Identities among relations for polygraphic rewriting

Transcription

1 Identities amon relations for polyraphic rewritin July 6, 2010 References: (P M, Y Guiraud) Identities amon relations for hiher-dimensional rewritin systems, arxiv: , to appear; (P M, Y Guiraud) Coherence in monoidal track cateories, arxiv: , submitted; (P M, Y Guiraud) Hiher-dimensional cateories with finite derivation type, Theory and Applications of Cateories, 2009;

2 Contents Fox-Squier theory Two-dimensional homotopy for polyraphs - Polyraphs and hiher track cateories - Critical branchins in 3-polyraphs Identities amon relations for polyraphs - Abelian track extensions - Generatin identities amon relations

3 Fox-Squier theory

4 Fact Monoids havin a finite converent presentation are decidable (normal form alorithm)

5 o o o o Strin-rewritin and word problem Strin rewritin system (Thue 1914) : 2-polyraph Σ with one 0-cell : Σ 1 s s x t t x s x x { } Σ s 1 t t Σ 2 Σ 2 Σ 1 : enerators Σ 1 : strins Σ 2 : rules Σ 2 : reductions Normal form alorithm for finite and converent (terminatin + confluent) 2-polyraph : w w? ŵ ŵ

6 Strin-rewritin and word problem Kapur-Narendran 85, Jantzen 85 : examples of finite strin-rewritin systems - with decidable word problems, - that are not equivalent to any finite converent strin-rewritin system on the same alphabet Example Σ : Σ 1 = { a,b } Σ 2 = { aba bab } there does not exist any finite converent 2-polyraph on Σ 1 Tietze equivalent to Σ However, Σ is Tietze equivalent to the finite converent polyraph: Υ 1 = { a,b,c } Υ 2 = { ab c, ca bc, bcb cc, ccb acc }

7 Σ 1 = { a,b } Σ 2 = { aba bab }

8 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc

9 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc aba ca β bc bab

10 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc aba ca β bc bab bcb β c a b cc

11 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc, bcb cc aba ca β bc bab bcb β c a b cc

12 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc, bcb cc aba ca β bc bab bcb β c a b cc ccb abcb acc

13 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc, bcb cc, ccb δ acc aba ca β bc bab bcb β c a b cc ccb abcb δ acc

14 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc, bcb cc, ccb δ acc aba ca β bc bab bcb β c a b cc ccb abcb δ acc acccb δ ccbcb cccc

15 Σ 1 = { a,b } Σ 2 = { aba bab } Υ 1 = { a,b,c } Υ 2 = { ab c, ca β bc, bcb cc, ccb δ acc } aba ca β bc bab bcb β c a b cc ccb abcb δ acc δ ccbcb acccb cccc δ acacc abccc β

16 Strin-rewritin and word problem Questions (Book 85, Kapur-Narendran 85, Jantzen 85, ) - Does a finitely presented decidable monoid have a finite converent presentation? - What conditions a monoid must satisfy if it can be presented by a finite converent rewritin system?

17 Fact Rewritin is not universal to decide the word problem in finitely presented monoids

18 / / I / / / F / First Squier theorem Squier 87 : there are finitely presented decidable monoids which do not have a finite converent presentation Theorem (Squier 87) A monoid havin a finite converent presentation is of homoloical type FP 3 Proof : - Σ a converent strin-rewritin system, M = Σ 1 /Σ 2, - The M-module of homoloical 2-syzyies of Σ kerj ZM[Σ 2 ] is enerated by critical branchins: J ZM[Σ 1 ] ZM Z - There are finitely presented decidable monoids which are not of type FP 3

19 First Squier theorem Example A finitely presented decidable monoid which is not of type FP 3 : M = ( a,b,c,d,d ab = a, da = ac, d a = ac ) infinite converent presentation Σ 1 = { a,b,c,d,d } Σ 2 = { ac n b ac n, n N, da ac, d a ac } dac n b ac n+1 b C n dac n ac n+1 H 3 (M,Z) is the free abelian roup on { Cn C n, n N } d ac n b ac n+1 b C n d ac n ac n+1 Hence M is not of homoloical type FP 3

