Coherent presentations of Artin groups

Transcription

1 Coherent presentations of Artin groups Philippe Malbos INRIA - πr 2, Laboratoire Preuves, Programmes et Systèmes, Université Paris Diderot & Institut Camille Jordan, Université Claude Bernard Lyon 1 Joint work with Stéphane Gaussent and Yves Guiraud Colloque GDR Topologie Algébrique et Applications October 18, 2013, Angers

2 Motivation A Coxeter system (W, S) is a data made of a group W with a presentation by a (finite) set S of involutions, s 2 = 1, satisfying braid relations tstst... = ststs... Forgetting the involutive character of generators, one gets the Artin s presentation of the Artin group B(W) = S tstst... = ststs... Objective. - Push further Artin s presentation and study the relations among the braid relations. (Brieskorn-Saito, 1972, Deligne, 1972, Tits, 1981). - We introduce a rewriting method to compute generators of relations among relations.

3 Motivation Set W = S 4 the group of permutations of {1, 2, 3, 4}, with S = {r, s, t} where r = s = t = The associated Artin group is the group of braids on 4 strands B(S 4 ) = r, s, t rsr = srs, rt = tr, tst = sts = = = The relations among the braid relations on 4 strands are generated by the Zamolodchikov relation (Deligne, 1997). strsrt srtstr srstsr stsrst rsrtsr tstrst Z r,s,t rstrsr tsrtst tsrsts trsrts rtstrs rstsrs

4 Plan I. Coherent presentations of categories - Polygraphs as higher-dimensional rewriting systems - Coherent presentations as cofibrant approximation II. Homotopical completion-reduction procedure - Tietze transformations - Rewriting properties of 2-polygraphs - The homotopical completion-procedure III. Applications to Artin groups - Garside s coherent presentation - Artin s coherent presentation References - S. Gaussent, Y. Guiraud, P. M., Coherent presentations of Artin groups, ArXiv preprint, Y. Guiraud, P.M., Higher-dimensional normalisation strategies for acyclicity, Adv. Math.,2012.

5 Part I. Coherent presentations of categories

6 Polygraphs A 1-polygraph is an oriented graph (Σ 0, Σ 1 ) s 0 Σ 0 Σ 1 t 0 A 2-polygraph is a triple Σ = (Σ 0, Σ 1, Σ 2 ) where - (Σ 0, Σ 1 ) is a 1-polygraph, - Σ 2 is a globular extension of the free category Σ 1. s 1(α) s 0 Σ 0 t 0 Σ 1 s 1 t 1 Σ 2 s 0s 1(α) = s 0t 1(α) α t 0s 1(α) = t 0t 1(α) t 1(α) A rewriting step is a 2-cell of the free 2-category Σ 2 over Σ with shape u w α w v where α : u v is a 2-cell of Σ 2 and w, w are 1-cells of Σ 1.

7 Polygraphs A (3, 1)-polygraph is a pair Σ = (Σ 2, Σ 3 ) made of - a 2-polygraph Σ 2, - a globular extension Σ 3 of the free (2, 1)-category Σ 2. u s 0 Σ 0 t 0 Σ 1 s 1 t 1 Σ 2 s 2 Σ 3 t 2 α A β v Let C be a category. A presentation of C is a 2-polygraph Σ such that C Σ 1/Σ 2 An extended presentation of C is a (3, 1)-polygraph Σ such that C Σ 1/Σ 2

8 Coherent presentations of categories A coherent presentation of C is an extended presentation Σ of C such that the cellular extension Σ 3 is a homotopy basis. In other words - the quotient (2, 1)-category Σ 2 /Σ 3 is aspherical, - the congruence generated by Σ 3 on the (2, 1)-category Σ 2 parallel 2-cells. contains every pair of Example. The full coherent presentation contains all the 3-cells. Theorem. [Gaussent-Guiraud-M., 2013] Let Σ be an extended presentation of a category C. Consider the Lack s model structure for 2-categories. The following assertions are equivalent: i) The (3, 1)-polygraph Σ is a coherent presentation of C. ii) The (2, 1)-category presented by Σ is a cofibrant 2-category weakly equivalent to C, that is a cofibrant approximation of C.

