Higher-dimensional categories with ﬁnite derivation type · 2020-02-05 · arXiv:0810.1442v1 [math.CT] 8 Oct 2008 Higher-dimensional categories with ﬁnite derivation type Yves

arX

iv:0

810.

1442

v1 [

mat

h.C

T]

8 O

ct 2

008

Higher-dimensional categories with finite derivation type

Yves Guiraud Philippe MalbosINRIA Nancy Université Lyon 1

[email protected] [email protected]

Abstract –The finite derivation type property is a homotopical condition on monoids. Craig Squierhas proved that a monoid must satisfy it in order to admit a presentation by a finite and convergentrewriting system. We generalise the property ton-categories presented by polygraphs and we recoverSquier’s theorem whenn is 1. However, we prove that this result does not hold anymore forcategoriesof dimension2 and above. We study several examples of2-categories presented by finite convergentpolygraphs, with or without the property of finite derivation type, in order to illustrate sample cases.Keywords –n-category; polygraph; finite derivation type; rewriting;low-dimensional topology.

INTRODUCTION

Monoids with finite derivation type

Craig Squier has proved that, if a monoid admits a presentation by a finite and convergent (i.e. terminat-ing and confluent) word rewriting system [5], then it hasfinite derivation type[19]. This is a homotopicalproperty of the monoid, independent of the chosen finite presentation. Squier has used it to prove thatfinite convergent word rewriting systems did not provide a universal way to decide equality in monoidspresented by generators and relations.

The property of finite derivation type is defined on the Squier’s combinatorial2-complex associatedto the word rewriting system: its vertices are the words, itsarrows are generated by the rewriting stepsand their inverses, its2-cells are the equivalence of rewriting paths (by commutation of rules that areapplied on disjoint parts of a word). Then the presentation (and the monoid) has finite derivation typewhen one can fill all the holes of the complex by filling finitelymany.

The property of finite derivation type can be interpreted computationally: it means that, in a wordrewriting system with this property, there are only finitelymany choices that can be made during com-putation. These choices correspond to the critical branchings of the rewriting system, where two rulescan be applied at the same place of a given word. Thus, the finite derivation type property ensures thatmaking a finite number of choices is sufficient to build a rewriting strategy that makes the computationdeterministic.

Generalisation ton-categories

There exist many types of rewriting systems, acting, for example, on terms, on elements of some sortof algebra or on topological objects. For instance, permutations admit a graphical presentation by onegenerator and two (directed) relations:

→ and → .

http://arxiv.org/abs/0810.1442v1

Introduction

It turns out that word rewriting systems and this presentation of permutations are instances of the sameobjects: polygraphs (or computads), which are presentations by “generators” and “relations” of somehigher-dimensional categories [20, 6], see also [21, 22].

Indeed, monoids and word rewriting systems are special instances of1-categories and2-polygraphs,respectively. And many usual types of rewriting systems have interpretations as3-polygraphs, seen aspresentations of2-categories, including the presentation of permutations,term rewriting systems [6, 13,10], Petri nets [12] or formal proofs of propositional and linear logics [11].

The purpose of this document is to formalise the property of finite derivation type forn-categoriesand to answer the following question:

For an n-category, does having a finite convergent presentation by an (n + 1)-polygraphimplies that it has finite derivation type?

As we have seen, Squier has proved that this was true whenn is 1. Until now, several examples of finiteand convergent3-polygraphs were studied and it was observed that all of themhad finite derivation type.However, it turns out that the answer is negative.

Overview of the document

In this document, we assume basic knowledge of theories of categories, ofn-categories and of rewriting.Good references to these fields include the books written by Saunders Mac Lane [15], by Eugenia Chengand Aaron Lauda [7] and by Franz Baader and Tobias Nipkow [1].

In Section 1 we briefly recall notations and representationsof n-categories. Then, in 1.4, we definepolygraphs and presentations ofn-categories by polygraphs, as in [6] and [16].

Section 2 contains several notions and tools we need, not directly to define the property of finitederivation type or to prove the main results, but to study theexamples. In particular, we recall thedefinition of context of ann-category in 2.1 and introduce the notions of module over ann-category andof derivation of ann-category into a module, respectively in 2.3 and in 2.4.

In Section 3, we introduce the notion of trackn-category, which is ann-category whosen-cells areinvertible, or an(n − 1)-category enriched in groupoid. This generalises the notion of track category,used as an algebraic model of the homotopy2-type [2]. The fundamental groupoid of the Squier’scombinatorial2-complex of a word rewriting system is a monoid enriched in groupoid [8]. Forn-polygraphs, we use trackn-categories to define homotopy relations, yielding the notion of homotopybasis of ann-category: a set of(n + 1)-cells that identifies all the paralleln-cells. Then ann-categoryhas finite derivation type when it admits a finite homotopy basis. In practice, we define this property onpolygraphs and prove Proposition 3.3.4: if twon-polygraphs present the same(n − 1)-category, bothhave finite derivation type or neither has.

In Section 4, we study the links between the property of finitederivation type and the rewritingproperties of polygraphs. In particular, we prove Proposition 4.3.5: if a finite convergent polygraphhas a finite set of critical branchings, then it has finite derivation type. We get Squier’s result [19] as aconsequence since a finite convergent word rewriting systemhas a finite number of critical branchings.Finally, Theorem 4.3.9 asserts that this result does not generalise to higher dimensions, as proved by acounter-example that is studied in the final part:

2

1. Higher-dimensional categories presented by polygraphs

Theorem. For every natural numbern with n ≥ 2, there exists an-category which doesnot have finite derivation type and admits a presentation by afinite convergent(n + 1)-polygraph.

In Section 5, we examine the case of2-categories presented by3-polygraphs. We give a classificationof their critical branchings and we use it to get additional sufficient conditions for a finite convergent3-polygraph to have finite derivation type. Finally, we studythree examples of finite and convergent3-polygraph, in order to illustrate sample cases:

(5.2) The3-polygraph of monoids has a finite number of critical branchings and, thus, has finite deriva-tion type. We use this result to rediscover Mac Lane’s coherence theorem [15]: in a monoidalcategory, all the diagrams built using the monoidal structure commute.

(5.4) The3-polygraph of permutations has an infinite number of critical branchings. However, it wasobserved by Yves Lafont [13] that, for confluence study, onlya finite number of them were useful.Following the same intuition, we prove that, despite the infinite number of critical branchings, this3-polygraph has finite derivation type.

(5.5) The final3-polygraph we study is the proof of Theorem 4.3.9: it is finiteand convergent butdoes not have finite derivation type. This3-polygraph is an artificial counter-example, yet it hasa topological flavour: it presents a2-category whose morphisms could be interpreted as (open-closed, planar) necklaces with pearls, considered up to homotopy.

1. HIGHER-DIMENSIONAL CATEGORIES PRESENTED BY POLYGRAPHS

1.1. Generalities onn-categories andn-functors

Throughout this document, we only consider small, strictn-categories and strictn-functors betweenthem. We denote byCatn the (large) category they form.

1.1.1. Vocabulary and notations. If C is ann-category, we denote byCk the set ofk-cells ofC. IfF : C → D is ann-functor, we denote byFk the corresponding map fromCk to Dk.

In ann-category,sk and tk denote thek-source andk-target maps. Iff is a k-cell, sk−1(f) andtk−1(f) are respectively called itssourceand target and respectively denoted bys(f) and t(f). Thesource and target maps satisfy theglobular relations:

si ◦ si+1 = si ◦ ti+1 and ti ◦ si+1 = ti ◦ ti+1.

Two cellsf andg areparallel when they have same source and same target. A pair(f, g) of parallelk-cells is called ak-sphere. Theboundaryof ak-cell is the(k− 1)-sphere∂f = (s(f), t(f)). The sourceand target maps are extended to ak-sphereγ = (f, g) by s(γ) = f andt(γ) = g.

A pair (f, g) of k-cells ofC is i-composablewhenti(f) = si(g) holds; wheni = k − 1, one simplysayscomposable. The set ofi-composable pairs ofk-cells is denoted byCk ⋆i Ck. The i-compositeof (f, g) is denoted byf ⋆i g, i.e.in the diagrammatic direction. The compositions satisfy the exchangerelation given, for everyj < k and every possible cellsf, f ′, g andg ′ by:

(f ⋆j f′) ⋆k (g ⋆j g

′) = (f ⋆k g) ⋆j (f′⋆k g

′).

3


If f is ak-cell, we denote by Idf or 1f its identity (k + 1)-cell and, by abuse, all the higher-dimensionalidentity cells it generates. When1f is composed with cells of dimensionk + 1 or higher, we abusivelydenote it byf to make expressions easier to read. A cell isdegeneratewhen it is an identity cell. Fork ≤ n, ak-categoryC can be seen as ann-category, with only degenerate cells above dimensionk.

1.1.2. Graphical representations. Low-dimensional cells are writtenu : p→ q, f : u⇒ v,A : f⇛ g

and pictured as usual (and so aren-categories, omitting the degenerate cells):

p pu

// q p

u

!!

v

==f

��

q p

u

!!

v

==f

�*

g

s�

A_ %9 q.

For readability, we also depict3-cells as "rewriting rules" on2-cells:

p

u

v

>>f

��q

A≡⇛ p

u

v

>>g

��q.

For 2-cells, circuit-like diagrams are an alternative representations, where0-cells are parts of the plane,1-cells are lines and2-cells are points, inflated for emphasis:

v

u

fp q

v

u

f

v

u

gpA _ %9 qp q

1.2. Standard cells and spheres

1.2.1. The suspension functor.For every natural numbern, thesuspension functor

Sn : Catn → Catn+1

informally lifts all the cells by one dimension, adding a formal 0-source and a formal0-target for all ofthem; thus, in the(n + 1)-category one gets, one has exactly the same compositions asin the originalone. More formally, given ann-categoryC, the(n + 1)-categorySnC has the following cells:

(SnC)0 = {−,+} and (SnC)k+1 = Ck∐ {−,+} .

Every cell has0-source− and0-target+. The(k+1)-source and(k+1)-target of a non-degenerate cellare itsk-source andk-target inC. The(k + 1)-composable pairs are thek-composable ones ofC, pluspairs where at least one of the cells is an identity of− or+.

4

1.3. Adjoining and collapsing cells

1.2.2. The standardn-cell andn-sphere. By induction onn, we define then-categoriesEn andSn,respectively called thestandardn-cell and thestandardn-sphere. Informally, we consider them as then-categorical equivalents of the standard topologicaln-ball andn-sphere, used to build then-categoricalequivalents of (relative) CW-complexes.

The standard0-cell E0 is defined as any chosen single-element set and the standard0-sphere as anychosen set with two elements. Then, ifn ≥ 1, then-categoriesEn andSn are defined as the suspensionsof En−1 andSn−1:

En = Sn−1(En−1) and Sn = Sn−1(Sn−1).

For coherence, we defineS−1 as the empty set. Thus, the standardn-cell En andn-sphereSn havetwo non-degeneratek-cells e−k ande+k for everyk in {0, . . . , n − 1}, plus a non-degeneraten-cell enin En. Using the cellular representations, the standard cellsE0, E1, E2 andE3 are respectively picturedas follows (forS−1, S0, S1 andS2, one removes the top-dimensional cell):

e0 e−0e1

// e+0 e−0

e−1

""

e+1

<<e2

��

e+0 e−0

e−1

""

e+1

<<e−2

�*

e+2

s�

e3_ %9 e+0 ,

If C is ann-category then, for everyk in {0, . . . , n}, the k-cells andk-spheres ofC are in bijectivecorrespondence with then-functors fromEk to C and fromSk to C, respectively. When the context isclear, we use the same notation for ak-cell ork-sphere and its correspondingn-functor.

As a consequence, ifI is a set, theI-indexed families(fi)i∈I of k-cells andk-spheres ofC are inbijective correspondence with then-functors fromI ·Ek toC and fromI ·Sk toC, respectively. We recallthat, for a setX and ann-categoryD, the copowerX · D is the coproductn-category

∐x∈XD, whose

set ofk-cells is the productX×Dk.

1.2.3. The inclusion and collapsingn-functors. For everyn, the inclusionn-functor Jn and thecol-lapsingn-functorPn

Jn : Sn → En+1 and Pn : Sn → En

are respectively defined as the canonical inclusion ofSn into En+1 and as then-functor sending bothe−nande+n to en, leaving the other cells unchanged.

1.3. Adjoining and collapsing cells

1.3.1. Definition. Let C be ann-category, letk be in{0, . . . , n − 1}, let I be a set and letΓ : I · Sk → C

be ann-functor. Theadjoining ofΓ to C and thecollapsing ofΓ in C are then-categories respectivelydenoted byC[Γ ] andC/Γ and defined by the following pushouts inCatn:

I · SkΓ

//

I·Jk��

c©

C

��

I · Ek+1// C[Γ ]

I · SkΓ

//

I·Pk

��

c©

C

��

Ek// C/Γ .

5


Whenk = n, one definesC[Γ ] by seeingC as an(n + 1)-category with degenerate(n + 1)-cells only.By abuse, one denotes a sphere ofΓ and the induced cell ofC[Γ ] the same way.

To be more explicit, then-categoryC[Γ ] has the same cells asC up to dimensionk; its (k + 1)-cellsare all the formal composites made of the(k+ 1)-cells ofC, plus one extra(k+ 1)-cell from Γ(i, e−k ) toΓ(i, e+k ); above dimensionk+ 1, its cells are the ones ofC, plus the identities of each extra cell.

Then-categoryC/Γ has the same cells asC up to dimensionk − 1; its k-cells are the equivalenceclasses ofk-cells ofC, for the congruence relation generated byΓ(i, e−k ) ∼ Γ(i, e

+k ), for everyi; above

dimensionk, its cells are the formal composites of the ones ofC, but with sources and targets consideredmodulo the previous congruence.

1.3.2. Extension of ann-functor. Let C andD be n-categories and letΓ : I · Sk → C be ann-functor. Then, by universal property ofC[Γ ], one extends ann-functorF : C → D to a uniquen-functorF : C[Γ ] → D by fixing, for everyγ in Γ , a(k+ 1)-cell F(γ) in D, provided the following two equalitieshold:

s(F(γ)) = F(s(γ)) and t(F(γ)) = F(t(γ)).

1.3.3. Occurrences. Here we see the groupZ of integers as ann-category: it has one cell in eachdimension up ton− 1 andZ as set ofn-cells; all the compositions ofn-cells are given by the addition.

Let C be ann-category and letΓ : I · Sk → C be ann-functor. We denote by||·||Γ then-functorfrom C[Γ ] to Z defined by:

||f||Γ =

{1 if f ∈ Γ

0 otherwise.

For every cellf, one calls||f||Γ thenumber of occurrences of cells ofΓ in f. Indeed, intuitively,||f||Γ isthe number of(k+ 1)-cells ofΓ thatf contains.

