCat = ⋆ 2 and   − =   − ⋆ 2   − (11) which, in turn, using the in vers e relations, induce   − =   − ⋆ 2   − and = ⋆ 2 (12) Those four relations ha ve se veral consequen ces. The first one is that, in the deco mpositio n (10) , we can assume that each A i has sha pe m ⊗ ε ⊗ n =   ε 30 4.5. The coher ence theor em f or b raided monoidal categ ories with ε in { − , + } . In other terms, the 2 -pro B admits and − as generators . W e define a morphism Ψ : B ( 1 ) → Brd o of 2 -pro s by Ψ ( ) = and Ψ ( − ) = This morphis m is well-d efined if and onl y if it is compatible with the in verse relations and the exchang e relatio ns of BrCat / MonCat . Fo r the in verse relations, we use the fact that is the in verse of in the 2 -pro of braids . For the exch ange relation s between ⋆ 0 and ⋆ 2 , we use the relations (11) and (12) to deduce that the y are generat ed by the fou r relations   ε 1 ⋆ 2   ε 2 =   ε 2 ⋆ 2   ε 1 where ε 1 and ε 2 range ov er { − , + } . W e check that, for each one, Ψ sends both sides to the same braid. For e xample, in the case ε 1 = ε 2 = + , we get Ψ  ⋆ 2  = and Ψ  ⋆ 2  = The relations (11) and (12) also induc e that the e xchange relat ions bet ween ⋆ 1 and ⋆ 2 in BrCat / MonCat are generat ed by the eigh t relations   ε 1 ⋆ 2   ε 2 =   ε 2 ⋆ 2   ε 1 and   ε 1 ⋆ 2   ε 2 =   ε 2 ⋆ 2   ε 1 where ε 1 and ε 2 range o ver { − , + } . W e check that Ψ is compati ble with them. 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Y V E S G U I R AU D I N R I A , I N S T I T U T C A M I L L E J O R DA N , CNRS, Université de L yon, Université L yon 1 Bâtiment Braconnier, 43 boulev a rd du 11 novembre 191 8, 69622 V illeur banne Cedex, France E-mail address: guirau d@math.un iv- lyon1. fr P H I L I P P E M A L B O S I N S T I T U T C A M I L L E J O R DA N , CNRS, Université de L yon, Université L yon 1 Bâtiment Braconnier, 43 boulev a rd du 11 novembre 191 8, 69622 V illeur banne Cedex, France E-mail address: malbos@math .univ- ly on1.fr 32