Identities among relations for higher-dimensional rewriting systems
aa r X i v : . [ m a t h . C T ] A p r Identities among relationsfor higher-dimensional rewriting systems
Yves Guiraud Philippe Malbos
INRIA Nancy Université Lyon 1LORIA Institut Camille [email protected] [email protected]
Abstract –
We generalize the notion of identities among relations, well known for presentations of groups,to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category,generalizing the notion of crossed module for groups, in order to define the natural system of identitiesamong relations. We relate the facts that this natural system is finitely generated and that the polygraphhas finite derivation type.
Support –
This work has been partially supported by ANR Inval project (ANR-05-BLAN-0267). I NTRODUCTION
The notion of identity among relations originates in the work of Peiffer and Reidemeister, in combina-torial group theory [14, 17]. It is based on the notion of crossed module , introduced by Whitehead, inalgebraic topology, for the classification of homotopy -types [20, 21]. Crossed modules have also beendefined for other algebraic structures than groups, such as commutative algebras [16], Lie algebras [11]or categories [15]. Then Baues has introduced track -categories , which are categories enriched ingroupoids, as a model of homotopy -type [2, 1], together with linear track extensions , as generaliza-tions of crossed modules [4].There exist several interpretations of identities among relations for presentations of groups: as ho-mological -syzygies [5], as homotopical -syzygies [12] or as Igusa’s pictures [12, 10]. One can alsointerpret identities among relations as the critical pairs of a group presentation by a convergent wordrewriting system [7]. This point of view yields an algorithm based on Knuth-Bendix’s completion pro-cedure that computes a family of generators of the module of identities among relations [9].In this work, we define the notion of identities among relations for n -categories presented by higher-dimensional rewriting systems called polygraphs [6], using notions introduced in [8]. Given an n -polygraph Σ , we consider the free track n -category Σ ⊤ generated by Σ , that is, the free ( n − ) -categoryenriched in groupoid on Σ . We define identities among relations for Σ as the elements of an abeliannatural system Π ( Σ ) on the n -category Σ it presents. For that, we extend a result proved by Baues andJibladze [3] for the case n = . Theorem 2.2.3.
A track n -category T is abelian if and only if there exists a unique (up to isomorphism)abelian natural system Π ( T ) on T such that [ Π ( T ) is isomorphic to Aut T . We define Π ( Σ ) as the natural system associated by that result to the abelianized track n -category Σ ⊤ ab .In Section 2.2, we give an explicit description of the natural system Π ( Σ ) . . Preliminaries Then, in Section 2.4, we interpret generators of Π ( Σ ) as elements of a homotopy basis of the track n -category Σ ⊤ , see [8]. More precisely, we prove: Theorem 2.4.1.
If an n -polygraph Σ has finite derivation type then the natural system Π ( Σ ) is finitelygenerated. To prove this result, we give a way to compute generators of Π ( Σ ) from the critical pairs of a convergentpolygraph Σ . Indeed, there exists, for every critical branching ( f, g ) of Σ , a confluence diagram: · g (cid:31) (cid:31) ????? f (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) · k (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) · h (cid:31) (cid:31) ????? · An ( n + ) -cell filling such a diagram is called a generating confluence of Σ . It is proved in [8] that thegenerating confluences of Σ form a homotopy basis of Σ ⊤ . We show here that they also form a generatingset for the natural system Π ( Σ ) of identities among relations.
1. P
RELIMINARIES
In this section, we recall several notions from [8]: presentations of n -categories by polygraphs (1.1),rewriting properties of polygraphs (1.2), track n -categories and homotopy bases (1.3). We fix an n -category C throughout this section. We denote by C k the set (and the k -category) of k -cells of C . If f is in C k , then s i ( f ) and t i ( f ) respectively denote the i -source and i -target of f ; we drop the suffix i when i = k − . Thesource and target maps satisfy the globular relations : s i s i + = s i t i + and t i s i + = t i t i + . (1)If f and g are i -composable k -cells, that is when t i ( f ) = s i ( g ) , we denote by f ⋆ i g their i -composite k -cell. We also write fg instead of f ⋆ g . The compositions satisfy the exchange relations given, forevery i = j and every possible cells f , g , h and k , by: ( f ⋆ i g ) ⋆ j ( h ⋆ i k ) = ( f ⋆ j h ) ⋆ i ( g ⋆ j k ) . (2)If f is a k -cell, we denote by f its identity ( k + ) -cell and, by abuse, all the higher-dimensional identitycells it generates. When f is composed with cells of dimension k + or higher, we simply denote itby f . A k -cell f with s ( f ) = t ( f ) = u is called a closed k -cell with base point u . Let C be an n -category and let k ∈ {