9 Examples Free monoid : no relation, an empty homotopy basis. Free commutative monoid: N 3 = r, s, t sr γ rs rs, ts γ st st, tr γ rt rt all the 3-cells - A homotopy basis can be made with only one 3-cell N 3 = r, s, t sr γ rs rs, ts γ st st, tr γ rt rt Z r,s,t is a coherent presentation, where Z r,s,t is the permutaedron γ st r str sγ rt srt γ rs t tsr Z r,s,t rst tγ rs trs γ rt s rts rγ st

10 Examples Artin s coherent presentation of the monoid B + 3 Art 3 (S 3 ) = s, t tst γ st sts where s = and t = Artin s coherent presentation of the monoid B + 4 Art 3 (S 4 ) = r, s, t rsr γ sr srs, rt γ tr tr, tst γ st sts Z r,s,t stγ rst stsrst γ strst tstrst tsγ rtst tsrtst strsrt sγrtsγ srtstr srγstr srstsr Z r,s,t γ rstsr rsrtsr rstrsr rsγ rtsr rstγ rs rstsrs tsrγ st tsrsts trsrts rtstrs tγrsts γrtsγ rts rγ strs

11 Coherent presentations Problems. 1. How to transform a coherent presentation? 2. How to compute a coherent presentation?

12 Part II. Homotopical completion-reduction procedure

13 Tietze transformations Two (3, 1)-polygraphs Σ and Υ are Tietze-equivalent if there is an equivalence of 2-categories F : Σ 2 /Σ 3 Υ 2 /Υ 3 inducing an isomorphism on presented categories. In particular, two coherent presentations of the same category are Tietze-equivalent. An elementary Tietze transformation of a (3, 1)-polygraph Σ is a 3-functor with source Σ that belongs to one of the following three pairs of dual operations: add a generator: for u Σ 1, add a generating 1-cell x and add a generating 2-cell u δ x remove a generator: for a generating 2-cell α in Σ 2 with x Σ 1, remove x and α u α α x x

14 Tietze transformations Two (3, 1)-polygraphs Σ and Υ are Tietze-equivalent if there is an equivalence of 2-categories F : Σ 2 /Σ 3 Υ 2 /Υ 3 inducing an isomorphism on presented categories. In particular, two coherent presentations of the same category are Tietze-equivalent. An elementary Tietze transformation of a (3, 1)-polygraph Σ is a 3-functor with source Σ that belongs to one of the following three pairs of dual operations: add a relation: for a 2-cell f Σ 2, add a generating 2-cell χ f add a generating 3-cell A f f u A f v χ f remove a relation: for a 3-cell A where α Σ 2, remove α and A f u A A α α v

15 Tietze transformations Two (3, 1)-polygraphs Σ and Υ are Tietze-equivalent if there is an equivalence of 2-categories F : Σ 2 /Σ 3 Υ 2 /Υ 3 inducing an isomorphism on presented categories. In particular, two coherent presentations of the same category are Tietze-equivalent. An elementary Tietze transformation of a (3, 1)-polygraph Σ is a 3-functor with source Σ that belongs to one of the following three pairs of dual operations: add a 3-cell: for equals 2-cells f g, u f add a generating 3-cell f A g g A remove a 3-cell: for a generating 3-cell f g with f g, remove A A v f u A v A

16 Tietze transformations Theorem. [Gaussent-Guiraud-M., 2013] Two (finite) (3, 1)-polygraphs Σ and Υ are Tietze equivalent if, and only if, there exists a (finite) Tietze transformation T : Σ Υ Consequence. If Σ is a coherent presentation of a category C and if there exists a Tietze transformation T : Σ Υ then Υ is a coherent presentation of C.

17 Rewriting properties of 2-polygraphs Let Σ = (Σ 0, Σ 1, Σ 2 ) be a 2-polygraph. Σ terminates if it does not generate any infinite reduction sequence u 1 u 2 u n A branching of Σ is a pair (f, g) of 2-cells of Σ 2 with a common source f v u g w Σ is confluent if all of its branchings are confluent: f v f u g w g u Σ is convergent if it terminates and it is confluent.