Theset of cells ofΓ in f is the subset{f}Γ of Γ that consists of all the cellsγ of Γ such that||f||γ ≥ 1

holds. It is, intuitively, the set of cells ofΓ required to writef. This construction can also be seen as ann-functor. One considers the setP(Γ) of parts ofΓ as ann-category this way: it has one cell in eachdimension up ton − 1 and parts ofΓ asn-cells; all the compositions ofn-cells are given by the union.Then{·}Γ is then-functor fromC[Γ ] to P(Γ) defined by:

{f}Γ =

{{f} if f ∈ Γ

∅ otherwise.

1.3.4. Then-category presented by an(n + 1)-category. Let C be an(n + 1)-category. Iff is an(n + 1)-cell of C, then∂f is ann-sphere ofC. Thus, the setCn+1 of (n + 1)-cells of C yields an(n + 1)-functor fromCn+1 · Sn to the underlyingn-category ofC: then-category presented byC is then-category denoted byC one gets by collapsing the(n + 1)-cells ofC in its underlyingn-category.

1.4. Polygraphs and presentations ofn-categories

Polygraphs(or computads) are presentations by “generators” and “relations” of somehigher-dimensionalcategories [20, 6], see also [21, 22]. We definen-polygraphs by induction on the natural numbern.

6

1.4. Polygraphs and presentations ofn-categories

The categoryPol0 of 0-polygraphs and morphisms between them is the one of sets andmaps. A0-polygraph isfinite when it is finite as a set. A0-cell of a 0-polygraph is one of its elements. Thefree0-category functoris the identity functorPol0 → Cat0. Now, let us fix a non-zero natural numbern andlet us assume that we have defined the categoryPoln−1 of (n − 1)-polygraphs and morphisms betweenthem, finite polygraphs, cells of a polygraph and the free(n − 1)-category functorPoln−1 → Catn−1,sending an(n − 1)-polygraphΣ to the(n − 1)-categoryΣ∗.

1.4.1. n-polygraphs. An n-polygraphis a pairΣ = (Σn−1, Σn) made of an(n − 1)-polygraphΣn−1

and a familyΣn of (n − 1)-spheres of the(n − 1)-categoryΣ∗n−1.

An n-cell of Σ is an element ofΣn and, if k < n, a k-cell of Σ is a k-cell of the (n − 1)-polygraphΣn−1. The set ofk-cells ofΣ is abusively denoted byΣk, thus identifying it to thek-polygraphunderlyingΣ. An n-polygraph isfinite when it has a finite number of cells in every dimension. Thesizeof ak-cell f in Σ∗, denoted by||f||, is the natural number||f||Σk

, giving the number ofk-cells ofΣ thatfis made of. For1-cells and3-cells, we also use|·| and|||·||| instead of||·||.

The original paper [6] contains an equivalent description of n-polygraphs, where they are defined asdiagrams

Σ0 Σ1

s0,t0qqqqq

xxqqqqq ι1

��

(· · · )

s1,t1ppppp

xxppppp

Σn−1

sn−1,tn−1ooooo

wwoooooιn−1

��

Σn

sn,tnppppp

wwppppp

Σ0 Σ∗1

s0,t0

oo (· · · )s1,t1

oo Σ∗n−1

sn−1,tn−1

oo

of sets and maps such that, for anyk in {0, . . . , n − 1}, the following two conditions hold:

• The diagramΣ∗0 Σ∗

1s0

oo

t0oo

(· · · )s1

oo

t1oo Σ∗

ksk−1

oo

tk−1oo is ak-category.

• The diagramΣ∗0 Σ∗

1s0

oo

t0oo

(· · · )s1

oo

t1oo Σ∗

ksk−1

oo

tk−1oo Σk+1

skoo

tkoo is a(k+ 1)-graph.

1.4.2. Morphisms ofn-polygraphs. Let Σ andΞ be twon-polygraphs. Amorphism ofn-polygraphsfromΣ toΞ is a pairF = (Fn−1, Fn) whereFn−1 is a morphism of(n−1)-polygraphs fromΣn−1 toΞn−1

and whereFn is a map fromΣn to Ξn such that the following two diagrams commute:

ΣnFn

//

sn−1

��

c©

Ξn

sn−1

��

Σ∗n−1 F∗n−1

// Ξ∗n−1.

ΣnFn

//

tn−1

��

c©

Ξn

tn−1

��

Σ∗n−1 F∗n−1

// Ξ∗n−1.

Alternatively, ifΣn : I · Sn−1 → Σ∗n−1 andΞn : J · Sn−1 → Σ∗

n−1 are seen as(n − 1)-functors, thenFn

7


is a map fromI to J such that the following diagram commutes inCatn−1:

I · Sn−1Σn

//

Fn ·1Sn−1

��

c©

Σ∗n−1

F∗n−1

��

J · Sn−1Ξn

// Ξ∗n−1.

We denote byPoln the category of polygraphs and morphisms between them.

1.4.3. The freen-category functor. Let Σ be ann-polygraph. Then-category freely generated byΣis then-categoryΣ∗ defined as follows:

Σ∗ = Σ∗n−1[Σn].

This construction extends to ann-functor(·)∗ : Poln → Catn called thefreen-category functor.

1.4.4. Then-category presented by an(n + 1)-polygraph. Let Σ be a(n + 1)-polygraph. Then-category presented byΣ is then-category denoted byΣ and defined as follows:

Σ = Σ∗n/Σn+1.

Twon-polygraphs areTietze-equivalentwhen the(n−1)-categories they present are isomorphic. IfC isann-category, apresentation ofC is an(n+1)-polygraphΣ such thatC is isomorphic to then-categoryΣpresented byΣ. One says that ann-categoryC is finitely generatedwhen it admits a presentation by an(n + 1)-polygraphΣ whose underlyingn-polygraphΣn is finite. One says thatC is finitely presentedwhen it admits a finite presentation.

1.4.5. Example : a presentation of the2-category of permutations. The2-categoryPerm of permu-tations has one0-cell, one1-cell for each natural number and, for each pair(m,n) of natural number, itsset of2-cells fromm to n is the groupSn of permutations ifm = n and the empty set otherwise. The0-composition of1-cells is the addition of natural numbers. The0-composition of two2-cellsσ ∈ Smandτ ∈ Sn is the permutationσ ⋆0 τ defined by:

σ ⋆0 τ(i) =

{σ(i) if 1 ≤ i ≤ n,

τ(i− n) otherwise.

Finally the1-composition of2-cells is the composition of permutations. The2-categoryPerm is pre-sented by the3-polygraph with one0-cell, one1-cell, one2-cell, pictured by , and the following two3-cells:

⇛ and ⇛ .

8

2. Contexts, modules and derivations ofn-categories

2. CONTEXTS, MODULES AND DERIVATIONS OFn-CATEGORIES

2.1. The category of contexts of ann-category

Throughout this section,n is a fixed natural number andC is a fixedn-category.

2.1.1. Contexts of ann-category. A context ofC is a pair(x,C) made of ann-spherex of C and ann-cell C in C[x] such that||C||x = 1. We often denote byC[x], or simply byC, such a context.

Let x andy ben-spheres ofC and letf be ann-cell in C[x] such that∂f = y holds. We denote byC[f] the image of a contextC[y] of C by the functor

C[y] → C[x]

defined by extending the identityn-functor ofC with y 7→ f.

2.1.2. The category of contexts.Thecategory of contexts ofC is denoted byCC and defined as follows:

• Its objects are then-cells ofC.

• If f andg aren-cells ofC, then the morphisms ofCC from f to g are the contextsC[∂f] of C suchthatC[f] = g holds.

• If C : f→ g andD : g→ h are morphisms ofCC thenD ◦ C : f→ h isD[C].

• The identity contextIf on ann-cell f of C is the context∂f[∂f].

WhenΣ is ann-polygraph, one usesCΣ instead ofCΣ∗.

2.1.3. Proposition. Every contextC of C[x] decomposes as follows:

C[x] = fn ⋆n−1 (fn−1 ⋆n−2 · · · (f1 ⋆0 x ⋆0 g1) · · · ⋆n−2 gn−1) ⋆n−1 gn,

where, for everyk in {1, . . . , n}, fk and gk are n-cells inC. Moreover, one can chose these cells sothat fk andgk are (the identities of)k-cells.

Proof. The set ofn-cellsf of C[x] such that||f||x = 1 is a quotient of the following inductively definedsetX:

• Then-cell x is inX.

• If C is inX andf is ann-cell of C such thatti(f) = si(C) holds for somei, thenf ⋆i C is inX.

• If C is inX andf is an-cell of C such thatti(C) = si(f) holds for somei, thenC ⋆i f is in X.

Using the associativity and exchange relations satisfied bythe compositions ofC, one can order thesesuccessive compositions to reach the required shape, or to reach the same shape withfk andgk beingidentities ofk-cells.

9


2.1.4. Whiskers. A whisker ofC is a contextC[x] with a decomposition

C[x] = fn−1 ⋆n−2 · · · (f1 ⋆0 x ⋆0 g1) · · · ⋆n−2 gn−1

such that, for everyk in {1, . . . , n − 1}, fk andgk arek-cells. We denote byWC the subcategory ofCC

with the same objects and with whiskers as morphisms. WhenΣ is ann-polygraph, we writeWΣ insteadof WΣ∗.

2.1.5. Proposition. LetΣ be ann-polygraph and letf be ann-cell ofΣ∗ with sizek ≥ 1. Then thereexistn-cellsγ1, . . . , γk in Σ and whiskersC1, . . . , Ck ofΣ∗ such thatf decomposes as follows:

f = C1[γ1] ⋆n−1 · · · ⋆n−1Ck[γk].

Proof. We proceed by induction on the size of then-cell f.If it has size1, then it contains exactly onen-cell γ of Σ, possibly composed with other ones of

lower dimension. Using the relations satisfied by compositions in ann-category, one can writef asC[γ],with C a context ofΣ∗. Moreover, this context must be a whisker, sincef has size1.

Now, let us assume that we have proved that everyn-cell with size at mostk, for a fixed non-zeronatural numberk, admits a decomposition as in Proposition 2.1.5. Then let usconsider ann-cell f withsizek+ 1. Since||f|| ≥ 2 and by construction ofΣ∗ = Σ∗

n−1[Σn], one gets thatf can be writteng ⋆i h,where(g, h) is a pair ofi-composablen-cells ofΣ∗, for somei in {0, . . . , n − 1}, with ||g|| and ||h||at least1. One can assume thati = n − 1 since, otherwise, one considers the following alternativedecomposition off, thanks to the exchange relation between⋆i and⋆n−1:

f = (g ⋆i s(h)) ⋆n−1 (t(g) ⋆i h) .

Since ||f|| = ||g|| + ||h||, one must have||g|| ≤ k and ||h|| ≤ k. We use the induction hypothesis todecomposeg andh as in 2.1.5, wherej denotes||g||:

g = C1[γ1] ⋆n−1 · · · ⋆n−1Cj[γj] and h = Cj+1[γj+1] ⋆n−1 · · · ⋆n−1Ck[γk].

We compose the right members and use the associativity of⋆n−1 to conclude.

2.2. Contexts in low dimensions

2.2.1. Contexts of a1-category as factorizations. From Proposition 2.1.3, the contexts of a1-categoryChave the following shape:

f ⋆0 x ⋆0 g,

wheref andg are1-cells ofC. The morphisms inCC from h : u → v to h ′ : u ′ → v ′ are the pairs(f : u ′ → u, g : v→ v ′) of 1-cells ofC such that the following diagram commutes inC:

u

h

��

c©

u ′foo

h′

��

vg

// v ′.

10

2.2. Contexts in low dimensions

WhenC is freely generated by a1-polygraph, the1-cells f andg are uniquely defined by the context.Moreover, the contexts fromh to h ′ are in bijective correspondence with the occurrences of thewordhin the wordh ′.

The categoryCC has been introduced by Quillen under the namecategory of factorizations ofC [18].It has been used by Leech to introduce cohomological properties of congruences on monoids [14] andby Baues and Wirsching for the cohomology of small categories [3].

2.2.2. Contexts of2-categories. Let C be a2-category. From Proposition 2.1.3, a context ofC has thefollowing shape:

h ⋆1 (g1 ⋆0 x ⋆0 g2) ⋆1 k

whereg1, g2, h andk are 2-cells. Morphisms inCC from a 2-cell f to a 2-cell f ′ are the contextsC = h ⋆1 (g1 ⋆0 x ⋆0 g2) ⋆1 k such thath ⋆1 (g1 ⋆0 f ⋆0 g2) ⋆1 k = f ′ holds:

•��

EE

��

CCg1

��•

��

CCf��

h��

k��

•��

CCg2

��• = •

��

CCf′

��•

Note that this decomposition is not unique. One proves that two decompositions

h ⋆1 (g1 ⋆0 x ⋆0 g2) ⋆1 k and h ′⋆1 (g

′1 ⋆0 x

′⋆0 g

′2) ⋆1 k

′

represent the same context if and only ifx = x ′ and there exist2-cells l1, l2, m1, m2 such that thefollowing four relations are defined and hold inC:

•$$

// • // • // •h��

•$$))

CC•// • ))

CC•h′

��l1�� l2��

=

•��##

;; JJ

l1��

g1��

m1��

• = •��

CCg′

1

��• •

��

CCg′

2

��• = •

��##

;; JJ

l2��

g2��

m2��

•

• ::

��

55 • // •��

55 •k′

��

m1��m2��

• ::// • // • // •

k��

=

11


Thus, informally, two decompositions represent the same context when one can pass from one to theother one by moving up and down2-cells that one finds at the left side or the right side of the spherex.

2.3. Modules overn-categories

2.3.1. Definition. Let C be ann-category. Amodule overC or C-moduleis a functor from the categoryof contextsCC to the categoryAb of abelian groups. Hence, aC-moduleM is given by:

• An abelian groupM(f), for everyn-cell f in C.

• A morphismM(C) : M(f) → M(g) of groups, for every contextC(x) : f → g of C. When thecontextC(x) is of the shapeh⋆ix (resp.x⋆ih) and when no confusion may occur, then one writesh ⋆im (resp.m ⋆i h) instead ofM(C)(m).

2.3.2. Proposition. LetΣ be ann-polygraph. AΣ∗-moduleM is entirely and uniquely defined by itsvalues on the following contexts ofC

C[ϕ] ⋆i x and x ⋆iC[ϕ]

for everyi in {0, . . . , n − 1}, every generating(i + 1)-cellϕ and everyi-contextC[∂ϕ].