20 Fact The homoloical finiteness condition FP 3 is not sufficient for a finitely presented decidable monoid to admit a finite converent presentation

21 Second Squier Theorem S(Σ) 2-dimensional combinatorial complex on a strin-rewritin system Σ: - 0-cells are strins in Σ 1, - 1-cells are derivations in Σ 2, - 2-cells are Peiffer elements : ulwl v uawl v ulwa v urwl v Peiff ulwr urwa v v uawr v urwr v Definition Σ has finite derivation type (FDT) if Σ is finite and S(Σ) has a finite set of homotopy trivializers Theorem (Squier 94) i) The property FDT is Tietze invariant for finite strin-rewritin systems ii) A monoid havin a finite converent presentation has FDT There are finitely presented decidable monoids which do not have FDT and which are left-fp 3

22 Two-dimensional homotopy for polyraphs

23 Track n-cateories and homotopy bases Definitions - A track n-cateory is an (n 1)-cateory enriched in roupoids - A homotopy basis of an n-cateory C is a cellular extension Γ such that the track n-cateory C/Γ is aspherical ie, for every n-spheres f A in C, there exists an (n + 1)-cell f in A C(Γ), ie, f = in C/Γ Definition An n-polyraph Σ has finite derivation type (FDT) if i) Σ is finite, ii) the free track n-cateory Σ = Σ n 1 (Σ n) admits a finite homotopy basis Proposition The property FDT is Tietze invariant for finite polyraphs

24 Critical branchins A branchin is critical when it is a "minimal overlappin" of n-cells, such as: Newman s diamond lemma ( 42) : Termination + confluence of critical branchins Converence

25 The homotopy basis of eneratin confluences A eneratin confluence of an n-polyraph Σ is an (n + 1)-cell with (f, ) critical branchin v u f ~! w f ~ u Theorem Let Σ be a converent n-polyraph A cellular extension of Σ made of one eneratin confluence for each critical branchin of Σ is a homotopy basis of Σ

26 / Example I : Mac Lane s coherence theorem Monoidal cateory (C,, I,, λ, ρ) with natural isomorphisms x,y,z : (x y) z x (y z) λ x : I x x ρ x : x I x such that the followin diarams commute: ((x y) z) t (x (y z)) t < / x ((y z) t) c / (x y) (z t) " x (y (z t)) (x I) y? x (I y) c λ ρ / x y Theorem (Mac Lane s coherence theorem 63) In a monoidal cateory, all the diarams built from C,, I,, λ and ρ are commutative

27 Example I : Mac Lane s coherence theorem Let Mon be the 3-polyraph: - one 0-cell, one 1-cell, two 2-cell :,, - three 3-cells : Lemma Mon terminates and is locally confluent, with the followin five eneratin confluences:

28 Example I : Mac Lane s coherence theorem Let Γ = {, } be the cellular extension of Mon with : The track 3-cateory Mon /Γ is the theory of monoidal cateories, ie, there is an equivalence: Monoidal cateories 3-functors Mon /Γ Cat Theorem The 3-cateory Mon /Γ is aspherical Corollary : Mac Lane s coherence theorem

29 Example II : polyraph of permutations Perm 3-polyraph of permutations : - one 0-cell, one 1-cell, one 2-cell : - two 3-cells : and β Theorem Perm is finite, converent and has FDT Perm has an infinite set of critical branchins three reular critical branchins a riht-indexed critical branchin k

30 Example II : polyraph of permutations reular critical branchins are confluent β β β β β β riht-indexed have two normal instances β β β β ββ ( ) β β ββ ( ) β β β β {, β, β, ββ ( ), ββ ( )} is a homotopy base

31 / / / F! / Generatin confluences and finite derivation type Generatin confluences of a converent n-polyraph Σ form a homotopy basis of Σ Corollary A finite converent n-polyraph with a finite number of critical branchins has FDT Corollary (Squier 94) A finite converent 2-polyraph has FDT - Two shapes of branchin in a 2-polyraph : Reular critical branchin Inclusion critical branchin / D / Theorem For n 3, there exist finite converent n-polyraphs without FDT