0, . . . , n } . A k -sphere of C is a pair γ = ( f, g ) ofparallel k -cells of C , that is, with s ( f ) = s ( g ) and t ( f ) = t ( g ) ; we call f the source of γ and g its target .We denote by S C the set of n -spheres of C . An n -category is aspherical when all of its n -spheres haveshape ( f, f ) . .1. Higher-dimensional categories and polygraphs1.1.3. Cellular extensions. A cellular extension of C is a pair Γ = ( Γ n + , ∂ ) made of a set Γ n + and amap ∂ : Γ n + → S C . By considering all the formal compositions of elements of Γ , seen as ( n + ) -cellswith source and target in C , one builds the free ( n + ) -category generated by Γ , denoted by C [ Γ ] .The quotient of C by Γ , denoted by C /Γ , is the n -category one gets from C by identification of n -cells s ( γ ) and t ( γ ) , for every n -sphere γ of Γ . We usually denote by f the equivalence class of an n -cell f of C in C /Γ . We write f ≡ Γ g when f = g holds. We define n -polygraphs and free n -categories by induction on n . A -polygraph isa graph, with the usual notion of free category.An ( n + ) -polygraph is a pair Σ = ( Σ n , Σ n + ) made of an n -polygraph Σ n and a cellular exten-sion Σ n + of the free n -category generated by Σ n . The free ( n + ) -category generated by Σ and the n -category presented by Σ are respectively denoted by Σ ∗ and Σ and defined by: Σ ∗ = Σ ∗ n [ Σ n + ] and Σ = Σ ∗ n /Σ n + . An n -polygraph Σ is finite when each set Σ k is finite, ≤ k ≤ n . Two n -polygraphs whose presented ( n − ) -categories are isomorphic are Tietze-equivalent . A property on n -polygraphs that is preservedup to Tietze-equivalence is Tietze-invariant .An n -category C is presented by an ( n + ) -polygraph Σ when it is isomorphic to Σ . It is finitelygenerated when it is presented by an ( n + ) -polygraph Σ whose underlying n -polygraph Σ n is finite. Itis finitely presented when it is presented by a finite ( n + ) -polygraph. Let us consider the monoid As = { a , a } with unit a and product a a = a . Wesee As as a ( -)category with one -cell a and one non-degenerate -cell a : a → a . As such,it is presented by the -polygraph Σ with one -cell a , one -cell a : a → a and one -cell a : a a ⇒ a . Thus As is finitely generated and presented. In what follows, we use graphicalnotations for those cells, where the -cell a is pictured as a vertical “string” and the -cell a as . A context of C is a pair ( x, C ) made of an ( n − ) -sphere x of C and an n -cell C in C [ x ] such that C contains exactly one occurrence of x . We denote by C [ x ] , or simply by C ,such a context. If f is an n -cell which is parallel to x , then C [ f ] is the n -cell of C one gets by replacing x by f in C .Every context C of C has a decomposition C = f n ⋆ n − ( f n − ⋆ n − ( · · · ⋆ f xg ⋆ · · · ) ⋆ n − g n − ) ⋆ n − g n , where, for every k in {
1, . . . , n } , f k and g k are k -cells of C . A whisker of C is a context that admits sucha decomposition with f n and g n being identities. Every context C of C n − yields a whisker of C suchthat C [ f ⋆ n − g ] = C [ f ] ⋆ n − C [ g ] holds.If Γ is a cellular extension of C , then every non-degenerate ( n + ) -cell f of C [ Γ ] has a decomposition f = C [ ϕ ] ⋆ n · · · ⋆ n C k [ ϕ k ] , with k ≥ and, for every i in {
1, . . . , k } , ϕ i in Γ and C i a context of C .The category of contexts of C is denoted by C C , its objects are the n -cells of C and its morphismsfrom f to g are the contexts C of C such that C [ f ] = g holds. We denote by W C the subcategory of C C with the same objects and with whiskers as morphisms. . Preliminaries1.1.7. Natural systems. A natural system on C is a functor D from C C to the category of groups. Wedenote by D u and D C the images of an n -cell u and of a context C of C by the functor D . When noconfusion arise, we write C [ a ] instead of D C ( a ) . A natural system D on C is abelian when D u is anabelian group for every n -cell u . We fix an ( n + ) -polygraph Σ throughout this section. One says that