Proof. Let f, g be twon-cells ofΣ∗ and letC[x] : f→ g be a morphism ofCΣ. We decomposeC[x] asfollows:

C[x] = fn ⋆n−1 · · · ⋆1 (f1 ⋆0 x ⋆0 g1) ⋆1 · · · ⋆n−1 gn,

in such a way that, for everyk in {1, . . . , n}, fk andgk arek-cells. Thus, in the categoryCΣ, the contextC[x] decomposes as

C[x] = Cn[xn] ◦ · · · ◦ C1[x1],

wherex1 = x and, for everyi in {1, . . . , n}, one hasCi[xi] = fi ⋆i−1 xi ⋆i−1 gi andxi+1 = ∂Ci[xi].Moreover, eachCi[xi] splits into:

Ci[xi] = (yi ⋆i−1 gi) ◦ (fi ⋆i−1 xi) ,

whereyi = ∂(fi ⋆i−1 xi). Thus, sinceM is a functor, it is entirely defined by its values on the contextswith shapef ⋆i x or x ⋆i f, with i in {0, . . . , n − 1} andf a non-degenerate(i + 1)-cell (indeed, whenfis degenerate as ai-cell, one hasx ⋆i f = x andM(x) is always an identity).

Now, let us consider then-contextf ⋆i x, wheref is an(i + 1)-cell of sizek ≥ 1. We decompose itas in 2.1.5:

f = C1[ϕ1] ⋆i · · · ⋆i Ck[ϕk],

whereϕ1, . . . , ϕk are generating(i + 1)-cells andC1, . . . , Ck arei-contexts. Thus, a contextf ⋆i xdecomposes intoCΣ as follows:

f ⋆i x = (C1[ϕ1] ⋆i x1) ◦ · · · ◦ (Ck[ϕk] ⋆i xk) ,

wherexk = x andxj = ∂(Cj+1[ϕj+1] ⋆i xj+1).Proceeding similarly with contexts of the shapex ⋆i f, one gets the result.

12

2.3. Modules overn-categories

2.3.3. Example: the trivial module. Let C be ann-category. Thetrivial C-modulesends eachn-cellof C toZ and each context ofC to the identity ofZ.

2.3.4. Example of modules over2-categories. Let V be a concrete category,i.e. a category equippedwith a faithful functor intoSet. We view it as a2-category with:

• One0-cell.

• Objects as1-cells. Their0-composition is given by the cartesian product.

• Morphisms as2-cells. Their0-composition is given by the cartesian product of morphisms. Their1-composition is the composition of morphisms inV, written in reverse order.

Let us fix a2-categoryC and let us consider2-functorsX : C → V andY : Cco → V, whereCco is C

where one has exchanged the source and target of every2-cell. Then, for every internal abelian groupGin V, the following assignments yield aC-moduleM(X, Y,G):

• Every2-cell f : u⇒ v is sent to the abelian group of morphisms:

M(X, Y,G)(f) = V(X(u)× Y(v), G

).

• Let C be a context fromf : u ⇒ v to f ′ : u ′ ⇒ v ′ and letC = g ⋆1 (w ⋆0 x ⋆0 w′) ⋆1 g

′ be adecomposition ofC, whereg, g ′ are2-cells andw,w ′ are1-cells. By hypothesis,X(g) andY(g ′)

are morphisms inV:

X(g) : X(u ′) → X(w)× X(u)× X(w ′) Y(g ′) : Y(v ′) → Y(w)× Y(v)× Y(w ′).

We denote byX(g) : X(u ′) → X(u) and byY(g ′) : Y(v ′) → Y(v) the morphisms they induce,thanks to the projections given by the cartesian structure.ThenM(X, Y,G)(C) is defined as themorphism of groups fromM(X, Y,G)(f) to M(X, Y,G)(f ′) that sends a morphismϕ : X(u) ×

Y(v) → G of V to the following one:

ϕ ◦(X(g), Y(g ′)

): X(u ′)× Y(v ′) → G .

WhenX or Y is trivial, i.e. sends all the cells ofC to the terminal object ofV, one denotes the corre-spondingC-module byM(∗, Y,G) orM(X, ∗, G). In particular,M(∗, ∗,Z) is the trivialC-module.

By construction, aC-moduleM(X, Y,G) is uniquely and entirely defined by its values on the contextsh ⋆1 x andx ⋆1 h, for every2-cell h in C, i.e. by the objectsX(u) andY(u), for every1-cell u, andby the morphismsX(f) andY(f) for every2-cell f. As a consequence, whenC is freely generated by a2-polygraphΣ, theC-moduleM(X, Y,G) is uniquely and entirely determined by:

• The objectsX(a) andY(a) of V for every1-cell a in Σ1.

• The morphismsX(γ) : X(u) → X(v) andY(γ) : Y(v) → Y(u) of V for every2-cell γ : u ⇒ v

in Σ2.

In the sequel, we consider this kind ofC-module withV being the categorySetof sets and maps or thecategoryOrd of partially ordered sets and monotone maps. In this last situation, an internal abeliangroup inOrd is a partially ordered set equipped with a structure of abelian group, such that group law ismonotone in both arguments.

13

3. Higher-dimensional categories with finite derivation type

2.4. Derivations ofn-categories

2.4.1. Definition. Let C be ann-category and letM be aC-module. Aderivation ofC intoM is a mapsending everyn-cell f of C to an elementd(f) of M(f) such that the following relation holds, for everyi-composable pair(f, g) of n-cells ofC:

d(f ⋆i g) = f ⋆i d(g) + d(f) ⋆i g.

Given a derivationd onC, we define its values on contexts by

d(C) =

n∑

i=−n

fn ⋆n−1 (fn−1 ⋆n−2 · · · (d(fi) ⋆i−1 · · · (f1 ⋆0 x ⋆0 f−1) · · · ⋆n−1 f−n,

for any contextC = fn ⋆n−1 · · · (f1 ⋆0 x ⋆0 f−1) · · · ⋆n−1 f−n in C[x]. This gives a mappingd(C) fromthe set of2-cells ofC with boundary∂x to the abelian groupM(∂C). In this way a derivation fromCintoM satisfies:

d(C[f]) = d(C)[f] + C[d(f)].

Given a derivationd of ann-categoryC and ann-functor Γ : I · Sn → C, we defined on (n + 1)-cellsof C[Γ ] by d(γ) = d(sγ) − d(tγ). Let us note that this has no reason to be a derivation ofC[Γ ].

If f is a degeneraten-cell, thend(f) = 0 holds. Indeed, one has:

d(f) = d(f ⋆n−1 f) = f ⋆n−1 d(f) + d(f)⋆n−1 = 2 · d(f).

2.4.2. Example: occurrences.If C is ann-category andΓ : I ·Sn → C is ann-functor, we have definedthen-functor ||·||Γ counting the number of occurrences of cells ofΓ in a cell ofC[Γ ]. This construction isa derivation ofC into the trivialC-module, sending eachn-cell of C to 0 and eachn-cell of Γ to 1.

2.4.3. Example of derivations of free2-categories. Let us consider a2-polygraphΣ, a concrete cat-egoryV and a module of the shapeM(X, Y,G), as defined in 2.3.4. Then, by construction ofΣ∗, aderivationd of Σ∗ intoM(X, Y,G) is entirely and uniquely determined by a family(dγ)γ∈Σ2

made of amorphism

dγ : X(u)× Y(v) → G

of V for each2-cell γ : u ⇒ v of Σ. Indeed, every2-cell of Σ∗ can be written as a formal composite of2-cells ofΣ and1-cells ofΣ∗. Moreover, the relations satisfied by a derivation imply that such a map iscompatible with the axioms ofn-category.

3. HIGHER-DIMENSIONAL CATEGORIES WITH FINITE DERIVATION TYPE

3.1. Track n-categories

3.1.1. Definitions. In ann-categoryC, a k-cell f is invertible when there exists ak-cell g from t(f)

to s(f) in C such that bothf⋆k−1g = s(f) andg⋆k−1f = t(f) hold. In that case,g is unique and denotedby f−1. The following relations are satisfied:

(1x)−1 = 1x and (f ⋆i g)

−1 =

{f−1

⋆i g−1 if i < k− 1

g−1⋆k−1 f

−1 otherwise.

14

3.2. Homotopy bases

Moreover, ifF : C → D is ann-functor, one has:

F(f−1) = F(f)−1.

A track n-categoryis ann-category whosen-cells are invertible,i.e. an (n − 1)-category enriched ingroupoid. We denote byTckn the category of trackn-categories andn-functors between them.

3.1.2. Example. Let C be ann-category. Given twon-cells f from u to v andg from v to u in C, wedenote byIf,g the followingn-sphere ofC:

If,g = (f ⋆n−1 g, 1u) .

If γ = (f, g) is ann-sphere ofC, we denote byγ−1 then-sphere(g, f) of C. Then we define the track(n + 1)-categoryC(γ) by:

C(γ) = C[γ, γ−1]/{Iγ,γ−1 ,Iγ−1,γ

}.

This construction is extended to a setΓ of n-spheres, yielding a track(n + 1)-categoryC(Γ).

3.1.3. The free track category functor. Given ann-polygraphΣ, thetrackn-category freely generatedbyΣ is then-category denoted byΣ⊤ and defined by:

Σ⊤ = Σ∗n−1(Σn),

This construction extends into a functor(·)⊤ : Poln → Tckn called thefree trackn-category functor.

3.2. Homotopy bases

3.2.1. Homotopy relation. Let C be an-category. Ahomotopy relationon C is a track(n + 1)-categoryT with C as underlyingn-category. Given ann-sphere(f, g) in C, one denotes byf ≈T g thefact that there exists an(n+ 1)-cell from f to g in T. If Γ is a set ofn-spheres ofC, one simply uses≈Γ

instead of≈C(Γ) and calls it thehomotopy relation onC generated byΓ .One hasf ≈T g if and only if π(f) = π(g) holds, whereπ is the canonical projection fromT to the

n-categoryT presented byT, i.e. C/Tn+1. As a consequence, the relation≈T is a congruence relationon the paralleln-cells ofC, i.e. it is an equivalence relation compatible with every composition of C.

3.2.2. Homotopy basis. A set Γ of n-spheres ofC is a homotopy basis ofC when, for everyn-sphere(f, g) of C, one hasf ≈Γ g. In other words,Γ is a homotopy basis if and only if, for everyn-sphereγ of C, there exists an(n + 1)-cell γ such that∂γ = γ holds, i.e. such that the followingdiagram commutes inCatn+1:

Snγ

//

Jn��

c©

C

��

En+1γ

// C(Γ).

15


3.2.3. Proposition. Let C be ann-category and letΓ be a homotopy basis ofC. If C admits a finitehomotopy basis, then there exists a finite subset ofΓ that is a homotopy basis ofC.

Proof. Let Γ ′ be a finite homotopy basis ofC. Let γ be ann-sphere ofC in Γ ′. SinceΓ is a homotopybasis ofC, there exists an(n+1)-cellϕγ in C(Γ) with boundaryγ. This defines an(n+1)-functorF fromC(Γ ′) to C(Γ) which is the identity on cells ofC and which sends eachγ in Γ ′ toϕγ. For eachϕγ, wefix a representative inC[Γ, Γ−1] and denote by{ϕγ}Γ the set of cells ofΓ occurring in this representative.Let us denote byΓ0 the following subset ofΓ

Γ0 =⋃

γ∈Γ ′

{ϕγ}Γ ,

consisting of all the cells ofΓ contained in the cellsϕγ. The subsetΓ0 is finite sinceΓ ′ and every{ϕγ}

are. Now let us see that it is an homotopy basis ofC. Let us fix ann-sphere(f, g) of C. By hypothesis,there exists an(n + 1)-cell A in C(Γ ′) with boundary(f, g). By application ofF, one gets an(n + 1)-cell F(A) in C(Γ) with boundary(f, g). Moreover, the(n + 1)-cell F(A) is a composite of cells of theshapeϕγ: hence, it lives inC(Γ0). As a consequence, one getsf ≈Γ0 g, which concludes the proof.

3.3. Polygraphs with finite derivation type

3.3.1. Definition. One says that ann-polygraphΣ hasfinite derivation typewhen it is finite and whenthe trackn-categoryΣ⊤ it generates admits a finite homotopy basis.

3.3.2. Lemma. LetΣ andΣ ′ ben-polygraphs. We denote byπ : Σ∗n−1 → Σ and byπ ′ : Σ ′∗

n−1 → Σ′

the canonical(n − 1)-functors. Then every(n − 1)-functorF fromΣ to Σ′can be lifted to ann-functor

F : Σ⊤ → Σ ′⊤ such that the following diagram commutes inCatn−1:

Σ∗n−1

π//

eF��

c©

Σ

F

��

Σ ′∗n−1 π′

//Σ

′.

Proof. For everyk-cell u in Σ∗, with k in {0, . . . , n − 2}, we takeF(u) = F(u). Sinceπ andπ ′ areidentities on cells up to dimensionn− 2, we have the relationF ◦ π(u) = π ′ ◦ F(u).

Now, let us consider an(n−1)-cell u in Σn−1. One arbitrarily chooses an(n−1)-cell ofΣ ′∗, henceof Σ ′⊤, that is sent onF ◦ π(u) by π ′, and one fixesF(u) to that(n − 1)-cell. One extendsF to every(n − 1)-cell ofΣ∗ thanks to the universal property ofΣ∗.

Then, letf be ann-cell fromu to v in Σn. Thenπ(u) = π(v) holds by definition ofπ. Applying Fon both members and using the property satisfied byF, one getsπ ′ ◦ F(u) = π ′ ◦ F(v). By definitionof π ′ and ofΣ ′⊤, this means that there exists ann-cell from F(u) to F(v) in Σ ′⊤. One takes one suchn-cell for F(f). Finally, one extendsF to everyn-cell of Σ⊤.

16

3.3. Polygraphs with finite derivation type

3.3.3. Lemma. LetΣ andΣ ′ ben-polygraphs and letF : Σ⊤ → Σ ′⊤ be ann-functor. Given a setΓ of

n-spheres ofΣ⊤, we defineF(Γ) as the following set ofn-spheres ofΣ ′⊤:

F(Γ) ={(F(g), F(g ′))

∣∣ (g, g ′) ∈ Γ}.

Then, for everyn-sphere(f, f ′) ofΣ⊤ such thatf ≈Γ f holds, we haveF(f) ≈F(Γ) F(f′).

Proof. It is an immediate consequence of functoriality ofF.

3.3.4. Proposition. LetΣ andΣ ′ be Tietze-equivalent finiten-polygraphs. ThenΣ has finite derivationtype if and only ifΣ ′ has.