18 Rewriting properties of 2-polygraphs A branching f v u is local if f and g are rewriting steps. g w Local branchings are classified as follows: - aspherical branchings have shape (f, f ), - Peiffer branchings have shape uu fv vu u g vv fv 1 u g = ug 1 fv ug uv fv - overlap branchings are all the other cases. A critical branching is a minimal (for inclusion of source) overlap branching. Theorem. [Newman s diammond lemma, 1942] For terminating 2-polygraphs, local confluence and confluence are equivalent properties.

19 Rewriting properties of 2-polygraphs Example. Consider the 2-polygraph Art 2 (S 3 ) = s, t tst γ st sts A Peiffer branching: γ st tst ststst stsγ ts tsttst tstγ st tststs γ st sts stssts A critical branching: γ st st stsst tstst tsγ st tssts

20 Homotopical completion procedure The homotopical completion procedure is defined as the combinaison of the - Knuth-Bendix procedure computing a convergent presentation from a terminating presentation (Knuth-Bendix, 1970). - Squier procedure, computing a coherent presentation from a convergent presentation (Squier, 1994).

21 Homotopical completion procedure Let Σ be a terminating 2-polygraph (with a total termination order). The homotopical completion of Σ is the (3, 1)-polygraph S(Σ) obtained from Σ by successive application of following Tietze transformations - for every critical pair u f v g w f v g ŵ compute f and g reducing to some normal forms. f v f - if v = ŵ, add a 3-cell A f,g u A f,g v = ŵ g w g - if v < ŵ, add the 2-cell χ and the 3-cell A f,g u f v g w f v A f,g g ŵ χ

22 Homotopical completion procedure Potential adjunction of additional 2-cells χ can create new critical branchings, - whose confluence must also be examined, - possibly generating the adjunction of additional 2-cells and 3-cells -... This defines an increasing sequence of (3, 1)-polygraphs Σ = Σ 0 Σ 1 Σ n Σ n+1 The homotopical completion of Σ is the (3, 1)-polygraph S(Σ) = Σ n. n0 Theorem. [Gaussent-Guiraud-M., 2013] For every terminating presentation Σ of a category C, the homotopical completion S(Σ) of Σ is a coherent convergent presentation of C.

23 Homotopical completion procedure Example. The Kapur-Narendran presentation of B + 3, obtained from Artin s presentation by coherent adjunction of the Coxeter element st Σ KN 2 = s, t, a ta α as, st β a The deglex order generated by t > s > a proves the termination of Σ KN 2. S(Σ KN 2 ) = s, t, a ta βa sta A α as, st aa sα sas γ sast β a, sas γt aat B saβ saa δ γ aa, saa aaaβ γaa aaaa aaast sasaa D saδ saaat δat aatat δ aat A, B, C, D γas aaas aaα sasas C aata saγ saaa δa aaαt However. The extended presentation S(Σ KN 2 ) obtained is bigger than necessary.

24 Homotopical completion-reduction procedure INPUT: A terminating 2-polygraph Σ. Step 1. Compute the homotopical completion S(Σ) (convergent and coherent). Step 2. Apply the homotopical reduction to S(Σ) with a collapsible part Γ made of - 3-spheres induced by some of the generating triple confluences of S(Σ), f 1 v f A f,g g 1 u g w g A g,h 2 h x h 2 x v B f 1 v h f x h f 2 C û u A w g f,h 3 û k h h 1 D x v k h 2

25 Homotopical completion-reduction procedure INPUT: A terminating 2-polygraph Σ. Step 1. Compute the homotopical completion S(Σ) (convergent and coherent). Step 2. Apply the homotopical reduction to S(Σ) with a collapsible part Γ made of - 3-spheres induced by some of the generating triple confluences of S(Σ), - the 3-cells adjoined with a 2-cell by homotopical completion to reach confluence, - some collapsible 2-cells or 3-cells already present in the initial presentation Σ. u f v g w f v A f,g g ŵ χ

26 Homotopical completion-reduction procedure INPUT: A terminating 2-polygraph Σ. Step 1. Compute the homotopical completion S(Σ) (convergent and coherent). Step 2. Apply the homotopical reduction to S(Σ) with a collapsible part Γ made of - 3-spheres induced by some of the generating triple confluences of S(Σ), - the 3-cells adjoined with a 2-cell by homotopical completion to reach confluence, - some collapsible 2-cells or 3-cells already present in the initial presentation Σ. The homotopical completion-reduction of terminating 2-polygraph Σ is the (3, 1)-polygraph R(Σ) = π Γ (S(Σ)) Theorem. [Gaussent-Guiraud-M., 2013] For every terminating presentation Σ of a category C, the homotopical completion-reduction R(Σ) of Σ is a coherent convergent presentation of C.