an n -cell u of Σ ∗ n reduces into an n -cell v when Σ ∗ contains a non-identity ( n + ) -cell with source u and target v . One says that u is a normal form when it does not reduceinto an n -cell. A normal form of u is an n -cell v which is a normal form and such that u reduces into v .A reduction sequence is a countable family ( u n ) n ∈ I of n -cells such that each u n reduces into u n + ; it is finite or infinite when the indexing set I is.One says that Σ terminates when it does not generate any infinite reduction sequence. In that case,every n -cell has at least one normal form and one can use Noetherian induction : one can prove propertieson n -cells by induction on the length of reduction sequences. A branching (resp. confluence ) is a pair ( f, g ) of ( n + ) -cells of Σ ∗ with samesource (resp. target), considered up to permutation. A branching ( f, g ) is local when f and g containexactly one generating ( n + ) -cell of Σ . It is confluent when there exists a confluence ( f ′ , g ′ ) with t ( f ) = s ( f ′ ) and t ( g ) = s ( g ′ ) . A local branching ( f, g ) is critical when the common source of f and g is a minimal overlapping of the sources of the ( n + ) -cells contained in f and g . A confluence diagram of a branching ( f, g ) is an ( n + ) -sphere with shape ( f ⋆ n f ′ , g ⋆ n g ′ ) , where ( f ′ , g ′ ) is a confluence. Aconfluence diagram of a critical branching is called a generating confluence of Σ .One says that Σ is (locally) confluent when each of its (local) branchings is confluent. A localbranching ( f, g ) is critical when the common source of f and g is a minimal overlapping of the sourcesof the generating ( n + ) -cells of f and g . In a confluent ( n + ) -polygraph, every n -cell has at mostone normal form. For terminating ( n + ) -polygraphs, Newman’s lemma ensures that local confluenceand confluence are equivalent properties [13]. One says that Σ is convergent when it terminates and it is confluent. In that case,every n -cell u has a unique normal form, denoted by b u . Moreover, we have u ≡ Σ n + v if and only if b u = b v . As a consequence, a finite and convergent ( n + ) -polygraph yields a syntax for the n -cells ofthe category it presents, together with a decision procedure for the corresponding word problem. The -polygraph Σ = ( a , a , a ) presenting As is convergent and has exactly onecritical pair ( a a , a a ) , with corresponding generating confluence a : a a a a a (cid:21) ) ????? ????? a a u (cid:9) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) a a a u (cid:9) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) a a a (cid:21) ) ????? ????? a _ % a .3. Track n -categories and homotopy bases Alternatively, this -cell a can be pictured as follows: _ % In turn, the -polygraph Σ = ( a , a , a , a ) , which is a part of a presentation of the theory of monoids,is convergent and has exactly one critical pair, with corresponding generating confluence a : a a a a a a a l (cid:0) llllllllllllllll a a a (cid:5) (cid:25) a a a (cid:30) RRRRRRRR RRRRRRRR a a a a a a a l (cid:0) llllllllllllllll a a a (cid:30) RRRRRRRR RRRRRRRR c (cid:13) a a a a a (cid:5) (cid:25) a a a a a (cid:5) (cid:25) a a a a a (cid:5) (cid:25) a a PPP PPP (cid:29) PPPPPP a a a a a (cid:5) (cid:25) a a nnnnnn m (cid:1) nnnnnn a a a a a pppppppp m (cid:1) pppppp a a NNNN NNNN (cid:29) NNN NNN a (cid:31) ? a a a (cid:5) (cid:25) a a a (cid:29) PPPPPPPPPP PPPPPPPPPP a _ % a a a m (cid:1) nnnnnnnnnnnnnnnnnnnn a a a (cid:29) PPPPPPPPPP PPPPPPPPPP a a a m (cid:1) nnnnnnnnnnnnnnnnnnnn a a a a _ % a a _ % a _ % a _ % In fact, this -cell a is Mac Lane’s pentagon [8]: % (cid:24) , FFFFFF FFFFFF (cid:127) (cid:31) F xxxxxx xxxxxx (cid:29) PPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPP - A nnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnn n -categories and homotopy bases n -categories. A track n -category is an n -category T whose n -cells are invertible, that is,for n ≥ , an ( n − ) -category enriched in groupoid. In a track n -category, we denote by f − the inverseof the n -cell f . A track n -category is acyclic when, for every ( n − ) -sphere ( u, v ) , there exists an n -cell f with source u and target v .The n -category presented by a track ( n + ) -category T is the n -category T = T n / T n + , where T n + is seen as a cellular extension of T n . Two track ( n + ) -categories are Tietze-equivalent if the n -categoriesthey present are isomorphic. Given an n -category C and a cellular extension Γ of C , the track ( n + ) -category