Proof. Let us assume thatΣ andΣ ′ aren-polygraphs which present the same(n − 1)-category, sayC.Let us assume thatΣ has finite derivation type, so that we can fix a finite homotopy basisΓ of Σ⊤.Using Lemma 3.3.2 twice on the(n − 1)-functor Id(C), we get twon-functorsF : Σ⊤ → Σ ′⊤ andG : Σ ′⊤ → Σ⊤ such that the following diagrams commute inCatn−1:

Σ∗n−1

π//

F��

c©

C

Id(C)

��

Σ ′∗n−1 π′

// C

Σ∗n−1

π//

c©

C

Σ ′∗n−1 π′

//

G

OO

C.

Id(C)

OO

In particular, bothπ andπ ′ are the identity onk-cells, for everyk < n − 1, hence so areF andG.Let us consider an(n− 1)-cell a in Σ ′

n−1. Thenπ ′ ◦ FG(a) = π ◦G(a) = π ′(a). Thus, there exists

ann-cell denoted byfa from a to FG(a) in Σ ′⊤. From these cells, we definefu for every(n− 1)-cell uin Σ ′∗, hence ofΣ ′⊤, using the following relations:

• For every degenerate(n − 1)-cell u, fu is defined asu.

• For everyi-composable pair(u, v) of (n − 1)-cells,fu⋆iv is defined asfu ⋆i fv.

We have that, for every(n − 1)-cell u, then-cell fu goes fromu to FG(u): to check this, we arguethatFG is ann-functor which is the identity on degenerate(n − 1)-cells.

Now, let us consider ann-cell g from u to v in Σ ′⊤. We denote byfg the followingn-cell fromu

tou in Σ ′⊤, with a cellular representation giving the intuition for the casen = 2:

f(g) = g ⋆n−1 fv ⋆n−1 FG(g)−1

⋆n−1 f−1u •

FG(u)

��

u

$$

FG(v)::

v

GG

fu

+?

g

�/fv

EY

FG(g)−1

]q

•

17


Let us prove that, for any composable pair(g, h) of n-cells inΣ ′⊤, we have:

fg⋆n−1h = g ⋆n−1 fh ⋆n−1 g−1

⋆n−1 fg.

For that, we assume thatg has sourceu and targetv, while h has sourcev and targetw. Then wecompute:

g ⋆n−1 fh ⋆n−1 g−1

⋆n−1 fg

= g ⋆n−1

(h ⋆n−1 fw ⋆n−1 FG(h)

−1⋆n−1 f

−1v

)

⋆n−1 g−1

⋆n−1

(g ⋆n−1 fv ⋆n−1 FG(g)

−1⋆n−1 f

−1u

)

= g ⋆n−1 h ⋆n−1 fw ⋆n−1 FG(h)−1

⋆n−1 FG(g)−1

⋆n−1 f−1u

= (g ⋆n−1 h) ⋆n−1 fw ⋆n−1 FG(g ⋆n−1 h)−1

⋆n−1 f−1u

= fg⋆n−1h.

Now, let us consider ann-cell g and a whiskerC[x] in Σ⊤ such thatx = ∂(g−n−1). We note that, bydefinition offg, it has the same(n − 1)-source and(n − 1)-target asg, so thatC[fg] is defined. Let usprove that the following relation holds:

fC[g] = C[fg].

From the decomposition of contexts, it is sufficient to provethat the following relation holds

fu⋆ig⋆iv = u ⋆i fg ⋆i v

for everyn-cell g, every possiblek-cellsu andv, with k < n − 1, and everyi < k such thatu ⋆i g ⋆i v

is defined. Let us assume thatg has sourcew and targetw ′ and compute, from the left-hand side of thisrelation:

fu⋆ig⋆iv = (u ⋆i g ⋆i v) ⋆n−1 fu⋆iw′⋆iv ⋆n−1 FG(u ⋆i g ⋆i v)

−1⋆n−1 f

−1u⋆iw⋆iv

= (u ⋆i g ⋆i v) ⋆n−1 (u ⋆i fw′ ⋆i v) ⋆n−1 (u ⋆i FG(g)−1

⋆i v) ⋆n−1 (u ⋆i f−1w ⋆i v)

= u ⋆i fg ⋆i v.

Now, we denote byΓ ′ the set ofn-spheres(fg, Ids(g)), for everyn-cell g in Σ ′n. Then, it follows from

the previous relations that, for everyn-cell g in Σ ′⊤, one has:

fg ≈Γ Ids(g) .

Let us consider ann-sphere(g, g ′) in Σ ′⊤. Then(G(g), G(g ′)) is ann-sphere inΣ⊤. SinceΓ is a ho-motopy basis forΣ⊤, we haveG(g) ≈Γ G(g

′), so that, by Lemma 3.3.3, one getsFG(g) ≈F(Γ) FG(g′).

Finally, let us denoteΓ ′′ the set ofn-spheres ofΣ ′⊤ defined byΓ ′′ = Γ ′ ∪ F(Γ) and let us provethat Γ ′′ is a finite homotopy basis ofΣ ′⊤. Since bothΣ ′

n andΓ are finite, so isΓ ′′. Let us consider ann-sphere(g, g ′) in Σ ′⊤, with sourcew and targetw ′, and let us prove thatg ≈Γ ′′ g ′ holds. We start byusing the definition offg to get:

g = fg ⋆n−1 f−1w′ ⋆n−1 FG(g) ⋆n−1 f

−1w .

Using the definition offg′ , one gets a similar formula forg ′. We have seen thatfg ≈Γ ′ w, fg′ ≈Γ ′ w

andFG(g) ≈Γ ′′ FG(g ′) hold. Thus one getsg ≈Γ ′′ g ′.

18

4. Critical branchings and finite derivation type

Proposition 3.3.4 shows that the property is invariant by Tietze-equivalence for finite polygraphs. Wewill illustrate in Example 4.3.10 that this is not the case for infinite ones. Thus we define

3.3.5. Definition. An n-category hasfinite derivation typewhen it admits a presentation by an(n+1)-polygraph with finite derivation type.

4. CRITICAL BRANCHINGS AND FINITE DERIVATION TYPE

4.1. Rewriting properties of polygraphs

In this section we review abstract rewriting properties on polygraphs. Proof of claims in this section areintuitively the same as the ones for abstract rewriting systems, which can be found in [1]. We fix an(n + 1)-polygraphΣ and ann-cell f in Σ∗.

4.1.1. Reductions and normal forms. One says thatf reducesinto somen-cell g when there exists anon-degenerate(n+ 1)-cellA from f to g in Σ∗. A reduction sequenceis a family(fk)k of n-cells suchthat eachfk reduces intofk+1. One says thatf is a normal form (forΣn+1) when every(n+1)-cell withsourcef is degenerate,i.e. it reduces only in itself. Anormal form forf is a normal formg such thatfreduces intog. The polygraphΣ is normalizing atf whenf admits a normal form. It isnormalizingwhen it is at everyn-cell of Σ∗.

4.1.2. Termination. One says thatΣ terminates atf when there exists no infinite reduction sequencestarting atf is stationary. One says thatΣ terminateswhen it does at everyn-cell ofΣ∗.

If Σ terminates atf, then it is normalizing atf, i.e. everyn-cell has at least one normal form.Moreover, in case of termination, one can prove properties using a form of induction, usually calledNoetherian induction: for that, one proves the property on normal forms; then one fixes ann-cell f, oneassumes that the result holds for everyg such thatf reduces intog and one proves that, under thosehypotheses, then-cell f satisfies the property.

4.1.3. Confluence.A branching ofΣ is a pair(A,B) of (n + 1)-cells ofΣ∗ with same source; thisn-cell is called thesourceof the branching(A,B). A branching(A,B) is local when||A|| = ||B|| = 1. Aconfluence ofΣ is a pair(A,B) of (n+ 1)-cells ofΣ∗ with same target. A branching(A,B) is confluentwhen there exists a confluence(A ′, B ′) such that bothtn(A) = sn(A ′) andtn(B) = sn(B ′) hold, as inthe following diagram:

A

��B

��???????

A′

��???????

B′

��

Such a pair(A ′, B ′) is called aconfluence for(A,B). Branchings and confluences are only consideredup to symmetry, so that(A,B) and(B,A) are considered equal. The polygraphΣ is (locally) confluentat f when every (local) branching with sourcef is confluent. It is(locally) confluentwhen it is at everyn-cell.

19


If Σ is confluent then everyn-cell of Σ∗ has at most one normal form. Thus, normalization andconfluence imply that then-cell f has exactly one normal form, writtenf. In a terminating polygraph,local confluence and confluence are equivalent: this was proved in the case of word rewriting systems (asubcase of2-polygraphs) by Newman [17] and, since, the result is calledNewman’s lemma.

4.1.4. Convergence.The polygraphΣ is convergent atf when it terminates and it is confluent atf. Itis convergentwhen it is at everyn-cell.

If Σ is convergent atf, thenf has exactly one normal form. Thanks to Newman’s lemma, one getsconvergence from termination and local confluence. IfΣ is convergent, we havef ≈Σn+1

g if and only if

f = g: thus, a convergent(n+ 1)-polygraph provides a decision procedure to the equivalence ofn-cellsin then-category it presents.

4.1.5. Critical branchings in polygraphs. Given a branchingb = (A,B) of Σ with sourcef and awhiskerC[∂f] of Σ∗, the pairC[b] = (C[A], C[B]) is a branching ofΣ, with sourceC[f]. Furthermore,if b is local andC is a whisker, thenC[b] is also local. We define by4 the order relation on branchingsof Σ given byb 4 b ′ when there exists a whiskerC such thatC[b] = b ′ holds.

A branching isminimalwhen it is minimal for the order relation4. A branching istrivial when itcan be written either as(A,A), for a(n+ 1)-cellA, or as(A ⋆i sn(B), sn(A) ⋆iB), for (n+ 1)-cellsAandB and ai in {0, . . . , n − 1}. A branching iscritical when it is minimal and not trivial.

A polygraph is locally confluent if and only if all of its critical branchings are confluent. To prove that,one checks that trivial branchings are always confluent and that a non-minimal branching is confluent ifand only if the corresponding minimal branching is (there isa minimal one by a size argument, allowedby the fact that we work in a polygraph).

4.2. Using derivations for proving termination of a3-polygraph

A method to prove termination of a3-polygraph has been introduced in [9], see also [10, 11]; in specialcases, it can also provide complexity bounds [4]. It turns out that the method uses interpretations that area special case of derivations, as described here. Here we only give an outline of the proof.

4.2.1. Theorem. LetΣ be a3-polygraph. LetX : Σ∗2 → Ord andY : (Σ∗)co

2 → Ord be2-functors,letG be an abelian group inOrd and letd be a derivation ofΣ∗

2 intoM(X, Y,G) such that the followingconditions hold:

• For every1-cell a in Σ1, the setsX(a) andY(a) are not empty.

• For every3-cell from f to g in Σ3, the inequalitiesX(f) ≥ X(g), Y(f) ≥ Y(g) andd(f) > d(g)are satisfied.

• The addition ofG is strictly monotone in both arguments.

• Every bounded decreasing sequence of elements ofG is stationary.

• There exists an elementz in G such that, for every2-cell f in Σ∗2, one hasd(f) ≥ z.

Then the3-polygraphΣ terminates.

20

4.3. Branchings and homotopy bases

Proof. Let us assume thatA : f ⇛ g is a3-cell of Σ∗ with size1. Then there exists a3-cell α : ϕ⇛ ψ

of Σ and a contextC of Σ∗2 such thatA = C[α] holds,i.e. such thatf = C[ϕ] andg = C[ψ] hold. Thus,

one gets:d(f) = d(C)[ϕ] + C[d(ϕ)] and d(g) = d(C)[ψ] + C[d(ψ)].

We use the factd(ϕ) > d(ψ) holds by hypothesis to getC[d(ϕ)] > C[d(ψ)]. Moreover, sinceX andYare2-functors intoOrd and sinced sends every2-cell to a monotone map, one getsd(C)[ϕ] ≥ d(C)[ψ].Finally, one uses the hypothesis on the strict monotony of addition in G to getd(f) > d(g). Then onededuces that, for every non-degenerate3-cell A : f ⇛ g, one hasd(f) > d(g). Thus, every infinitereduction sequence(fk)k would produce an infinite, strictly decreasing sequence(d(fk)k) in G, theexistence of which is prohibited by the last two hypotheses.

4.2.2. Special cases.The sequel contains several examples where derivations areused to prove termi-nation. Other examples can be found in [10] or [4]. In some of those, we take at the trivial2-functorfor at least one of the2-functorsX andY. Also, we often takeG to beZ, with the derivation satisfyingd(f) > 0 for everyf. One can check that those situations match the hypotheses ofTheorem 4.2.1.


In the case of convergent word rewriting systems,i.e. convergent2-polygraphs with exactly one0-cell,the critical branchings generate a homotopy basis [19]. In this section, we generalise this result to anypolygraph.

In particular, we recover Squier’s theorem as Corollary 4.3.7, stating that a finite convergent2-polygraph has finite derivation type. However, this result fails to generalise to higher-dimensionalpolygraphs, as stated in Theorem 4.3.9. Indeed, starting with n = 3, there exist finite convergentn-polygraphs with an infinite number of critical branchings. The detailed proof can be found in 5.5.

4.3.1. Notation. WhenΣ is a locally confluent(n + 1)-polygraph, we assume that, for every criticalbranchingb = (A,B), a confluence(A ′, B ′) has been chosen. We denote byΓΣ the set of all the(n + 1)-spheres(A ⋆nA

′, B ⋆n B′) of Σ, for each critical branchingb = (A,B).

4.3.2. Lemma. LetΣ be a locally confluent(n+1)-polygraph. Then every local branchingb = (A,B)

admits a confluence(A ′, B ′) such thatA ⋆nA′ ≈ΓΣ B ⋆n B

′ holds.

Proof. First, let us examine the case whereb is a trivial branching. IfA = B, then(tn(A), tn(B)) is aconfluence that satisfies the required property. Otherwise,let us assume that there exist(n+ 1)-cellsA1

andB1 in Σ∗ and ai in {0, . . . , n − 2} such thatA = A1 ⋆i sn(B1) andB = sn(A1) ⋆i B1 hold: then(tn(A1) ⋆i B1, A1 ⋆i tn(B1)) is a confluence that satisfies the required property.

Now, let us assume thatb is not trivial. Letb1 = (A1, B1) be a minimal branching such thatb1 4 b,with a whiskerC such thatb = C[b1] holds. Since(A,B) is not trivial, thenb1 cannot be trivial, sothat it is critical. Then we consider its fixed confluence(A ′, B ′). Then(C[A ′], C[B ′]) is a confluence for(A,B). Furthermore, one has

A ⋆nC[A′] = C[A1] ⋆nC[A

′] = C[A1 ⋆nA′]

21


and, similarly,B ⋆n C[B′] = C[B1 ⋆n B

′]. SinceC is a whisker and since, by definition ofC, one hasA1 ⋆nA

′ ≈ΓΣ B1 ⋆n B′, one gets that(C[A ′], C[B ′]) satisfies the required property.