32 Critical branchins in 3-polyraphs Reular critical branchin u s v = f v u h = u f sβ v Inclusion critical branchin s = sβ C Riht-indexed critical branchin s k f = h k f = k sβ ;

33 Critical branchins in 3-polyraphs Reular u s v = f v u h = u f sβ v u s v = u f h v = u f sβ v v s u v = h f u = v f u sβ u s v = u f v h = u f sβ v Inclusion s = sβ C Indexed s k f = h k f = k sβ ; k s = k f h = k f sβ k 0 s f k 0 h 1 k 1 h 2 h p k p = k 1 k p k n = h n k n k 0 k 1 f sβ k p k n

34 Critical branchins in 3-polyraphs Theorem Let Σ be a finite converent 3-polyraph i) If Σ does not have indexed critical branchin it has FDT ii) If Σ has indexed critical branchins with finitely many normal instances it has FDT ta f F ta A C Peiff A C A sa = sb f C B F Peiff sa = sb C B en conf B h C tb f F tb C C

35 Example III : pearl necklaces Pearl 3-polyraph presentin the 2-cateory of pearl necklaces: - one 0-cell, one 1-cell, three 2-cells : - four 3-cells : β δ Examples of 2-cells Proposition Pearl is finite and converent but does not have FDT

36 Example III : pearl necklaces Four reular critical branchin δ δ δ δ β β δ βδ δ One riht-indexed critical branchin k k {,,, } n n

37 Example III : pearl necklaces β β δ δ Peiff β Peiff δ βδ δ Peiff δ Peiff Peiff δ

38 Example III : pearl necklaces δ Peiff β βδ δ δ β Peiff β β n ( β n ) n + 1 β

39 Example III : pearl necklaces δ δ δ δ β β δ βδ δ n ( β { ( An infinite homotopy base : δ, δ,, βδ, β β n ) n ), n N } n + 1 The 3-polyraph Pearl is finite and converent but does not have finite derivation type

40 Identities amon relations for polyraphs

41 The special case of roups (Peiffer-Reidemester-Whitehead 49) the free crossed module : C(P ) F(X) on a presentation P = (X,R) of a roup G Whitehead s lemma ( 49) : the followin diaram commutes ker / C(P ) // F(X) / G ( ) ab / ker J ZG[R] J [ ] / ZG[X] / ZG Brown-Huebschmann ( 82) : the Abelianisation map induces an isomorphism of G-modules ker kerj Theorem (Cremanns-Otto 96) If P is finite, then G has FDT if and only if ker is finitely enerated Corollary A finitely presented roup is of type FP 3 if and only if it has FDT

42 f b b Identities amon relations Let C be a n-cateory presented by a (n + 1)-polyraph Σ Let Σ ab be the Abelianised free (n + 1)-track cateory on Σ Σ ab n+1 s / t / Σ n Theorem A track cateory is Abelian if and only if it is linear / / C There exists a C-module Π(Σ) of identities amon relations of Σ and an isomorphism of Σ n 1 -modules Π(Σ) Aut Σ ab Π(Σ) is defined as follows: - for an n-cell u of C, then Π(Σ)(u) is the quotient of { Z f } v f in Σ ab, v = u by : " f n = f + for every f v with v = u f n = n f for every v f ' w with v = w = u

43 Generatin identities amon relations Theorem Any homotopy basis of the free track cateory Σ forms a eneratin set for the module Π(Σ) Corollary If a polyraph Σ has FDT, then the module Π(Σ) is finitely enerated Corollary If Σ is converent, the module Π(Σ) is enerated by the eneratin confluences of Σ Example Let As be the 2-polyraph (,, ) - As is a finite converent presentation of the monoid a aa = a - As has one eneratin confluence: - Hence the followin element enerates Π(Σ): = 1 ( ) 1 = = =

44 Finite Abelian derivation type Definition An n-polyraph Σ has finite Abelian derivation type (TDF ab ) if i) Σ is finite, ii) Σ ab admits a finite homotopy basis TDF implies TDF ab, Proposition Let C be an n-cateory presented a polyraph Σ, then Σ has TDF ab if and only if the C-module Π(Σ) is finitely enerated Corollary A 1-cateory has TDF ab if and only if it is of homoloical type FP 3