generated by Γ is denoted by C ( Γ ) and defined as follows: C ( Γ ) = C (cid:2) Γ, Γ − (cid:3) (cid:14) Inv ( Γ ) . Preliminaries where Γ − contains the same ( n + ) -cells as Γ , with source and target reversed, and Inv ( Γ ) is made of the ( n + ) -cells ( γ ⋆ n γ − , 1 sγ ) and ( γ − ⋆ n γ, 1 tγ ) , where γ ranges over Γ . Let us note that, when f and g are n -cells of C , we have f ≡ Γ g if and only if there exists an ( n + ) -cell with source f and target g in C ( Γ ) . When Σ is an ( n + ) -polygraph, one writes Σ ⊤ instead of Σ ∗ n ( Σ n + ) . Let C be an n -category. A homotopy basis of C is a cellular extension Γ of C such that the track ( n + ) -category C ( Γ ) is acyclic or, equivalently, when the quotient n -category C /Γ is aspherical or, again equivalently, when every sphere ( f, g ) of C satisfies f ≡ Γ g . Let Σ be a convergent n -polygraph. Thegenerating confluences of Σ form a homotopy basis of Σ ⊤ .Remark. A complete proof of Lemma 1.3.3 is given in [8]. Squier has proved the same result for presen-tations of monoids by word rewriting systems [18, 19]. When formulated in terms of homotopy bases,Squier’s result is a subcase of the case n = of Lemma 1.3.3. The -polygraph Σ = ( a , a , a ) presenting As has exactly one generating con-fluence a and, thus, this -cell forms a homotopy basis of the track -category Σ ⊤ . The -polygraph Σ = ( a , a , a , a ) also has exactly one generating confluence a , with Mac Lane’s pentagon as shape,which forms a homotopy basis of the track -category Σ ⊤ .The resulting -polygraph Σ = ( a , a , a , a , a ) is a part of a presentation of the theory ofmonoidal categories. In [8], Mac Lane’s coherence theorem is reformulated in terms of homotopy basesand proved by an application of Lemma 1.3.3 to a convergent -polygraph containing Σ . Let T be a track n -category and let B be a family of closed n -cells of T . The followingassertions are equivalent:1. The cellular extension e B = (cid:10) e β : β → sβ , β ∈ B (cid:11) is a homotopy basis of T .2. Every closed n -cell f in T can be written f = (cid:0) g ⋆ n − C (cid:2) β ε (cid:3) ⋆ n − g − (cid:1) ⋆ n − · · · ⋆ n − (cid:0) g k ⋆ n − C k (cid:2) β ε k k (cid:3) ⋆ n − g − k (cid:1) (3) where, for every i ∈ {
1, . . . , k } , we have β i ∈ B , ε i ∈ { − , + } , C i ∈ W T and g i ∈ T n .Proof. Let us assume that e B is a homotopy basis of T and let us consider a closed n -cell f : w → w in T . Then, by definition of a homotopy basis, there exists an ( n + ) -cell A : f → w in T ( e B ) . Byconstruction of T ( e B ) , the ( n + ) -cell A decomposes into A = A ⋆ n · · · ⋆ n A k , where each A i is an ( n + ) -cell of T ( e B ) that contains exactly one generating ( n + ) -cell of B . As aconsequence, each A i has shape g i ⋆ n − C i he β ε i i i ⋆ n − h i with β i ∈ B , ε i ∈ { − , + } , C i ∈ W T and g i , h i ∈ T n , . By hypothesis on A , we have f = s ( A ) , hence: f = g ⋆ n − C [ s ( β ε )] ⋆ n − h .6 . Identities among relations We proceed by case analysis on ε . If ε = + , then we have: f = g ⋆ n − C [ β ] ⋆ n − h = (cid:0) g ⋆ n − C [ β ] ⋆ n − g − (cid:1) ⋆ n − ( g ⋆ n − h )= (cid:0) g ⋆ n − C [ β ] ⋆ n − g − (cid:1) ⋆ n − s ( A ) . And, if ε = − , we get: f = g ⋆ n − h = (cid:0) g ⋆ n − C [ β − ] ⋆ n − g − (cid:1) ⋆ n − ( g ⋆ n − C [ β ] ⋆ n − h )= (cid:0) g ⋆ n − C [ β − ] ⋆ n − g − (cid:1) ⋆ n − s ( A ) . An induction on the natural number k proves that f has a decomposition as in (3).Conversely, we assume that every closed n -cell f in T has a decomposition as in (3). Then we have f ≡ e B s ( f ) for every closed n -cell f in T . Let us consider two parallel n -cells f and g in T . Then f ⋆ n − g − is a closed n -cell, yielding f ⋆ n − g − ≡ e B s ( f ) . We compose both members by g on the right hand to get f ≡ e B g . Thus e B is a homotopy basis of T . One says that an n -polygraph Σ has finite derivation type when it is finiteand when the track n -category Σ ⊤ admits a finite homotopy basis. This property is Tietze-invariantfor finite n -polygraphs, so that one says that an n -category has finite derivation type when it admits apresentation by an ( n + ) -polygraph with finite derivation type. Let T be a track n -category and let Γ be a cellular extension of T . If T has finitederivation type, then so does T /Γ .Proof. Let B be a finite homotopy basis of T . Let us denote by B the cellular extension of T /Γ madeof one ( n + ) -cell A with source f and target g for each ( n + ) -cell A from f to g in B . Then B is ahomotopy basis of T /Γ .
