Higher-dimensional rewriting strategies and acyclic polygraphs Yves Guiraud Philippe Malbos Institut Camille Jordan, Lyon. Tianjin 6 July 2010

Transcription

1 Higher-dimensional rewriting strategies and acyclic polygraphs Yes Girad Philippe Malbos Institt Camille Jordan, Lyon Tianjin 6 Jly 2010

2 1 Introdction

3 11 Motiation Problem: Monoid M as a celllar object One 1-cell x for eery x in M Contract triangles x xy y for eery x, y in M Qestion: Bild a "complete celllar model" ie, free celllar object with the same homotopy type ie, cofibrant replacement (in a model category)

4 11 Motiation Problem: Monoid M as a celllar object One 1-cell x for eery x in M Contract triangles x xy y for eery x, y in M Qestion: Bild a "complete celllar model" ie, free celllar object with the same homotopy type ie, cofibrant replacement (in a model category) Lower dimensions: Use a presentation M = X R x One 1-cell for eery x in X One 2-cell for eery relation = in R Higher dimensions: Use rewriting theory

5 12 Rewriting theory Rewriting: Theory of comptational properties of presentations [The 14, Newman 42] Example: A = a aa = a A = a aa φ a ( ) aaaaa φaaa aφaa aaφa aaaφ aaaa φaa aφa aaφ aaa φa aφ aa φ a

6 12 Rewriting theory Rewriting: Theory of comptational properties of presentations [The 14, Newman 42] Example: A = a aa = a A = a aa φ a ( ) aaaaa φaaa aφaa aaφa aaaφ aaaa φaa aφa aaφ aaa φa aφ aa φ a "Good" comptational properties: Termination: Reach "normal form" a eentally Conflence: Eery path leads to a Conergence (= termination + conflence) Soltion for the word problem [The] Cofibrant replacement for A

7 13 The dimensions of rewriting Dimensions of the cofibrant replacement: 0 1 ( ) n n + 1 n + 2 ( ) n + p Syntactical dimensions Original object Homotopical dimensions Rewriting Syntactical dimensions: Comptads [Street, Batanin, Makkai] Polygraphs [Brroni, Métayer, Girad, Malbos] Homotopical dimensions: General setting for reslts by Sqier, Kobayashi, Otto, etc

8 2 The syntactical dimensions of rewriting

9 21 Presentations of categories by 2-polygraphs Monoid: category with one 0-cell Presentation of category C: Objects and generating morphisms (a graph X) Commtatie diagrams x c a a y b z y b (relations R in free category X with C X /R)

10 21 Presentations of categories by 2-polygraphs Monoid: category with one 0-cell Presentation of category C: Objects and generating morphisms (a graph X) Commtatie diagrams x c a a y b 2-Polygraph Σ: rewriting presentation of a category z y b (relations R in free category X with C X /R) Graph Σ 1 Free category Σ 1 Celllar extension Σ 2 of Σ 1 = Set of 2-cells Presented category: Σ = Σ 1 /Σ 2 φ with, Σ 1 Example: The 2-polygraph As 2 = (, a, aa φ a) is a presentation of A

11 22 Rewriting theory: normal forms and termination Let Σ be a 2-polygraph Rewriting step: w φw w with φ : in Σ 2 Normal form: 1-cell Σ 1 st f Intition: Normal forms sed to represent elements of Σ in Σ 1 Condition: Σ 1! comptable normal form in class Σ

12 22 Rewriting theory: normal forms and termination Let Σ be a 2-polygraph Rewriting step: w φw w with φ : in Σ 2 Normal form: 1-cell Σ 1 st f Intition: Normal forms sed to represent elements of Σ in Σ 1 Condition: Σ 1! comptable normal form in class Σ Termination: infinite seqence of rewriting steps 1 f 1 2 f 2 3 f 3 ( ) f n 1 n f n ( ) Conseqence: Termination Existence of normal forms Example: As 2 = (a, aa φ a) terminates a n a n 1 ( ) a 3 a 2 a

13 23 Rewriting theory: branchings and conflence f 1 ( ) f p p Branching: pair of rewriting seqences g 1 ( ) g q w q Local: p = q = 1 f 1 ( ) Conflent: g 1 ( ) f p p h 1 ( ) h r g q w q k1 ( ) k s

14 23 Rewriting theory: branchings and conflence f 1 ( ) f p p Branching: pair of rewriting seqences g 1 ( ) g q w q Local: p = q = 1 f 1 ( ) f p p h 1 ( ) h r Conflent: g 1 ( ) g q w q k1 ( ) k s φa aa φ Example: a conflent local branching in As 2 aaa a aφ aa φ (Local) conflence: eery (local) branching is conflent Conseqence: Conflence Unicity of normal forms

15 24 Rewriting theory: conergence Conergence: termination + conflence Conergence = Existence and nicity of normal forms Conergence + finiteness = idem + comptability Class FCP: Categories admitting a finite conergent presentation

16 24 Rewriting theory: conergence Conergence: termination + conflence Conergence = Existence and nicity of normal forms Conergence + finiteness = idem + comptability Class FCP: Categories admitting a finite conergent presentation Theorem [Newman s lemma] Termination + Local conflence = Conergence Critical branchings: local branching with a "minimal oerlapping" of 2-cells aaa φa aa a aφ aa Theorem Termination + Conflence of critical branchings = Conergence