4.3.3. Lemma. LetΣ be a convergent(n + 1)-polygraph and let(A,B) be a branching ofΣ such thatbothtn(A) andtn(B) are in normal form. Then one hastn(A) = tn(B) andA ≈ΓΣ B.

Proof. SinceΣ is terminating, we can prove the result by induction on the source of the branching.First, if this sourcef is a normal form, then by definition of normal form, bothA andB must be

identities. Hencetn(A) andtn(B) are equal, and so doA andB. ThusA ≈ΓΣ B holds.Now, we fix an-cell f, which is not a normal form. We assume that the result holds for every

branching(A,B) such that the targets ofA andB are normal forms and such that there exists a nontrivial (n + 1)-cell from f to their source. Let(A,B) be a branching with sourcef and such that thetargets ofA andB are normal forms. Sincef is not a normal form,A andB cannot be identities, henceone can decompose them intoA = A1 ⋆nA2 andB = B1 ⋆n B2 with A1 andB1 being(n + 1)-cells ofsize1.

The pair(A1, B1) is a local branching. Thus, using Lemma 4.3.2, one gets a confluence(A ′1, B

′1) for

(A1, B1) such thatA1 ⋆nA′1 ≈ΓΣ B1 ⋆n B

′1 holds. Let us denote byg the common target ofA ′

1 andB ′1,

by e its normal form and byA3 an-cell fromg to e.Then we consider the branching(A2, A

′1 ⋆n A3), whose source is denoted byh. The targets ofA2

andA ′1 ⋆n A3 are normal forms andA1 is a non trivial(n + 1)-cell from f to h: thus, the induction

hypothesis can be applied to this branching, yielding thatA2 has targete and thatA2 ≈ΓΣ A′1 ⋆n A3

holds.We proceed similarly to prove thatB2 satisfies the same properties, so that one gets thatA andB

have the same target and thatA ≈ΓΣ B holds. The constructions we have done are summarized in thefollowing diagram:

h

A′

1

>>>

��>>>

A2

##

f

A

��

B

EE

A1

88qqqqqqqqqqqqq

B1&&MMMMMMMMMMMMM ≈ΓΣ g A3

// e.

k

B′

1��

??��

B2

;;

=

=

≈ΓΣ

≈ΓΣ

4.3.4. Proposition. LetΣ be a convergent(n + 1)-polygraph. ThenΓΣ is a homotopy basis forΣ⊤.

Proof. Let (A,B) be an(n + 1)-sphere inΣ⊤, with source and target denoted byf andg, respectively.SinceΣ is convergent, thesen-cells share the same normal form, which we denote bye. LetA ′ andB ′

22


be(n+ 1)-cells inΣ∗, from f to e and fromg to e respectively:

f

A

��

B

��

A′

��

e.

gB′

@@

We apply the previous result to the branching(A ⋆n B′, A ′), yielding A ⋆n B

′ ≈ΓΣ A ′ and, hence,A ≈ΓΣ A

′⋆n B

′−1. We proceed similarly with the branching(B ⋆n B′, A ′) to getB ≈ΓΣ A

′⋆n B

′−1

and, thus,A ≈ΓΣ B.

4.3.5. Proposition. A finite convergent polygraph with a finite set of critical branchings has finitederivation type.

Proof. If Σ has a finite set of critical branchings, then the setΓ(Σ) is finite.

4.3.6. Corollary. A terminating polygraph with no critical branching has finite derivation type.

4.3.7. Corollary ([19]). A finite convergent2-polygraph has finite derivation type.

Proof. If Σ is a finite convergent2-polygraph with one0-cell, i.e. a word rewriting system, then its setof critical branchings is finite. Indeed, it is equal to the number of possible overlaps between the wordscorresponding to the sources of2-cells: there are finitely many2-cells and finitely many letters in eachword. If Σ has more than one0-cell, then the number of possible overlaps is bounded by thenumber ofoverlaps inΣ ′, built fromΣ by identification of all its0-cells.

From this result Squier has proved that, if a finitely presented monoid admits a presentation by afinite convergent word rewriting system, then it has finite derivation type, [19]. Now we prove that thisresult is false forn-categories whenn ≥ 2.

4.3.8. Proposition. For every natural numbernwithn ≥ 3, there exists a finite convergentn-polygraphwithout finite derivation type.

Proof. We consider the3-polygraphΣ with one0-cell, one1-cell, three2-cells , , and the fol-lowing four3-cells:

⇛ , ⇛ , ⇛ , ⇛ .

The 3-polygraphΣ is finite and convergent. However, the first and second3-cells create an infinitenumber of critical branchings whose confluence diagrams cannot be presented by a finite homotopybasis. These facts are proved in 5.5.

Then we apply suspension functors onΣ to get ann-polygraph, for somen ≥ 3. It has exactly thesame cells and compositions in dimensionsn − 3, n − 2, n− 1 andn asΣ in dimensions0, 1, 2 and3;on top of that, it has two cells in each dimension up ton− 4 and no other possible compositions, exceptwith degenerate cells. Thus, we conclude that then-polygraph we have built is finite and convergent, yetit still fails to have finite derivation type.

23


4.3.9. Theorem. For every natural numbern with n ≥ 2, there exists an-category which does nothave finite derivation type and admits a presentation by a finite convergent(n + 1)-polygraph.

Proof. Let n ≥ 2, by Proposition 4.3.8, there exists a finite convergent(n + 1)-polygraphΣ withoutfinite derivation type. Let denote byC then-category presented by the polygraphΣ. By proposition3.3.4 any finite(n + 1)-polygraph presentingC does not have finite derivation type. This proves thatC

does not have finite derivation type.

We end this section by an example which illustrates that the property of finite derivation type is notTietze-invariant for infinite polygraphs.

4.3.10. Example. Let C be the2-category presented by the3-polygraphΣ with one0-cell, one1-cell,three2-cells ,��, and the following two3-cells:

��

��α

⇛��

and ��β

⇛��.

The polygraphΣ terminates and does not have critical branching. By Corollary 4.3.6 it follows thatΣhas finite derivation type and, thus, so doesC.

Now let us consider another presentation of the2-categoryC, namely the3-polygraphΞ defined thesame way asΣ except for the3-cells:

��

��

α⇛ ��

��

and ��β

⇛��.

The polygraphΞ still terminates, but it has the following non-confluent critical branching:

��

��

α

E�,EEEEEEEEEEEEEE

EEEEEEEEEEEEEE

EEEEEEEEEEEEEE

��

β

y2Fyyyyyyyyyyyyyy

yyyyyyyyyyyyyy

yyyyyyyyyyyyyy

α

C�+CCCCCCCCCCCCCC

CCCCCCCCCCCCCC

CCCCCCCCCCCCCC��

��

��

��

We define, by induction on the natural numberk ≥ 1, the2-cellk

as follows:

1= and

k+1=

k.

24


Then, we complete the polygraphΞ into an infinite convergent polygraphΞ∞ = Ξ ∪ {βk, k ≥ 1}, whereβk is the3-cell:

��

��

��

��

��

k βk

⇛

��

�� k

We defineβ0 asβ. For any natural numberk, we have the confluence diagrams:

��

��

��

��

��

��

k+1

βk+1

C�+CCCCCCCCCCCCCC

CCCCCCCCCCCCCC

CCCCCCCCCCCCCC

��

��

��

��

��

k

��

��

k

α{3G{{{{{{{{{{{{{{

{{{{{{{{{{{{{{

{{{{{{{{{{{{{{

��βk C�+

CCCCCCCCCCCCCC

CCCCCCCCCCCCCC

CCCCCCCCCCCCCC

��

��k+1

��

��

�� k

��k

α

{3G{{{{{{{{{{{{{{

{{{{{{{{{{{{{{

{{{{{{{{{{{{{{

αβk

��

By Proposition 4.3.4, the setΓ = {αβk | k ∈ N} form a homotopy basis ofΞ⊤∞ .Let us prove that the polygraphΞ∞ does not have finite derivation type. On the contrary, suppose

thatΞ∞ does have finite derivation type. Then, following Proposition 3.2.3, there is a finite subsetΓ0 ofΓ which is a homotopy base ofΞ⊤∞ . SubsetΓ0 beeing finite, there is a natural numberl > max{k | αβk ∈

Γ0} such that the4-cell αβl is not inΓ0. However, sinceΓ0 is a homotopy base we have

s (αβl) ≈Γ0 t (αβl) .

That is there is a4-cellΦ in Ξ⊤∞(Γ0) such thatsΦ = s (αβl) andtΦ = t (αβl) hold. Thus there exist4-cellsΦ1,Φ2 ∈ Ξ

⊤∞(Γ0) and a contextC of Ξ⊤ such that

Φ = Φ1 ⋆3C[Ψε] ⋆3Φ2,

whereΨ ∈ Γ0 andε ∈ {−1, 1}. We have

s2Φ = s2(C[Ψε]) = s2(C)[s2(Ψ)].

SinceΨ ∈ Γ0, there existk < l such thatΨ = αβk. Hences2Φ = s2[C][αβk]. In an other hand, wehaves2Φ = s2(αβl). This means that there are2-cellsf1, f2, g1, g2 such that

f2 ⋆1 (f1 ⋆0��

��

��

��

k⋆0 g1) ⋆1 g2 =

��

��

��

��

l,

holds inΞ∗∞ , which is impossible withk < l. This proves thatΞ∞ does not have finite derivation type.

25

5. The case of3-polygraphs

5. THE CASE OF3-POLYGRAPHS

5.1. Classification of critical branchings

5.1.1. Definitions. Let Σ be a3-polygraph and let(A,B) be a critical branching ofΣ. Let us denoteby α andβ the3-cells ofΣ that generateA andB. Then(A,B) falls in one of three cases.

The first possibility is that there exists a contextC of Σ∗2 such thatsα = C[sβ] holds. Then, the

source of the branching(A,B) is:

sα = sβ

C

.

In that case,(A,B) is aninclusioncritical branching.If the branching(A,B) is not an inclusion one, the second possibility is that thereexist1-cellsu, v

and2-cellsf, g, h such thatsα andsβ decompose in one of the following ways.

• One hassα = f ⋆1 (u ⋆0 h) andsβ = (h ⋆0 v) ⋆1 g, so that the source of(A,B) is:

g

vv

u

sα=

g

v

u

f

h =sβ

u

vf

.

• One hassα = f ⋆1 (h ⋆0 u) andsβ = (v ⋆0 h) ⋆1 g:

g u

vvsα

= h

gu

v f

=

fv

sβu .

• One hassα = f ⋆1 (u ⋆0 h ⋆0 v) andsβ = h ⋆1 g:

sα

gu v=

f

u h

gvv =

sβu vv

f

.

• One hassα = f ⋆1 h andsβ = (u ⋆0 h ⋆0 v) ⋆1 g:

g

u vsα=

u vf

g

h =

v

sβ

fu

.

If (A,B) matches one of these cases, then it is called aregular critical branching.Finally, when the branching(A,B) is not an inclusion or regular one, there exist1-cellsu, v and

2-cellsf, g, h suchsα andsβ decompose in one of the following ways.

26

5.1. Classification of critical branchings

• One hassα = f ⋆1 (h ⋆0 u) andsβ = (h ⋆0 v) ⋆1 g, so that there exists a2-cell k such that thesource of(A,B) is:

k

g

sα

= k

f

h

g= k

sβ

f

.

In that case, one can write(A,B) = (C[k],D[k]) for appropriate contextsC andD of Σ∗. Thefamily (C[k],D[k])k, wherek ranges over the2-cells with appropriate boundary and such that(C[k],D[k]) is a minimal branching, is called aright-indexedcritical branching.

• One hassα = f ⋆1 (u ⋆0 h) andsβ = (v ⋆0 h) ⋆1 g, so that there exists a2-cell k such that thesource of(A,B) is:

k

g

sα

= k

g

f

h = ksβ

f

.

In that case, one can write(A,B) = (C[k],D[k]) for appropriate contextsC andD of Σ∗. Thefamily (C[k],D[k])k, wherek ranges over the2-cells with appropriate boundary and such that(C[k],D[k]) is a minimal branching, is called aleft-indexedcritical branching.

• One is not in the right-indexed or left-indexed cases and onehas

sα = f ⋆1 (u0 ⋆0 h1 ⋆0 u1 ⋆0 h2 ⋆0 · · · ⋆0 un−1 ⋆0 hn ⋆0 un)

andsβ = (v0 ⋆0 h1 ⋆0 v1 ⋆0 h2 ⋆0 · · · ⋆0 vn−1 ⋆0 hn ⋆0 vn) ⋆1 g ,

so that there exist2-cellsk0, . . . , kn such that the source of(A,B) is as follows, where we writepinstead ofn − 1 for size reasons:

k0 kn

sα

g

kpk1

= knk0 h1 k1 h2

g

f

hnhp kp

=

f

knk0

sβ

kpk1 .

27


In that case, one can write(A,B) = (C[k0, . . . , kn],D[k0, . . . , kn]) for appropriate3-cells CandD in someΣ∗[x0, . . . , xn]. The family(C[k0, . . . , kn],D[k0, . . . , kn])k0,...,kn , where theki’srange over the2-cells with appropriate boundary and such that(C[k0, . . . , kn],D[k0, . . . , kn]) isa minimal branching, is called amulti-indexedcritical branching.

In all those cases, the branching(A,B) is said to be aninstanceof the corresponding right-indexed orleft-indexed or multi-indexed one. It is anormal instance when the indexing2-cell k (resp.2-cellsk0,. . . , kn) is a normal form (resp. are normal forms).

A 3-polygraph isnon-indexedwhen each of its critical branchings is an inclusion one or a regularone. It isright-indexed(resp.left-indexed) when each of its critical branchings is either an inclusionone,a regular one or an instance of a right-indexed (resp. left-indexed) one.

5.1.2. Proposition. A3-polygraph with a finite set of3-cells has a finite number of inclusion and regularcritical branchings.

Proof. Let Σ be a polygraph withΣ3 = {α1, . . . , αp} finite. Giveni ∈ {1, . . . , p}, Σ3 being finite, thesetJ of j such that there is a morphisms(αj) → s(αi) in WΣ is finite. Moreover, for aj ∈ J, the set ofmorphisms ofWΣ from s(αj) to s(αi) is finite. This proves that there is a finite set of inclusion criticalbranchings.

Now, let us prove thatΣ has a finite number of regular critical branchings. Let us fix somei andjin {1, . . . , p} and let us assume that there exist two whiskersC andD of Σ∗ such that(C[αi],D[αj])

is a regular branching, with sourcef. Then there exist a2-cell h and whiskersC ′ andD ′ of Σ∗

such thatC[s(α)i] = C ′[h] = D ′[h] = D[s(αj)] holds. But the setsWΣ(s(αi), f), WΣ(s(αj), f),WΣ(h,C[s(αi)]) andWΣ(h,C[s(αj)]) are finite. Hence there exist finitely regular branchings of thisform, with i andj fixed. SinceΣ3 is finite, the3-polygraphΣ has finitely many regular branchings.