2. I
DENTITIES AMONG RELATIONS n -categories Let T be a track n -category. For every ( n − ) -cell u in T , we denote by Aut T u the group of closed n -cells of T with base u . This mapping extends to a natural system Aut T on the ( n − ) -category T n − , sending a context C of T n − to the morphism of groups that maps f to C [ f ] .A track n -category T is abelian when, for every ( n − ) -cell u of T , the group Aut T u is abelian. The abelianized of a track n -category T is the track n -category denoted by T ab and defined as the quotientof T by the n -spheres ( f ⋆ n − g, g ⋆ n − f ) , where f and g are closed n -cells with the same base. Each
Aut T ab u is the abelianized group of Aut T u . As a consequence, a track n -category T is abelian if and only if the natural system Aut T on T n − is abelian. . Identities among relations2.1.3. Lemma. Let T be a track n -category. For every n -cell g : v → u , the mapping ( · ) g from Aut T u to Aut T v and sending f to f g = g − ⋆ n − f ⋆ n − g is an isomorphism of groups. Moreover, if T is abelian and g, h : v → u are n -cells of T , then theisomorphisms ( · ) g and ( · ) h are equal.Proof. We have: ( u ) g = g − ⋆ n − u ⋆ n − g = v . Let f and f be closed n -cells of T with base u . Then: ( f ⋆ n − f ) g = g − ⋆ n − f ⋆ n − f ⋆ n − g = g − ⋆ n − f ⋆ n − g ⋆ n − g − ⋆ n − f ⋆ n − g = f g1 ⋆ n − f g2 . Hence ( · ) g is a morphism of groups and it admits ( · ) g − as inverse. Now, if T is abelian and g, h : v → u are parallel n -cells, we have: f g = g − ⋆ n − f ⋆ n − g = ( g − ⋆ n − h ) ⋆ n − ( h − ⋆ n − f ⋆ n − h ) ⋆ n − ( h − ⋆ n − g )= ( h − ⋆ n − f ⋆ n − h ) ⋆ n − ( g − ⋆ n − h ) ⋆ n − ( h − ⋆ n − g )= f h . If a track n -category T has finite derivation type, then so does T ab .Proof. We apply Lemma 1.3.8 to the quotient T ab of T . Let T be a track n -category and let D be a natural system on T . We denote by b D the natural system on T n − defined by b D u = D u . A track n -category T is linear when there exists anabelian natural system Π ( T ) on T such that [ Π ( T ) is isomorphic to Aut T . Remark.
If such an abelian natural system D exists, then it is unique up to isomorphism. Indeed, bydefinition of b D , we have b D u = b D v whenever u and v are ( n − ) -cells of T such that u = v holds. Thus,if u is an ( n − ) -cell of T , then D u = b D w for every ( n − ) -cell w of T with w = u . As a consequence,if D and E are abelian natural systems on T such that both b D and b E are isomorphic to Aut T , then D and E are isomorphic. A track n -category is abelian if and only if it is linear.Proof. If T is linear, then each group Aut T u is isomorphic to an abelian group. Thus T is abelian.Conversely, let us assume that T is abelian and let us define the abelian natural system Π ( T ) on T .For an ( n − ) -cell u of T , the abelian group Π ( T ) u is defined as follows, by generators and relations: • It has one generator ⌊ f ⌋ for every n -cell f : a → a with a = u . .2. Defining identities among relations • Its defining relations are: i) ⌊ f ⋆ n − g ⌋ = ⌊ f ⌋ + ⌊ g ⌋ , for f : a → a and g : a → a with a = u ; ii) ⌊ f ⋆ n − g ⌋ = ⌊ g ⋆ n − f ⌋ , for f : a → b and g : b → a with a = b = u .If u and u ′ are ( n − ) -cells of T and if C is a context of T from u to u ′ , then the action Π ( T ) C : Π ( T ) u − → Π ( T ) u ′ is defined, on a generator ⌊ f ⌋ , with f a closed n -cell of T with base a such that a = u , by C ⌊ f ⌋ = ⌊ B [ f ] ⌋ , where B is a context of T n − , from a to some a ′ with a ′ = u ′ , such that B = C holds. We note that B [ f ] is a closed n -cell of T with base some a ′ such that a ′ = u ′ , so that ⌊ B [ f ] ⌋ is a generating elementof Π ( T ) u ′ . Now, let us check that this action is well-defined, that is, it does not depend on the choice ofthe representatives f and B .For f , we check that Π ( T ) C is compatible with the relations defining Π ( T ) u . If f and g are closed n -cells of T with base a such that a = u , then we have: ⌊ B [ f ⋆ n − g ] ⌋ = ⌊ B [ f ] ⋆ n − B [ g ] ⌋ = ⌊ B [ f ] ⌋ + ⌊ B [ g ] ⌋ . And, for n -cells f : a → b and g : b → a , with a = b = u , we have: ⌊ B [ f ⋆ n − g ] ⌋ = ⌊ B [ f ] ⋆ n − B [ g ] ⌋ = ⌊ B [ g ] ⋆ n − B [ f ] ⌋ = ⌊ B [ g ⋆ n − f ] ⌋ . For B , we decompose C in v ⋆ n − C ′ ⋆ n − w , where v and w are ( n − ) -cells of T and C ′ is awhisker of T . Since T and T n − coincide up to dimension n − , any representative B of C can bewritten B = b ⋆ n − C ′ ⋆ n − c , where b and c are respective representatives of v and w in T n − . As aconsequence, it is sufficient (and, in fact, equivalent) to prove that the definition of Π ( T ) C is invariantwith respect to the choice of the representative B of C when C has shape v ⋆ n − x or x ⋆ n − w .We examine the case C = v ⋆ n − x , the other one being symmetric. We consider two representatives b and b ′ of v in T n − . By definition of T , there exists an n -cell g : b → b ′ in T , as in the following diagram,drawn for the case n = : b ! ! b ′ = = a / / g (cid:5) (cid:25) f L Thanks to the exchange relation, we have: ( g ⋆ n − a ) ⋆ n − ( b ′ ⋆ n − f ) = g ⋆ n − f = ( b ⋆ n − f ) ⋆ n − ( g ⋆ n − a ) . Hence: b ′ ⋆ n − f = ( g − ⋆ n − a ) ⋆ n − ( b ⋆ n − f ) ⋆ n − ( g ⋆ n − a ) . 9 . Identities among relations As, a consequence, one gets, using the second defining relation of Π ( T ) v ⋆ n − u : (cid:4) b ′ ⋆ n − f (cid:5) = (cid:4) ( g − ⋆ n − a ) ⋆ n − ( b ⋆ n − f ) ⋆ n − ( g ⋆ n − a ) (cid:5) = (cid:4) ( b ⋆ n − f ) ⋆ n − ( g ⋆ n − a ) ⋆ n − ( g − ⋆ n − a ) (cid:5) = ⌊ b ⋆ n − f ⌋ . Now, let us prove that the abelian natural systems [ Π ( T ) and Aut T are isomorphic. For an ( n − ) -cell u of T , we define Φ u : Π ( T ) u → Aut T u as the morphism of groups given on generators by Φ u ( ⌊ f ⌋ ) = f g , where f is a closed n -cell of T with base v such that v = u and g is any n -cell of T with source v andtarget u . Let us check that Φ u is well-defined. We already know that Φ u is independent of the choiceof g . Let us prove that this definition is compatible with the relations defining Π ( T ) u .For the first relation, let f and f be closed n -cells of T with base v such that v = u and let g : v → u be an n -cell of T . Then: Φ u ( ⌊ f ⋆ n − f ⌋ ) = ( f ⋆ n − f ) g = f g1 ⋆ n − f g2 = Φ u ( ⌊ f ⌋ ) ⋆ n − Φ u ( ⌊ f ⌋ )= Φ u ( ⌊ f ⌋ + ⌊ f ⌋ ) . For the second relation, we fix n -cells f : v → v , f : v → v and g : v → u , with v = v = u .Then: Φ u ( ⌊ f ⋆ n − f ⌋ ) = ( f ⋆ n − f ) g = ( g − ⋆ n − f ) ⋆ n − ( f ⋆ n − f ) ⋆ n − ( f − ⋆ n − g )= ( f ⋆ n − f ) g − ⋆ n − f = Φ u ( ⌊ f ⋆ n − f ⌋ ) . Thus Φ u is a morphism of groups from Π ( T ) u to Aut T u . Moreover, it admits f → ⌊ f ⌋ as inverse and, asa consequence, is an isomorphism.Finally, let us prove that Φ u is natural in u . Let C be a context of T n − from u to v . Let us checkthat the morphisms of groups Φ v ◦ Π ( T ) C and Aut T C ◦ Φ u coincide. Let f be a closed n -cell of T withbase point u ′ such that u ′ = u . We fix an n -cell g : u ′ → u in T and we note that C [ g ] is an n -cell of T with source C [ u ′ ] and target C [ u ] = v . Then we have: Φ v ◦ Π ( T ) C ( ⌊ f ⌋ ) = ( C [ f ]) C [ g ] = C [ g − ] ⋆ n − C [ f ] ⋆ n − C [ g ]= C (cid:2) g − ⋆ n − f ⋆ n − g (cid:3) = C [ f g ]= Aut T C ◦ Φ u ( ⌊ f ⌋ ) .10 .3. Identities among relations of Tietze-equivalent polygraphs Remark.
Theorem 2.2.3 is proved in [2, 3] for the case n = . Let Σ be an n -polygraph. The natural system of identities among relations of Σ isthe abelian natural system Π ( Σ ⊤ ab ) , which we simply denote by Π ( Σ ) . If w is an ( n − ) -cell of Σ , anelement of the abelian group Π ( Σ ) w is called an identity among relations associated to w . Let Σ and Υ be two Tietze-equivalent n -polygraphs. Then there exist n -functors F : Σ ⊤ ab → Υ ⊤ ab and G : Υ ⊤ ab → Σ ⊤ ab such that the following two diagrams commute: Σ ⊤ ab F / / π Σ (cid:15) (cid:15) (cid:15) (cid:15) c (cid:13) Υ ⊤ ab π Υ (cid:15) (cid:15) (cid:15) (cid:15) Σ Υ Υ ⊤ ab G / / π Υ (cid:15) (cid:15) (cid:15) (cid:15) c (cid:13) Σ ⊤ ab π Σ (cid:15) (cid:15) (cid:15) (cid:15) Υ Σ
Proof.