27 The homotopical completion-reduction procedure Example. Σ KN 2 = s, t, a ta α as, st β a S(Σ KN 2 ) = s, t, a ta α as, st β a, sas γ aa, saa δ aat A, B, C, D s, t, a ta α as, st β a, sas γ aa, saa δ aat A, B, C, C, D There are four critical triple branchings, overlapping on sasta, sasast, sasasas, sasasaa. - Critical triple branching on sasta proves that C is superfluous: γta aata aaα aaas γta aata sasta Ba δa saβa saaa saa saγ sasα sasas aaα sasta aaas γas sasα sasas C saγ saaa aaα δa aata C = sasα 1 1 (Ba 1 aaα) 2 (saa 1 δa 1 aaα)

28 The homotopical completion-reduction procedure Example. Σ KN 2 = s, t, a ta α as, st β a S(Σ KN 2 ) = s, t, a ta α as, st β a, sas γ aa, saa δ aat A, B, C, D s, t, a ta α as, st β a, sas γ aa, saa δ aat A, B, C, D - Critical triple branching on sasast proves that D is superfluous: aaaβ γast aaast aaaa γast aaast aaaβ Ct aaaβ aaαt sasast aaaa aaast sasast saγt saaat aatat δat sab saδ sasaβ γaa sasaa D aaαt sasaβ sasaa saδ saaat aatat δat D = sasaβ 1 1 ( (Ct 1 aaaβ) 2 (sab 1 δat 1 aaαt 1 aaaβ) )

29 The homotopical completion-reduction procedure Example. S(Σ KN 2 ) = s, t, a ta s, t, a ta α as, st Σ KN 2 = s, t, a ta α as, st β a, sas α as, st β a γ aa, saa δ aat A, B, C, D β a, γ sas aa,, saa δ aat A, B, C, D - The 3-cells A and B correspond to the adjunction of the rules γ and δ during the completion βa aa γt aat sta A sα sas γ sast B saβ saa δ

30 The homotopical completion-reduction procedure Example. Σ KN 2 = s, t, a ta α as, st β a S(Σ KN 2 ) = s, t, a ta α as, st β a, sas γ aa, saa δ aat A, B, C, D s, t, a tst α sts, β st a, sas γ aa,, saa δ aat A, B, C, D - The generator a is superflous. R(Σ KN 2 ) = s, t tst α sts = Art 3 (S 3 )

31 Part III. Applications to Artin groups

32 Garside s presentation Let W be a Coxeter group W = S s 2 = 1, ts mst = st mst, with m st and s, t S where ts mst stands for the word tsts... with m st letters. Artin s presentation of the Artin monoid B + (W): Art 2 (W) = S ts mst = st mst, with m st and s, t S Garside s presentation of B + (W) Gar 2 (W) = W \ {1} u v α u,v uv, whenever u v where uv is the product in W, u v is the product in the free monoid over W. Notations : - u v whenever l(uv) = l(u) + l(v). - u v whenever l(uv) < l(u) + l(v).

33 Garside s coherent presentation The Garside s coherent presentation of B + (W) is the extended presentation Gar 3 (W) obtained from Gar 2 (W) by adjunction of one 3-cell α u,v w uv w α uv,w u v w A u,v,w uvw u α v,w α u vw u,vw for every u, v, w in W \ {1} with u v w. Theorem. [Gaussent-Guiraud-M., 2013] For every Coxeter group W, the Artin monoid B + (W) admits Gar 3 (W) as a coherent presentation. Proof. By homotopical completion-reduction of the 2-polygraph Gar 2 (W).