5.1.3. Theorem. A finite convergent non-indexed3-polygraph has finite derivation type.

Proof. We use the previous result and, then, we apply Proposition 4.3.5.

5.2. Mac Lane’s coherence theorem revisited

5.2.1. Monoidal categories. A monoidal categoryis a data(C,⊗, e, a, l, r) made of a categoryC, abifunctor⊗ : C× C → C, an objecte of C and three natural isomorphisms

ax,y,z : (x⊗ y)⊗ z → x⊗ (y⊗ z) , lx : e⊗ x → x, rx : x⊗ e → x,

such thatle = re holds and such that the following two diagrams commute inC:

(x⊗(y⊗z))⊗ta

// x⊗((y⊗z)⊗t)

a

''PPPPPPPPPPPP

((x⊗y)⊗z)⊗t

a77nnnnnnnnnnnn

a++WWWWWWWWWWWWWWWWWWWWWW

c© x⊗(y⊗(z⊗t))

(x⊗y)⊗(z⊗t)

a

33gggggggggggggggggggggg

28


x⊗(e⊗y)

l

$$JJJJJJJJJJ

(x⊗e)⊗y

a99rrrrrrrrrr

r

55x⊗y .

c©

Mac Lane’s coherence theorem [15] states that, in a such monoidal category, all the diagrams whosearrows are built from⊗, e, l and r commute. Thereafter, we give a proof of this fact by buildingahomotopy basis of a3-polygraph.

5.2.2. The3-polygraph of monoids. We denote byΣMon the3-polygraph with one0-cell, one1-cell,two 2-cells and and the following three3-cells:

⇛α , ⇛λ , ⇛ρ .

We denote byΓ the set made of the following4-cellsαα, αρ andλρ, where we commit the abuse ofdenoting a3-cell ofΣ∗ with size1 like its generating3-cell:

α _ %9

α

B�*BBBBBBB

BBBBBBB

BBBBBBBα

|4H|||||||

|||||||

|||||||

α

Q�2QQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQ

α

m,@mmmmmmmmmmmmmmmmmmmm

mmmmmmmmmmmmmmmmmmmm


αα

��

λ

9�&99999999

99999999

99999999α

�6J��

��

��

ρ

n,@

αρ

��

��

��

��

��

��

��

��

λ

7�%

ρ

�9Mλρ

��

5.2.3. Theorem. The setΓ of 4-cells form a homotopy basis of the track3-categoryΣ⊤Mon.

Proof. Let us prove thatΣMon terminates by using a derivation. We consider theΣ∗2-moduleM(X, ∗,Z)

generated by the following values:

X( ) = N \ {0} , X( )(i, j) = i+ j , X( ) = 1 .

29


The functorX satisfies the following equalities:

X

( )(i, j, k) = i+ j + k = X

( ),

X

( )(i) = i = X

( )(i) and X

( )(i) = i = X

( )(i) .

Now, letd be the derivation ofΣ∗2 intoM(X, ∗,Z) generated by the following values:

d (i, j) = i and d = 0 .

The derivationd satisfies the following strict inequalities:

d

( )(i, j, k) = 2i + j > i+ j = d

( )(i, j, k) ,

d

( )(i) = 1 > 0 = d

( )(i) and d

( )(i) = i > 0 = d

( )(i) .

We apply Theorem 4.2.1 to conclude.

The3-polygraph has five critical branchings, all of the regular type, and all of them are confluent.Their confluence diagrams are given by the boundaries of the4-cells ofΓ , plus the following two ones:

λ

<�'<<<<<<<<<

<<<<<<<<<

<<<<<<<<<α

~5I~~~~~~~

~~~~~~~

~~~~~~~

λ

n,@

λα

��

��

��

� ��

��

��

� ��

��

ρ

<�'<<<<<<<<

<<<<<<<<

<<<<<<<<α

~5I~~~~~~~

~~~~~~~

~~~~~~~

ρ

n,@

ρα

��

� ��

��

��

��

��

��

��

SinceΣ terminates and has all its critical branchings confluent, itis convergent as a consequence ofNewman’s lemma. Thus we know that the set{αα, λρ, λα, αρ, ρα} of 4-cells is a (finite) homotopybasis ofΣ⊤.

To get the result, we check thatλα andρα are superfluous in this homotopy basis, since their bound-aries are also the boundaries of4-cells ofΣ⊤(Γ). Let us detail the proof forλα.

We consider the4-cell( )

⋆1αα of Σ⊤(Γ). We partially fill its boundary with other4-cells of

30


Σ⊤(Γ) and equalities, yielding a3-sphere ofΣ⊤ denoted byγ:

α _%9

λ��

=α

I�.IIIIIIIIIIIII

IIIIIIIIIIIII

IIIIIIIIIIIII

λ��

α

u0Duuuuuuuuuuuuu

uuuuuuuuuuuuu

uuuuuuuuuuuuu

ρ _%9

α[#7

α _%9

=

λ_ey

ρ}}}}}}}}

}}}}

}4H}}}} }}}}}}}}

α

t0D

αρ γ

αρ

As a consequence of this construction, we havesγ ≈Γ tγ. Then we build the following diagram, provingthats(λα) ≈Γ t(λα) also holds:

λ

�~�

λ��

α

1�"

λ��

��

�u ��

α

CCCCCCCC

CCCC

C�+CCC

CCC CCC

λ_ey

λuuuuuuuuuuuu

uuuuuu

u0Duuuuuuuuuuuu

uuuuuuλLLLLLL

LLLLLLLLLLLL

L[oLLLLL

LLLLLLLLLL

λ_ey

= =

=

For the4-cell ρα, one proceeds in a similar way, starting with the4-cell( )

⋆1 αα.

5.2.4. Corollary (Mac Lane’s coherence theorem [15]).In a monoidal category(C,⊗, e, a, l, r), allthe diagrams whose arrows are built from⊗, e, a, l andr are commutative.

Proof. We seeCat1 as a (large)3-category with one0-cell, categories as1-cells, functors as2-cellsand natural transformations as3-cells. The0-composition is the cartesian product of categories, the1-composition is the composition of functors and the2-composition is the "vertical" composition of naturaltransformations.

31


Then monoidal categories are exactly the3-functors fromΣ⊤ to Cat1 such that the source and targetof every4-cell of Γ are identified. The correspondence between a monoidal category (C,⊗, e, a, l, r) andsuch a3-functorM is given by:

M( ) = C, M( ) = ⊗, M( ) = e, M(α) = a, M(λ) = l, M(ρ) = r.

As a consequence, a diagramD in C whose arrows are built from⊗, e, a, l andr is the image byM ofa3-sphereγ of Σ⊤. SinceΓ is a homotopy basis ofΣ⊤, we havesγ ≈Γ tγ. SinceM is a3-functor andsince it identifies the source and the target of every3-sphere ofΓ , it does the same with the ones ofγ,i.e. we haveM(sγ) =M(tγ), which means that the diagramD commutes.

5.3. Right-indexed and left-indexed polygraphs

5.3.1. Proposition. Let Σ be a terminating right-indexed (resp. left-indexed)3-polygraph. ThenΣ isconfluent if and only if the following two conditions are satisfied:

1. Each of its inclusion and regular critical branchings is confluent.

2. Each normal instance of each of its right-indexed (resp. left-indexed) critical branchings is con-fluent.

Proof. If Σ is confluent then, by definition, all of its branchings are confluent: in particular, its inclusionand regular critical branchings and the normal instances ofits right-indexed or left-indexed ones.

Conversely, let us assume thatΣ has all of its inclusion and regular critical branchings andall of thenormal instances of its right-indexed (resp. left-indexed) critical branchings that are confluent. SinceΣ isterminating, we know that it is sufficient to prove that it is locally confluent. Moreover, sinceΣ is right-indexed (resp. left-indexed), there remain to prove that every non normal instance of the right-indexed(resp. left-indexed) critical branching is confluent.

From now on, we assume thatΣ is right-indexed, the proof in the left-indexed case being similar. Letus consider a right-indexed critical branching(A[k], B[k])k, which has the following shape by definition:

D k

C k

A[k] ~4H~~~~~~~

~~~~~~~

~~~~~~~

B[k] @�*@@@@@@@

@@@@@@@

@@@@@@@

E k

Now, let f be a2-cell such that(A[f], B[f]) is a non normal instance of(A[k], B[k])k. SinceΣ is termi-nating,f admits a normal form, sayg. We denote byF a 3-cell from f to g. Sinceg is in normal form,

32

5.3. Right-indexed and left-indexed polygraphs

the branching(A[g], B[g]) is a normal instance of(A[k], B[k])k so that, by hypothesis, it is confluent: letus denote by(G,H) a confluence for this branching, with targeth. With all those ingredients, one buildsthe following confluence diagram for the critical branching(A[f], B[f]), thus concluding the proof:

fDD[F] _ %9 gD

G

<�'<<<<<<<<<

<<<<<<<<<

<<<<<<<<<

fC

A[f] ~4H~~~~~~~

~~~~~~~

~~~~~~~

B[f] @�*@@@@@@@

@@@@@@@

@@@@@@@

C[F] _%9 gC

A[g] ~4H~~~~~~~

~~~~~~~

~~~~~~~

B[g] @�*@@@@@@@

@@@@@@@

@@@@@@@

h .

E fE[F]

_%9 gE

H

�7K��

��

��

5.3.2. Homotopy bases of indexed3-polygraphs. LetΣ be a locally confluent and right-indexed (resp.left-indexed)3-polygraph. We assume that a confluence has been chosen for:

• Each inclusion and regular critical branching.

• Each normal instance of each right-indexed (resp. left-indexed) critical branching.

We denote byΓΣ the collection of the2-spheres ofΣ∗ corresponding to these confluence diagrams.

5.3.3. Proposition. LetΣ be a convergent right-indexed (resp. left-indexed)3-polygraph. ThenΓΣ is ahomotopy basis ofΣ⊤.

Proof. The proof follows the same scheme than the results of 4.3, where it was proved that the family of3-spheres associated to the confluence diagrams of all the critical branchings was a homotopy basis.

First, we prove that every local branching of(A,B) of Σ admits a confluence(A ′, B ′) such thatA ⋆2 A

′ ≈ΓΣ B ⋆2 B′ holds. The proof is the same as in 4.3 when(A,B) is a trivial or when it is

generated by an inclusion or a regular critical branching.There remains to check the cases of local branchings of the shapeC(A[f], B[f]), where(A[k], B[k])k)

is a right-indexed (resp. left-indexed) critical branching and whereC is a context. For that, we proceedby Noetherian induction on the indexing2-cell f, thanks to the termination ofΣ.

Whenf is a normal form, then(A[f], B[f]) is a normal instance of the branching(A[k], B[k])k. Tobuild ΓΣ we have fixed a confluence for this branching, say(A ′, B ′). Then we have:

C[A[f]] ⋆2A′ ≈ΓΣ C[B[f]] ⋆2 B

′.

33


Let us assume thatf is a non normal2-cell such that(A[f], B[f]) is an instance of the branching(A[k], B[k])k. Moreover, we assume that, for every2-cell g such thatf reduces intog and(A[g], B[g]) isan instance of(A[k], B[k])k, there exists a confluence(A ′, B ′) for (A[g], B[g]) such thatA[g]⋆2A ′ ≈ΓΣ

B[g] ⋆2 B′ holds.

Sincef in not in normal form, we can choose a2-cell g such thatf reduces intog, through a3-cell F.Sincef andg have the same boundary, we have an instance(A[g], B[g]) of the branching(A[k], B[k])k.We apply the induction hypothesis tog to get a confluence(A ′, B ′), with target denoted byh, such thatA[g] ⋆2A

′ ≈ΓΣ B[g] ⋆2 B′ holds. Moreover, the branchings(C[A[f]], C[sA[F]]) and(C[B[f]], C[sB[F]])

are trivial branchings, yielding:

C[A[f]] ⋆2C[tA[F]] ≈ΓΣ C[sA[F]] ⋆2C[A[g]]

andC[B[f]] ⋆2C[tB[F]] ≈ΓΣ C[sB[F]] ⋆2C[B[g]].

With these constructions, we build the following diagram, where we have assumed that the consideredbranching was right-indexed – the case of a left-indexed critical branching is similar:

C

tA fC[tA[F]] _%9

≈ΓΣ

C

tA g

C[A′]

=�(=========

=========

=========

C

sA

sBf=

C[A[f]] ~4H~~~~~~~

~~~~~~~

~~~~~~~

C[B[f]] @�*@@@@@@@

@@@@@@@

@@@@@@@

C[sA[F]]

C[sB[F]]_ %9

C

sBg

sA=

C[A[g]] ~4H~~~~~~~

~~~~~~~

~~~~~~~

C[B[g]] @�*@@@@@@@

@@@@@@@

@@@@@@@

≈ΓΣ

C

h

.

C

tB fC[tB[F]]

_%9

≈ΓΣ

C

tB g

C[B′]

�6J��

��

��

Hence, composing the three4-cells ofΣ⊤(ΓΣ) of that diagram, one proves that the confluence(C[tA[F]]⋆2C[A ′], C[tB[F]] ⋆2C[B

′]) satisfies the required

C[A[f]] ⋆2C[tA[F]] ⋆2C[A′] ≈ΓΣ C[B[f]] ⋆2C[tB[F]] ⋆2C[B

′],

thus concluding the first part of the proof.The remainder of the proof is exactly the same as in 4.3, with the following two steps:

• For every branching(A,B) such thattA andtB are in normal form, one hastA = tB andA ≈ΓΣ

B. For that, we use the first part of the proof and we proceed by noetherian induction on the sourcefo the branching(A,B).

34

5.4. The3-polygraph of permutations

• We consider two parallel3-cellsA andB in Σ⊤ and prove thatA ≈ΓΣ B holds.

5.3.4. Theorem. A finite convergent3-polygraph with finitely many normal instances of indexed criticalbranchings has finite derivation type.


Here we see an example of a3-polygraph that is finite, convergent, right-indexed and, thus, with aninfinite number of critical branchings, yet with finite derivation type. Another proof for termination andthe ideas for proving confluence we use here can be found in [13].

5.4.1. Definition. The3-polygraphΣPermhas one0-cell ∗, one1-cell , one2-cell , and the followingtwo 3-cells:

α⇛ and

β

⇛ .

5.4.2. Termination. We consider the(ΣPerm)∗2-moduleM(X, ∗,Z) with:

X( )

= N and X( )

(i, j) = (j + 1, i).

Then we consider the derivationd of (ΣPerm)∗2 intoM(X, ∗,Z) given by:

d( )

(i, j) = i.