To simplify notations, we consider that the ( n − ) -categories Σ and Υ are equal, instead ofsimply isomorphic. Let us build F , the construction of G being symmetric.First, we define an n -functor F from Σ ⊤ to Υ ⊤ . On i -cells, with i ≤ n − , F is the identity, whichmakes the diagram commute up to dimension n − since π Σ and π Υ are also identities on the samedimensions.If a is an ( n − ) -cell in Σ , we arbitrarily choose an ( n − ) -cell in π − π Σ ( a ) for F ( a ) . Since F isthe identity up to dimension n − , we have that the source and target of F ( a ) are equal to the source andtarget of a , respectively.Then, F is extended to any ( n − ) -cell of Σ ⊤ by functoriality. Let ϕ : u → v be an n -cell of Σ . Wehave, by definition of F ( u ) and F ( v ) : π Υ ◦ F ( u ) = π Σ ( u ) = π Σ ( v ) = π Υ ◦ F ( v ) . Thus, there exists an n -cell from F ( u ) to F ( v ) in Σ ⊤ . We arbitrarily choose F ( ϕ ) to be one of those n -cells and, then, we extend F to any n -cell of Σ ⊤ by functoriality.Let f and g be closed n -cells in Σ ⊤ . We have F ( f ⋆ n − g ) = F ( f ) ⋆ n − F ( g ) by definition of F .As a consequence, F induces a n -functor from Σ ⊤ ab to Υ ⊤ ab that satisfies, by construction, the relation π Υ ◦ F = π Σ . We fix two Tietze-equivalent n -polygraphs Σ and Υ , together with n -functors F and G as in Lemma 2.3.1. We denote by e G the morphism of natural systems on Σ = Υ , from Π ( Υ ) to Π ( Σ ) ,defined by e G ( ⌊ f ⌋ ) = ⌊ G ( f ) ⌋ .For every ( n − ) -cell w in Σ ⊤ ab , we define an n -cell Λ w from w to GF ( w ) in Σ ⊤ ab , by structuralinduction on w . If w is an identity, then Λ w = w . Now, let w be an ( n − ) -cell in Σ . By hypothesison F and G , we have: π Σ ◦ GF ( w ) = π Υ ◦ F ( w ) = π Σ ( w ) . 11 . Identities among relations As a consequence, there exists an n -cell from w to GF ( w ) in Σ ⊤ ab and we arbitrarily choose Λ w to besuch an n -cell. Finally, if w = w ⋆ i w , for some i ∈ {
0, . . . , n − } , then Λ w = Λ w ⋆ i Λ w . If f : u → v is an n -cell of Σ ⊤ ab , we denote by Λ f the closed n -cell with basis u defined by: Λ f = f ⋆ n − Λ v ⋆ n − GF ( f ) − ⋆ n − Λ − u . Finally, we define: Λ Σ = (cid:8) ⌊ Λ ϕ ⌋ (cid:12)(cid:12) ϕ ∈ Σ n (cid:9) . Let f be an n -cell in Σ ⊤ ab with a decomposition f = C [ ϕ ε ] ⋆ n − · · · ⋆ n − C k [ ϕ ε k k ] , with ϕ i ∈ Σ n , ε i ∈ { − , + } and C i ∈ W Σ ∗ . Then we have: ⌊ Λ f ⌋ = k X i = ε i C i ⌊ Λ ϕ i ⌋ . (4) Proof.
Let f : u → v and g : v → w be n -cells in Σ ⊤ ab . We have: Λ f ⋆ n − g = ( f ⋆ n − g ) ⋆ n − Λ w ⋆ n − GF ( f ⋆ n − g ) − ⋆ n − Λ − u = f ⋆ n − (cid:0) g ⋆ n − Λ w ⋆ n − GF ( g ) − ⋆ n − Λ − v (cid:1) ⋆ n − Λ v ⋆ n − GF ( f ) − ⋆ n − Λ − u = f ⋆ n − Λ g ⋆ n − Λ v ⋆ n − GF ( f ) − ⋆ n − Λ − u = f ⋆ n − Λ g ⋆ n − f − ⋆ n − Λ f . Hence: (cid:4) Λ f ⋆ n − g (cid:5) = (cid:4) f ⋆ n − Λ g ⋆ n − f − ⋆ n − Λ f (cid:5) = ⌊ Λ f ⌋ + ⌊ Λ g ⌋ . (5)Now, let f : w → w ′ be an n -cell and u be an i -cell, i ≤ n − , of Σ ⊤ ab such that u ⋆ i w is defined. Thenwe have: Λ u ⋆ i f = ( u ⋆ i f ) ⋆ n − Λ u ⋆ i w ′ ⋆ n − GF ( u ⋆ i f ) − ⋆ n − Λ − u ⋆ i w = ( u ⋆ i f ) ⋆ n − ( Λ u ⋆ i Λ w ′ ) ⋆ n − ( GF ( u ) ⋆ i GF ( f ) − ) ⋆ n − ( Λ − u ⋆ i Λ − w )= ( u ⋆ n − Λ u ⋆ n − GF ( u ) ⋆ n − Λ − u ) ⋆ i ( f ⋆ n − Λ w ′ ⋆ n − GF ( f ) − ⋆ n − Λ − w )= u ⋆ i Λ f . Similarly, we prove that Λ f ⋆ i v = Λ f ⋆ i v if v is an i -cell, i ≤ n − , such that w ⋆ i v is defined. As aconsequence, we get Λ C [ f ] = C [ Λ f ] , for every whisker C of Σ ∗ , hence: (cid:4) Λ C [ f ] (cid:5) = C ⌊ Λ f ⌋ . (6)We prove (4) by induction on k , using (5) and (6). Let B be a generating set for the natural system Π ( Υ ) . Then the set Λ Σ ∐ e G ( B ) is agenerating set for the natural system Π ( Σ ) . .4. Generating identities among relations Proof.