34 Garside s coherent presentation Step 1. We compute the coherent convergent presentation S(Gar 2 (W)) - The 2-polygraph Gar 2 (W) has one critical branching for every u, v, w in W \ {1} when α u,v w uv w u v w u v w u α v,w u vw - There are two possibilities. α u,v w uv w α uv,w if u v w u v w A u,v,w uvw u α v,w α u vw u,vw

35 Garside s coherent presentation Step 1. We compute the coherent convergent presentation S(Gar 2 (W)) - The 2-polygraph Gar 2 (W) has one critical branching for every u, v, w in W \ {1} when α u,v w uv w u v w u v w u α v,w u vw - There are two possibilities. u v w otherwise u v w α u,v w u α v,w u vw B u,v,w uv w β u,v,w

36 Garside s coherent presentation α u,v w uv w α uv,w u v w A u,v,w uvw u α v,w α u vw u,vw u v w α u,v w u α v,w u vw B u,v,w uv w β u,v,w α u,v wx uv wx u v wx C u,v,w,x β uv,w,x uvw x u β v,w,x u vw x α u,vw x u v wx u β v,w,x α u,v wx D u,v,w,x u vw x uv w x βu,v,w x uv wx uv α w,x β u,v,w xy uv w xy uv α w,xy u vw xy F u,v,w,x,y uv wxy u β vw,x,y u vwx y uv wx y uv α vw,x β u,v,wx y β u,v,w x uv w x uv α w,x u vw x E u,v,w,x uv wx u α vw,x u vwx β u,v,wx u vw = u v w β u,v,w xy uv w xy u vw xy G u,v,w,x,y uv β w,x,y uv wx y u β vw,x,y u vwx y β u,v,wx y β u,v,w uv w = uv xy I u,v,w,v,w β u,v,w uv w = uv x y β uv,x,y uvx y = uv x y β uv,x,y uv xy β u,v,xy u vxy β uv,x,y H u,v,x,y uvx y β u,vx,y

37 Garside s coherent presentation Proposition. For every Coxeter group W, the Artin monoid B + (W) admits, as a coherent convergent presentation the (3, 1)-polygrap S(Gar 2 (W)) where - the 1-cells are the elements of W \ {1}, - there is a 2-cell u v α u,v uv for every u, v in W \ {1} with u v, - the 2-cells u vw β u,v,w uv w, for every u, v, w in W \ {1} with u v w, - the nine families of 3-cells A, B, C, D, E, F, G, H, I.

38 Garside s coherent presentation Step 2. Homotopical reduction of S(Gar 2 (W)). - We consider some triple critical branchings of S(Gar 2 (W)) In the case u v w x α u,v w x u v w x α uv,w x uv w x uvw x A u,v,w x α u,vw x u α v,w x u vw x u B v,w,x u v α w,x u β v,w,x u v wx α u,v w x uv w x u v w x = uv α w,x α u,v wx u v α w,x u v wx α uv,w x B uv,w,x uv wx β uv,w,x uvw x C u,v,w,x α u,vw x u β v,w,x u vw x and similar 3-spheres for the following cases u v w x u v w x u v w x y u v w x y u v w x u v w and u v w with vw = v w

39 Artin s coherent presentation Theorem. [Gaussent-Guiraud-M., 2013] For every Coxeter group W, the Artin monoid B + (W) admits the coherent presentation Art 3 (W) made of - Artin s presentation Art 2 (W) = S ts mst = st mst, - one 3-cell Z r,s,t for every elements t > s > r of S such that the subgroup W {r,s,t} is finite. Theorem. [Gaussent-Guiraud-M., 2013] For every Coxeter group W, the Artin group B(W) admits Gar 3 (W) and Art 3 (W) as coherent presentation.