The2-functorsX, Y and the derivationd satisfy the conditions of Theorem 4.2.1. Indeed, the derivationd sends every2-cell ofΣ∗

Perm to a map with values inN and the following inequalities hold:

X

( )(i, j) = (i+ 1, j + 1) ≥ (i, j) = X

( )(i, j),

X

( )(i, j, k) = (k+ 2, j + 1, i) = X

( )(i, j, k),

d

( )(i, j) = i+ j+ 1 > 0 = d

( )(i, j),

d

( )(i, j, k) = 2i+ j+ 1 = 2i + j = d

( )(i, j, k).

35


5.4.3. Normal forms. Let f be a2-cell of Σ∗Permsatisfying:

d(f)(0, . . . , 0) = 0.

Thenf is in normal form. Otherwise, there exists a contextC and a2-cell g such thatf = C[g] holdsandg is the source of one of the two3-cells ofΣPerm. As a consequence, there exists a family(i1, . . . , in)

of natural numbers, withn = 2 or n = 3, such that the following inequalities hold:

d(f)(0, . . . , 0) ≥ d(g)(i1, . . . , in) ≥ 1.

Let us defineN0 as the set of2-cells given by the following inductive construction:

= or .

First, we check that the following holds:

X( )

(i1, . . . , in, j) = (j + n, i1, . . . , in).

We proceed by structural induction, using the definition andthe functoriality ofX:

X( )

(i, j) = (j+ 1, i)

and

X

( )(i1, . . . , in, in+1, j) =

(X( )

× IdN

)(i1, . . . , in, j + 1, in+1)

= (j + n+ 1, i1, . . . , in+1).

Now, let us prove that the2-cells ofN0 are in normal form, still by structural induction. For the basecase, we have, by definition ofd:

d( )

(0, 0) = 0.

Then, for the inductive case, we have, using the fact thatd is a derivation:

d

( )(0, . . . , 0) = d

( )(0, . . . , 0) + d

( )(0, 0) = 0.

Let us denote byN the set of2-cells ofΣ∗Permgiven by the following inductive graphical scheme:

= ∗ or or .

Let us prove that the2-cells ofN are in normal form, by structural induction. First, one hasd(∗) = 0

sinced is a derivation. Then, one gets:

d( )

(i1, . . . , in+ 1) = d( )

(i1) + d( )

(i2, . . . , in) = 0.

36


And, finally, using the values ofX onN0:

d

( )(i1, . . . , im, j, k1, . . . , kn)

= d( )

(i1, . . . , im, j) + d( )

(i1, . . . , im, k1, . . . , kn) = 0.

Finally, we prove that every2-cell of Σ∗Perm in normal form is inN. We proceed by induction on the pair

(m,n) of natural numbers, wherem is the size of the2-cells andn is the size of their source.The only2-cells ofΣ∗

Permwith size0 are the1n, wheren denotes the1-cell with sizen. All of themare in normal form. Moreover, they belong toN: 10 is∗ and, for every natural numbern, 1n+1 = 11⋆01n.

The only2-cell of Σ∗Perm whose source has size0 is 10 = ∗, which is in normal form and belongs

toN.Then, let us fix two non-zero natural numbersm andn. We assume that, every2-cell g of Σ∗

Permwhich is in normal form and such that(||g|| , |sg|) < (m,n) holds is inN, where compare pairs ofnatural numbers with the product order.

Let us consider a2-cell f of Σ∗Perm in normal form, with sizem and whose source has sizen. Since

||f|| = m ≥ 1 and since is the only2-cell ofΣPerm, there exists a2-cell g such thatf decomposes into:

f =g

.

Sincef is in normal form, then so doesg. Moreover,g has sizem − 1 and its source has sizen. Weapply the induction hypothesis tog: this2-cell is inN. Its source isn ≥ 1, so thatg 6= ∗; there remainstwo possibilities, by definition ofN:

g = h or g =h

.

In the first case, the2-cell h is in normal form, has sizem−1 and its source has sizen−1. By inductionhypothesis, we know thath is inN. There are two subcases for the decomposition off:

f =h

or f =h

.

The first decomposition is a proof thatf is inN, sinceh is inN and is inN0. The second decom-position tells us thatf = ⋆0 f

′, wheref ′ is in normal form (otherwisef would not), has sizem and itssource has sizen− 1; we apply the induction hypothesis to get thatf ′ is inN; then we get thatf is inN.

Now, let us examine the second case: the2-cell h is in normal form, has size at mostm − 2 and itssource has sizen − 1; hence, by induction hypothesis,h is inN. There are three subpossibilities:

f =

h

or f =h

or f =h

.

The first subcase is, in fact, impossible sincef would contain the source of a3-cell, which contradictsthe assumption thatf is in normal form. The second case gives thatf is inN. In the third case, we have

37


a decomposition off into (f ′ ⋆0 1p) ⋆1 (11 ⋆0 f′′) wheref ′ is inN0 andf ′′ is a normal form (otherwisef

would not), has size at mostm− 1 and has sourcen− 1: thus, we apply the induction hypothesis to getthatf ′′ and, hence,f are inN.

5.4.4. Confluence. The 3-polygraphΣPerm has three regular critical branchings, with the followingsources:

, , ,

plus a right-indexed critical branching, with source:

k .

From Theorem 5.3.1, we know that, to get confluence ofΣPerm, it is sufficient to prove that the threeregular critical branchings are confluent and that each normal instance of the right-indexed one is.

In the following confluence diagrams, we commit the abuse of naming3-cells with their generatoronly. First, we check that the three regular critical branchings are confluent:

α

7�%

α

�9Mαα

��

α _ %9

β 9�&99999999

99999999

99999999

β_%9

αβ

��

α

�8L��

��

��

β _%9

α9�&999999999

999999999

999999999β

�8L��

��

��

α_ %9

βα

��

From the inductive characterization of the setN of normal forms we have given, we deduce that thereare two normal instances of the right-indexed critical branching: fork = andk = . We check that

38


both are confluent:

α _%9

α

5�$555555555555

555555555555

555555555555

β

�:N��

��

��

β6�$66666666

66666666

66666666

α_%9

α

:N

ββ( )

��

and:

β _ %9 β _ %9

β

9�&99999999

99999999

99999999β

�8L��

��

��

β 9�&99999999

99999999

99999999

β_ %9

β_ %9

β

�8L��

��

��

ββ( )

��

5.4.5. Theorem. The3-polygraphΣPerm has finite derivation type.

Proof. Indeed, the3-polygraphΣPerm is finite, convergent, right-indexed and has finitely many normalinstances of its right-indexed critical branchings. Thus Theorem 5.3.3 tells us thatΣPermhas finite deriva-tion type. More precisely, the five4-cellsαα, αβ, βα, ββ

( )andββ

( )form a homotopy basis of

the track3-categoryΣ⊤Perm.

39


5.5. Main counterexample

Let us consider the3-polygraphΣ with one0-cell, one1-cell, three2-cells , and and the follow-ing four3-cells:

α⇛ ,

β

⇛ ,γ

⇛ ,δ⇛ .

We define by induction on the natural numberk the2-cellk

as follows:

0= and

k+1=

k⋆1 .

5.5.1. Termination. To prove that the3-polygraphΣ terminates, we use two subsequent derivationsof Σ∗

2, as in Theorem 4.2.1.We consider the derivation||·|| , into the trivial moduleM(∗, ∗,Z). One checks that the following

holds:

||α|| = 0, ||β|| = 0, ||γ|| = 1, ||δ|| = 1,

Let us note that one would have the same results with the derivation ||·|| . As a consequence, one getsthat the3-polygraphΣ terminates if and only if the3-polygraph{α,β} terminates. To prove the latter,we consider the derivationd into theΣ∗

2-moduleM(X, Y,Z), with:

X( )

= N, X( )

= (0, 0), X( )

(i) = i+ 1.

Y( )

= N, Y( )

= (0, 0), Y( )

(i) = i+ 1.

d( )

(i, j) = i, d( )

(i, j) = i, d( )

(i, j) = 0.

Sinced is a derivation, one gets:

dα = d( )

− d( )

= d( )

⋆1

( )+ ⋆1

(d( )

⋆0

)− d

( )⋆1

( )− ⋆1

(⋆0 d

( )).

Thus, for every natural numbersi andj, one gets:

dα(i, j) = d( )

(i + 1, j) + d( )

(0, i) − d( )

(i, j + 1) − d( )

(0, j)

= (i + 1) + 0 − i − 0

= 1.

Similarly, one getsdβ(i, j) = 1 for every natural numbersi andj, yielding the termination of{α,β} and,thus, ofΣ.

40


5.5.2. Normal forms. Let f be a2-cell of Σ∗, that cannot be reduced by the3-cellsγ andδ and whichsatisfies:

d(f)(0, . . . , 0) = 0.

Thenf is in normal form. Indeed, otherwise there exists a contextC such thatf = C[g], with eitherg = sα or g = sβ. As a consequence, there exist two natural numbersi andj such that the followinginequalities hold:

df(0, . . . , 0) ≥ dg(i, j) ≥ 1.

Now, we defineN as the set of2-cells given by the following inductive construction scheme:

= (a) ∗ or (b)k

or (c) k

or (d)k

or (e)k

.

We use the special graphical representations, and for 2-cells ofN which have, respectively,degenerate source and target, degenerate source, degenerate target.

We start by checking that the2-cells ofN are in normal form. For that, one proceeds by structuralinduction, using the construction scheme, in order to provetwo properties.

The first one is that each2-cell ofN is irreducible by the3-cellsγ andδ: this is an observation that

the given construction scheme do not allow any2-cell ofN to contain either or .The second one is that, for a2-cell f of N, one hasdf(0, . . . , 0) = 0. For the base case (a), one has

d(∗) = 0 sinced is a derivation. Then, for the induction, there are four cases:

(b) d

(k

)(0, . . . , 0)

= d( )

(0, k) + d( )

(0, k) + k · d( )

(0, 0) + d( )

+ d( )

(0, . . . , 0)

= 0.

(c) d(

k)(0, . . . , 0)

= d( )

(0, k) + k · d( )

(0, 0) + d( )

(0, . . . , 0) + d( )

(0, . . . , 0)

= 0.

(d) d

(k

)(0, . . . , 0)

= d( )

(0, k) + k · d( )

(0, 0) + d( )

(0, . . . , 0) + d( )

(0, . . . , 0)

= 0.

(e) d(

k)(0, . . . , 0)

= k · d( )

(0, 0) + d( )

(0, . . . , 0)

= 0.

41


Now, let us prove that every2-cell of Σ∗ that is in normal form is contained in the setN. We proceedby induction on the triple(m,n, p) of natural numbers, wherem is the size of the2-cells,n the size oftheir source,p the size of their target.

The only2-cells ofΣ∗ with size0 are the1n, wheren denotes the1-cell with sizen. All of them arein normal form and belong toN. Indeed, each1n can be formed, from∗, byn subsequent applicationsof the construction rule (e) withk = 0.

The2-cells ofΣ∗ with size1 are the1p ⋆0ϕ ⋆0 1q, whereϕ is one of , and . Such a2-cell isalways in normal form and belongs toN. Indeed, we have seen that1q is inN. Thenϕ ⋆0 1q is inN by

using the following case, with = 1q, depending onϕ:

• If ϕ = : case (c) with = ∗ andk = 0.

• if ϕ = : case (d) with = ∗ andk = 0.

• If ϕ = : case (e) withk = 1.

Finally, 1p ⋆0ϕ ⋆0 1q is inN, thanks to case (e), appliedp times in sequence, each time withk = 0.Now, let us fix a non-zero natural numberm and two natural numbersn andp. We assume that we

have proved the result for eachg in normal form of size at mostm − 1 or with sizem and such that theinequality(|sg| , |tg|) < (n, p) holds.

Let us consider a2-cell f, in normal form and such that||f|| = m, |sf| = n and|tf| = p hold. Sincefhas size at least1, there exists a2-cell g such thatf decomposes in one of the three following ways:

f = g or g or g

One denotes byϕ the corresponding generating2-cell. Sincef is in normal form, so doesg andg hassizem− 1: we apply the induction hypothesis to it, so that we know thatg is inN. Thus,g decomposesinto one of the five following ways:

g = (a) ∗ or (b)k

h or (c) k h

or (d)k

h or (e)k

h .

We study all the possible decompositions off, depending on the one ofg and onϕ. In case (a),i.e. wheng = ∗, we haveϕ = , since this is the only possibility to havetϕ degenerate. We have already seenthat is inN. In case (b), one has the following possibilities, depending onϕ:

f =k

h ork

h ork

h

The following2-cells must be in normal form, sincef is, and they have size at mostm− 2:

h , h , h .

We apply the induction hypothesis to each one: they belong toN. Thusf is in N by case (b). The

proof for case (c) is obtained from case (b) by replacing the2-cellk

by k . In case (d), the

reasoning depends onϕ:

42


• Whenϕ = , one has the following possibilities, depending whereϕ connects tog:

f =k

h ork

h ork

h

ork

h ork

h .

The first and third case cannot occur. Indeed, one proves, by structural induction that a normalform with source of size at least1 and with degenerate target has the following shape:

k· · ·

k k.

As a consequence, such a decomposition off would contain either or , preventing itfrom being in normal form.

For the second case, one applies the induction hypothesis tothe 2-cell : indeed, it is a2-cell with size at mostm− 1 that must be in normal form, otherwisef would not. Thus,f is builtfrom 2-cells ofN following case (d) and, as such, is inN.

The fourth decomposition contains either or , respectively whenk ≥ 1 andk = 0. Thusit is not possible thatf decomposes this way, since it is a normal form.

For the fifth decomposition, one applies the induction hypothesis to h , which is a2-cell

that must be in normal form, with size at mostm− 1.

• Whenϕ = , one has the following possible decompositions off:

f =k

h ork

h ork

h .

The first case shows thatf is in N: it is built with case (d), applied with =k

h ,

which isg that we already know to belong toN, with = ∗ andk = 0.

In the second case, we apply the induction hypothesis to : it is a normal form of size atmostm− 1. Thusf is built from case (d).

In the third case, one applies the induction hypothesis toh : it is a normal form of size at

mostm− 1. Thusf is built from case (d).

• Whenϕ = , the2-cell can decompose as follows:

f =k

h ork

h

ork+1

h ork

h .

43


The first case cannot occur: otherwise,f would contain and, thus, it would not be in normalform.

In the second case, we apply the induction hypothesis to : this is a normal form with size atmostm− 1. This proves thatf is inN, built following case (d).

In the third case,f is inN, built following case (d).

In the fourth case, we apply the induction hypothesis toh : this is a normal form with size at

mostm− 1. Thusf is inN, built from case (d).