Let f be a closed n -cell with basis w in Σ ⊤ . By definition of Λ f , we have: ⌊ f ⌋ = (cid:4) Λ f ⋆ n − Λ w ⋆ n − GF ( f ) ⋆ n − Λ − w (cid:5) = ⌊ Λ f ⌋ + ⌊ GF ( f ) ⌋ . On the one hand, we consider a decomposition of f in generating n -cells of Σ n : f = C [ ϕ ε ] ⋆ n − · · · ⋆ n − C k [ ϕ ε k k ] . Hence: ⌊ Λ f ⌋ = k X i = ε i C i ⌊ Λ ϕ i ⌋ . On the other hand, the natural system Π ( Υ ) is generated by B , so that ⌊ F ( f ) ⌋ admits a decomposition ⌊ F ( f ) ⌋ = P j ∈ J η j B j ⌊ g j ⌋ , with ⌊ g j ⌋ ∈ B . Hence: ⌊ GF ( f ) ⌋ = X j ∈ J B j ⌊ G ( g j ) ⌋ = X j ∈ J B j [ e G ( ⌊ g j ⌋ )] . Thus, ⌊ f ⌋ can be written as a linear combination of elements of Λ Σ and of B , proving the result. Let Σ and Υ be two Tietze-equivalent n -polygraphs such that Σ n and Υ n are finite.Then the natural system Π ( Σ ) is finitely generated if and only if the natural system Π ( Υ ) is finitelygenerated.Proof. We use Lemma 2.3.4 with B and Σ n finite. If an n -polygraph Σ has finite derivation type then the natural system Π ( Σ ) is finitelygenerated.Proof. Let us assume that the n -polygraph Σ has finite derivation type. By Proposition 2.1.4, the abeliantrack category Σ ⊤ ab has finite derivation type. Let B be a finite homotopy basis of Σ ⊤ ab and let e B be the setof closed n -cells of Σ ⊤ ab defined by: e B = (cid:8) s ( β ) ⋆ n − t ( β ) − (cid:12)(cid:12) β ∈ B (cid:9) . By Lemma 1.3.6, any closed n -cell f in Σ ⊤ ab can be written f = (cid:0) g ⋆ n − C [ β ε ] ⋆ n − g − (cid:1) ⋆ n − · · · ⋆ n − (cid:0) g k ⋆ n − C k [ β ε k k ] ⋆ n − g − k (cid:1) , where, for every i in {
1, . . . , k } , β i ∈ e B , ε i ∈ { − , + } , C i ∈ W Σ ∗ and g i ∈ Σ ∗ n . As a consequence, forany identity among relations ⌊ f ⌋ in Π ( Σ ) , we have: ⌊ f ⌋ = k X i = ε i (cid:4) g i ⋆ n − C i [ β i ] ⋆ n − g − i (cid:5) = k X i = ε i C i ⌊ β i ⌋ . Thus, the elements of je B k form a generating set for Π ( Σ ) . . Identities among relations2.4.2. Proposition. For a convergent n -polygraph Σ , the natural system Π ( Σ ) is generated by the gen-erating confluences of Σ .Proof. By Squier’s confluence lemma (Lemma 1.3.3), the set of generating confluences of Σ forms ahomotopy basis of Σ ⊤ . Following the proof of Theorem 2.4.1, we transform it into a generating set forthe natural system Π ( Σ ) . We consider the -polygraph Σ = ( a , a , a ) presenting the monoid As . Here is apart of the free -category Σ ∗ : a a a a e y a a a a a m (cid:1) a a ^ r a a a a a a a e y a a a r (cid:6) a a a X l a a a a a a a a a j ~ a a a a ` t a a a a w (cid:11) a a a a S g ( · · · ) The -polygraph Σ is convergent and has exactly one generating confluence, written with both notations: a a ⋆ a a _ % a a ⋆ a _ % Thus the natural system Π ( Σ ) on the category Σ = As is generated by following the element, where thelast equality uses the exchange relation: (cid:4) s ( a ) ⋆ t ( a ) − (cid:5) = (cid:4) ( a a ⋆ a ) ⋆ ( a − ⋆ a a − ) (cid:5) = (cid:4) a a ⋆ a a − (cid:5) = (cid:4) a a − (cid:5) . The graphical notations, where − is pictured as , make this last equality more clear: (cid:4) s ( ) ⋆ t ( ) − (cid:5) = $ % = j k = ⌊ ⌋ One can prove the same result by a combinatorial analysis. Indeed, one can note that the minimal -cellsfrom a n + to a n1 are the a i1 a a n − − i1 , for i in {
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