40 Artin s coherent presentation The 3-cells Z r,s,t for Coxeter types A 3 stγ rst stsrst γ strst tstrst tsγ rtst tsrtst strsrt sγrtsγ srtstr srγstr srstsr Z r,s,t γ rstsr rsrtsr rstrsr rsγ rtsr rstγ rs rstsrs tsrγ st tsrsts trsrts rtstrs tγrsts γrtsγ rts rγ strs

41 Artin s coherent presentation The 3-cells Z r,s,t for Coxeter types B 3 stγ rs tsr γ st rsrtsr srtsγ rt str srγst rsγrt srtsrtstr srtstrstr srstsrsrt srstγrs t srsγrt srst sγ rt srγ st r srstrsrst srsrtsrst γ rs tsrst strsrstsr stsrsrtsr tstrsrtsr tsγ rt sγ rt sr tsrtstrsr tsrγ st rsr tsrstsrsr tsrstγ rs Z r,s,t tsrstrsrs tsrsrtsrs trsrstsrs rtsrtstrs rtstrstrs tsrsγrt srs tγrs tsrs γ rt srγ st rs rtsγ rt strs rsrstsrst rsrtstrst rsrtsrtst rstrsrsts rstsrsrts rsrγ st rst rsrtsγ rt st rsγ rt srγ st rstγ rs ts rγ st rsγ rt s

42 Artin s coherent presentation The 3-cells Z r,s,t for Coxeter types A 1 A 1 A 1 γ st r str sγ rt srt γ rs t tsr Z r,s,t rst tγ rs trs γ rt s rts rγ st

43 Artin s coherent presentation The 3-cells Z r,s,t for Coxeter type H 3 srstsrsrstsrsrt srtstrsrtstrsrt srtsrtstrsrtstr srtsrstsrsrstsr srtsrstrsrsrtsr strsrsrtsrsrtsr stsrsrstsrsrtsr tstrsrstsrsrtsr tsrtsrstsrstrsr tsrtsrtstrstrsr tsrtstrsrtstrsr tsrstsrsrstsrsr tsrstrsrsrtsrsr tsrsrtsrstrsrsr tsrsrtsrstsrsrs srstrsrsrtsrsrt srsrtsrstrsrsrt srsrtsrstsrsrst srsrtsrtstrsrst Z r,s,t tsrsrtsrtstrsrs tsrsrtstrsrtsrs tsrsrstsrsrtsrs trsrsrtsrsrtsrs srsrtstrsrtsrst srsrstsrsrtsrst rsrsrtsrsrtsrst rsrstrsrsrtsrst rsrstsrsrstsrst rsrtstrsrtstrst rsrtsrtstrsrtst rsrtsrstsrsrsts rsrtsrstrsrsrts rstrsrsrtsrsrts rstsrsrstsrsrts rtstrsrtstrsrts rtsrtstrsrtstrs rtsrstsrsrstsrs rtsrstrsrsrtsrs

44 Artin s coherent presentation The 3-cells Z r,s,t for Coxeter type I 2 (p) A 1, p 3 st rs sγ rt rs p 2 p 1 p 1 ( ) γ sr p t st rs γ rs t t sr p Z r,s,t rs p t tγ rs t rs p rt sr p 1 ( ) γ rt sr p 1 rγ st sr p 2

45 Coherent presentations and actions on categories Definition. (Deligne, 1997) An action T of a monoid M on categories is specified by - a category C = T ( ) - an endofunctor T (u) : C C, for every element u of M, - natural isomorphisms T u,v : T (u)t (v) T (uv) and T : 1 C T (1) satisfying the following coherence conditions: T u,v T (w) T (uv)t (w) T uv,w T (u)t (v)t (w) = T (u)t v,w T (u)t (vw) T u,vw T (uvw) T (1)T (u) T T (u) T 1,u T (u)t (1) T (u)t T u,1 T (u) = T (u) T (u) = T (u)

46 Coherent presentations and actions on categories Theorem. [Gaussent-Guiraud-M., 2013] Let M be a monoid and let Σ be a coherent presentation of M. There is an equivalence of categories Act(M) 2Cat(Σ 2 /Σ 3, Cat) Such equivalence for the Garside s presentation of spherical Artin monoids (Deligne, 1997) Consequence. To determine an action of an Artin group B(W) on a category C, it suffices to attach to any generating 1-cell s S and endofunctor T (s) : C C, to any generating 2-cell an isomorphism of functors such that these satisfy coherence Zamolochikov relations.

47 Other applications Coherent presentation of Garside monoids [Gaussent-Guiraud-Malbos, 2013]. Coherent presentation of plactic and Chinese monoids [Guiraud-Malbos-Mimram, 2013].