Case (e) depends on the values ofϕ:

• Whenϕ = , we have the following possible decompositions off:

f =k

h ork

h .

In the first case, one must havek = 0: otherwise,f would contain which is not a normal form.Thus the2-cell h is a normal form of sizem − 1: we apply the induction hypothesis to get thath

is inN. Then, by structural induction onh, one shows that it decomposes in one of the followingtwo ways:

h =k

ork

.

The first decomposition is impossible since, otherwise,f would contain and, thus, it wouldnot be a normal form. The second decomposition gives thatf is inN, built from case (c).


f =k

h ork

h .

In the first case,f is inN, built fromh in two subsequent steps: with case (e), then with case (d).

In the second case, one applies the induction hypothesis toh which is a2-cell in normal

form, with either size at mostm − 1, whenk > 0, or with sizem and source of sizen − 1. Thusthis2-cell is inN, and so doesf, which is built following case (e).


f =k+1

h ork

h .

In the first case,f is built fromh by application of case (e) and, as such, is inN.

In the second case, one applies the induction hypothesis toh , which is a2-cell in normal

form, with either size at mostm − 1, whenk > 0, or with sizem and source of sizen − 1. As aconsequence, this2-cell is inN, proving thatf is built following case (e) and, thus, it is inN.

44


We have proved that the2-cell of Σ∗ that are in normal form are exactly the2-cells ofN. In particular,we denote byN0 the set of normal forms with degenerate source and target, which are defined by thefollowing inductive scheme:

= ∗ ork

.

5.5.3. Confluence.Let us examine the critical branchings ofΣ. The3-polygraphΣ has four regularcritical branchings whose sources are:

, , , ,

plus one right-indexed critical branching, generated byα andβ, with source:

k .

ThusΣ is a terminating and right-indexed3-polygraph. By application of Theorem 5.3.1, we get conflu-ence ofΣ by proving that its four regular critical branchings and allnormal instances of its right-indexedcritical branchings are confluent.

For the regular ones, we have the following confluence diagrams:

γ

;�'

δ

�7Kγδ

��

δ

;�'

γ

�7Kδγ

��

β _ %9

γ

@�)@@@@@@@@

@@@@@@@@

@@@@@@@@α

y2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

γ

j*>

αγ

�

α _%9

δ

@�)@@@@@@@@

@@@@@@@@

@@@@@@@@β

y2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

δ

j*>

βδ

�

From the characterization of normal forms ofΣ, the normal instances of the right-indexed critical branch-ing αβ are the instances corresponding to the following2-cells:

= , = , = ,

plus:=

n,

for every inN0 andn in N.

45


Now we check that, for each one of these2-cells, the corresponding critical branchingαβ isconfluent. Let us note that, for the first three, there are several possible confluence diagrams, because

they also contain regular critical branchings ofΣ. For = , we choose the following one:

δ _ %9 β _%9

γ

6�$66666666666

66666666666

66666666666

α�7K��

��

��

β :�':::::::

:::::::

:::::::

γ_ %9

α_%9

δ

�:N��

��

��

αβ

„ «

��

For = :

δ

M�0MMMMMMMMMMMMMMMM

MMMMMMMMMMMMMMMM

MMMMMMMMMMMMMMMM

α

m,@mmmmmmmmmmmmmmmmmmmm



β D�,DDDDDDD

DDDDDDD

DDDDDDD

α_%9

δ_ %9

α

�8L��

��

��

αβ( )

��

For = :

β _%9 γ _ %9

β

9�&999999999

999999999

999999999

αz3Gzzzzzzz

zzzzzzz

zzzzzzz

βR�2RRRRRRRRRRRRRRRRRRR

RRRRRRRRRRRRRRRRRRR

RRRRRRRRRRRRRRRRRRR

γ

q.Bqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqq

αβ( )

��

46


Finally, for =n

:

n

αL�/

β

r/C

n+1

.αβ

“ n”

��

5.5.4. Homotopy basis.The3-polygraphΣ is convergent and right-indexed. Thus, Theorem 5.3.3 tellsus that the following4-cells form a homotopy basis ofΣ⊤:

γδ, δγ, αγ, βδ, αβ

( ), αβ

( ), αβ

( ),

plus, for every inN0 andn in N, the4-cell

αβ( n)

.

In fact, the4-cellsαβ

( ), αβ

( )andαβ

( )are superfluous. Indeed, their source and target

are the boundary of4-cells inΣ⊤({αγ,βδ}), as proved thereafter.

Forαβ

( ):

β _ %9

=γ

) �

=δ

�5I��

��

��

β _ %9

δ~~~ ~~~~~~

~5I~~~~ ~~~~~~~~

γ _ %9

δ

DDDDDD

DDDDDD

DDDDDD

D�+DDDDDDD

DDDDDDD

DDDDDDDα

�EY

β ��

γhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhh

h*>hhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhh

δVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVV

V 4VVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVV

=

α _ %9

γ?�)

????????

????????

????????δ _ %9

γ

@@@@@@ @@@

@�)@@@@

@@@@@@@@=

γzzzzzzzzzzzz

zzzzzz

z3Gzzzzzzz

zzzzzzz

zzzzzzz

α_ %9

δ

� @T

=

αγ

BBBBBBBB

�0BBBBBBBB

βδ||||||||

.N||||||||

47


Forαβ( )

:

δ _%9

αy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

β E�,EEEEEEEE

EEEEEEEE

EEEEEEEE

δ _ %9

α

>Ui>>>>>>>>>>

>>>>>>>>>>

>>>>>>>>>>

α_ %9

δ

�5I��

��

��βδ

@`

=

And, forαβ( )

:

β _ %9

γ

>�)>>>>>>>>

>>>>>>>>

>>>>>>>>α

y2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

β E�,EEEEEEEE

EEEEEEEE

EEEEEEEE

γ _ %9

β�u ��

��

��

γ_%9

αγ��

��

��

��

~� ��

=

We denote byΓ0 the family made of the4-cellsγδ, δγ, αγ andβδ. Then, for every natural numbern,one defines:

Γn+1 = Γn ∪{αβ( n)

, ∈ N0

}.

Thus, the following set of4-cells is a homotopy basis ofΣ:

Γ =⋃

n∈N

Γn.

For every natural numbern, we denote byξn the4-cell αβ( n)

.

5.5.5. Lemma. Letn be a natural number. There is no4-cell ofΣ⊤(Γ0) which is parallel toξn, i.e.:

sξn 6≈Γ0 tξn.

Proof. Let us assume, on the contrary, that there exists a4-cell Φ in Σ⊤(Γ0) such thatsΦ = sξn andtΦ = tξn hold. We build the derivationd of Σ∗ into the trivialΣ∗-module that takes the followingvalues on the generating3-cells:

dα = 1, dβ = −1, dγ = 0, dδ = 0.

48


Thend induces a derivation ofΣ⊤ into the trivialΣ⊤-module: we defined(f−1) = −d(f) and we verifythat, with those values,d is compatible with the inverse relations.

Then, we check that, for any4-cell Ψ of Σ⊤(Γ0), the extension ofd satisfiesdΨ = 0. Sinced is aderivation, it is sufficient to check this on the generating4-cells:

• d(γδ) = dγ− dδ = 0.

• d(δγ) = dδ− dγ = 0.

• d(αγ) = dα+ dβ + dγ − dγ = 1− 1 = 0.

• d(βδ) = dβ+ dα+ dδ− dδ = −1+ 1 = 0.

Thus, sinceΦ is inΣ⊤(Γ0), one must havedΦ = 0. However, one has:

dΦ = dα− dβ = 1 − (−1) = 2.

This proves thatΦ cannot be inΣ⊤(Γ0).

5.5.6. Lemma. Letn be a natural number. There is no4-cell ofΣ⊤(Γn) which is parallel toξn, i.e.:

sξn 6≈Γn tξn.

Proof. On the contrary, let us assume thatΦ is a4-cell of Σ⊤(Γn) such thatsΦ = sξn andtΦ = tξnhold.

First, we prove thatΦ cannot contain any occurrence of a generating4-cell αβ(

k)

or its

inverse, withk < n. For that, we define the derivationd of Σ∗2 into the moduleM(X, ∗, G) given

thereafter:

• The2-functorX : Σ∗2 → Set is generated by the values:

X( )

= N, X( )

= (0, 0), X( )

(i) = i+ 1.

• The abelian groupG is freely generated by the setN of natural numbers. The natural numbern,seen as a generator ofG, is denoted byan.

• The derivationd is given by:

d( )

= 0, d( )

(i, j) = aj, d( )

(i) = 0.

Now, let us assume that there exist4-cellsΨ1, Ψ2 in Σ⊤(Γn), a contextC of Σ⊤, anε in {−1, 1}, a2-celland ak in {0, . . . , n − 1} such that the4-cellΦ decomposes this way:

Φ = Ψ1 ⋆3C[αβ(

k)ε]

⋆3 Ψ2.

49


Then we haves2Φ = (s2C)[s2αβ

(k)]

. In particular, both2-cells are sent byd to the same

element ofG. But, on the one hand, by hypothesis, one gets:

s2Φ = s2ξn = n ,

so that the following holds:d (s2Φ) = an.

And, on the other hand, one has:

(s2C)[s2αβ

(k)]

= k

s2C

.

Then, using the fact thatd is a derivation, one computes:

d

k

s2C

= d

(∗

s2C

)+ d

(k

)+ d

( )= d(f) + ak ,

wheref = (s2C)[ ]

. Thus, we havean = ak + d(f), with k < n andf in Σ∗2. This is impossible

becauseG is freely generated andd sends any2-cell to an element ofG written using theai’s withpositive coefficients.

Thus, we conclude that the4-cellΦ must be built using the4-cells ofΓ0 and their inverses,i.e.Φ isa4-cell ofΣ⊤(Γ0). However, this would contradict the previous result.

5.5.7. Theorem. The3-polygraphΣ does not have finite derivation type.

Proof. On the contrary, let us assume thatΣ does have finite derivation type. Then, by application ofProposition 3.2.3, there exists a finite subfamilyΓ ′ of Γ which is a homotopy basis ofΣ⊤.

SinceΓ ′ is finite, there exists some natural numbern such thatΓ ′ is contained inΓn. In particular,the4-cell ξn is not inΓ ′. However, sinceΓ ′ is a homotopy basis and sinceΓ ′ is contained inΓn, we have:

sξn ≈Γn tξn.

We have seen that this is not possible, thus contradicting the fact that one can extract a finite homotopybasis fromΓ . As a consequence, the3-polygraphΣ does not have finite derivation type.

5.5.8. A variant of the counterexample. In the previous3-polygraph, one can think that the problemcomes from the complicated normal forms, especially from the fact that one can find normal formsof N0 everywhere in a given2-cell. Here we give another example, similar to the first one but withmore simple normal forms. It is a bit more contrived, which led us to prefer the other one for the mainexposition.

Let Ξ be the3-polygraph with the following generating cells:

50

REFERENCES

• Two 0-cells, denoted byξ andη and, in the diagrammatic representations, respectively picturedby a white background and by a gray one.

• Two 1-cells ξp

//η andηq

//ξ. By abuse, both are pictured by a wire, leaving the backgroundsdiscriminate them.

• Four2-cells , , , and .

• Two 3-cellsα′

⇛ andβ′

⇛ .

Following the same reasoning steps as for the previous example one proves that the finite3-polygraphΞis convergent and that it does not have finite derivation type. Indeed, the following family of4-cells,indexed by the natural numbern, form a homotopy basis ofΞ⊤ and cannot be reduced further:

n

α′

B�+

β′

|3G

n+1α′β′

„n+1

«

��

REFERENCES

[1] Franz Baader and Tobias Nipkow,Term rewriting and all that, Cambridge University Press, 1998.

[2] Hans Joachim Baues,Combinatorial homotopy and4-dimensional complexes, de Gruyter Expositions inMathematics, vol. 2, Walter de Gruyter & Co., Berlin, 1991.

[3] Hans Joachim Baues and Günther Wirsching,Cohomology of small categories, J. Pure Appl. Algebra38(1985), no. 2-3, 187–211.

[4] Guillaume Bonfante and Yves Guiraud,Polygraphic programs and polynomial-time functions, Logical Meth-ods in Computer Science (2007), Under revision.

[5] Ronald Vernon Book and Friedrich Otto,String-rewriting systems, Springer, 1993.

[6] Albert Burroni,Higher-dimensional word problems with applications to equational logic, Theoretical Com-puter Science115(1993), no. 1, 43–62.

[7] Eugenia Cheng and Aaron Lauda,Higher-dimensional categories: an illustrated guide book, 2004.

[8] Nick Gilbert, Monoid presentations and associated groupoids, Internat. J. Algebra Comput.8 (1998), no. 2,141–152.

[9] Yves Guiraud,Présentations d’opérades et systèmes de réécriture, Ph.D. thesis, Université Montpellier 2,June 2004.

[10] , Termination orders for 3-dimensional rewriting, J. Pure and Appl. Algebra207(2006), no. 2, 341–371.

[11] , The three dimensions of proofs, Annals of Pure and Applied Logic141(2006), no. 1-2, 266–295.

51

REFERENCES

[12] , Two polygraphic presentations of Petri nets, Theoretical Computer Science360 (2006), no. 1-3,124–146.

[13] Yves Lafont,Towards an algebraic theory of boolean circuits, J. Pure and Appl. Algebra184(2003), no. 2-3,257–310.

[14] Jonathan Leech,Cohomology theory for monoid congruences, Houston J. Math.11 (1985), no. 2, 207–223.

[15] Saunders Mac Lane,Categories for the working mathematician, 2nd ed., Springer, 1998.

[16] François Métayer,Resolutions by polygraphs, Theory and Applications of Categories11 (2003), 148–184.

[17] Maxwell Herman Alexander Newman,On theories with a combinatorial definition of "equivalence", Annalsof Mathematics43 (1942), no. 2, 223–243.

[18] Daniel Quillen,Higher algebraicK-theory. I, AlgebraicK-theory, I: HigherK-theories (Proc. Conf., BattelleMemorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341.

[19] Craig Squier,A finiteness condition for rewriting systems, Theoretical Computer Science131(1994), no. 2,271–294, Revised by Friedrich Otto and Yuji Kobayashi.

[20] Ross Street,Limits indexed by category-valued2-functors, J. Pure Appl. Algebra8 (1976), no. 2, 149–181.

[21] , The algebra of oriented simplexes, J. Pure Appl. Algebra49 (1987), no. 3, 283–335.

[22] , Higher categories, strings, cubes and simplex equations, Appl. Categ. Structures3 (1995), no. 1,29–77.

52

Documents

Higher-dimensional categories with ﬁnite derivation type · 2020-02-05 · arXiv:0810.1442v1 [math.CT] 8 Oct 2008 Higher-dimensional categories with ﬁnite derivation type Yves