17 25 2-Categories 2-category: category enriched in categories Informally: 1-cells: x y with one composition 0 = x y z 2-cells: x f y with two compositions f 0 g = x f y g Exchange relation: (f 1 g) 0 (h 1 k) = (f 0 h) 1 (g 0 k) for z and f 1 g = f g h k x f g w y

18 25 2-Categories 2-category: category enriched in categories Informally: 1-cells: x y with one composition 0 = x y z 2-cells: x f y with two compositions f 0 g = x f y g Exchange relation: (f 1 g) 0 (h 1 k) = (f 0 h) 1 (g 0 k) for z and f 1 g = f g h k x f g w y Example: The free 2-category Σ generated by a 2-polygraph Σ Example: a pro(p) P one 0-cell 1-cells: natral nmbers with composition m 0 n = m + n 2-cells: morphisms of P with compositions f 0 g = f g f 1 g = g f

19 26 Presentations of 2-categories by 3-polygraphs 3-Polygraphs: 2-polygraph Σ 2 Free 2-category Σ 2 Celllar extension Σ 3 of Σ 2 = Set of 3-cells Presented 2-category: Σ = Σ 2 /Σ 3 f ω x g y with f, g Σ 2

20 26 Presentations of 2-categories by 3-polygraphs 3-Polygraphs: 2-polygraph Σ 2 Free 2-category Σ 2 Celllar extension Σ 3 of Σ 2 = Set of 3-cells Presented 2-category: Σ = Σ 2 /Σ 3 f ω x g y with f, g Σ 2 Diagrammatic representations: Generating 2-cells: Composed 2-cells: Generating 3-cells:

21 27 Examples of 3-polygraphs: presentations of pros Associatie algebras: a presentation As 3 of the pro As ( FCP) 0-cell: 1-cell: 2-cell: 3-cell:

22 27 Examples of 3-polygraphs: presentations of pros Associatie algebras: a presentation As 3 of the pro As ( FCP) 0-cell: 1-cell: 2-cell: 3-cell: Monoids: a presentation Mon 3 of the pro Mon ( FCP) 2-cells: 3-cells:

23 27 Examples of 3-polygraphs: presentations of pros Associatie algebras: a presentation As 3 of the pro As ( FCP) 0-cell: 1-cell: 2-cell: 3-cell: Monoids: a presentation Mon 3 of the pro Mon ( FCP) 2-cells: 3-cells: Bialgebras: a presentation of the pro of bialgebras ( FCP) 2-cells: (some) 3-cells:

24 27 Examples of 3-polygraphs: presentations of pros Associatie algebras: a presentation As 3 of the pro As ( FCP) 0-cell: 1-cell: 2-cell: 3-cell: Monoids: a presentation Mon 3 of the pro Mon ( FCP) 2-cells: 3-cells: Bialgebras: a presentation of the pro of bialgebras ( FCP) 2-cells: (some) 3-cells: Braids: a presentation of the pro of braids (FCP: open problem) 2-cells: 3-cells:

25 28 Examples of 3-polygraphs in compter science First-order fnctional program: type nat = 0 S(nat) fonction pls : nat * nat > nat pls(0, n) > n pls(s(m), n) > S(pls(m, n))

26 28 Examples of 3-polygraphs in compter science First-order fnctional program: type nat = 0 S(nat) fonction pls : nat * nat > nat pls(0, n) > n pls(s(m), n) > S(pls(m, n)) Polygraphic ersion: 2-cells: 3-cells: Reslts in program analysis: termination [Girad 06], complexity [Bonfante-Girad 09]

27 28 Examples of 3-polygraphs in compter science First-order fnctional program: type nat = 0 S(nat) fonction pls : nat * nat > nat pls(0, n) > n pls(s(m), n) > S(pls(m, n)) Polygraphic ersion: 2-cells: 3-cells: Reslts in program analysis: termination [Girad 06], complexity [Bonfante-Girad 09] Tring machines [Brroni, Bonfante-Girad 09] Petri nets [Girad 06]

28 29 Presentations of n-categories by polygraphs n-category: category enriched in (n 1)-categories k-cells between parallel (k 1)-cells with k compositions exchange relations

29 29 Presentations of n-categories by polygraphs n-category: category enriched in (n 1)-categories k-cells between parallel (k 1)-cells with k compositions exchange relations (n + 1)-Polygraph Σ: presentation of an n-category n-polygraph Σ n free n-category Σ n celllar extension Σ n+1 of Σ n

30 29 Presentations of n-categories by polygraphs n-category: category enriched in (n 1)-categories k-cells between parallel (k 1)-cells with k compositions exchange relations (n + 1)-Polygraph Σ: presentation of an n-category n-polygraph Σ n free n-category Σ n celllar extension Σ n+1 of Σ n Example: the 4-polygraph Mon 4 is Mon 3 extended with 4-cells Presentation of the 2-pro MonCat of monoidal categories [Girad-Malbos 10]

31 3 The homotopical dimensions of rewriting

32 31 The informal idea Problem: category C "homotopically eqialent" free n-category Σ Presentation of C: 2-polygraph Σ Graph Σ 1 Celllar extension Σ 2 of Σ 1 st C Σ 1 /Σ 2 = pasting, sch as φ ψ ξ with φ, ψ, ξ in Σ 2

33 31 The informal idea Problem: category C "homotopically eqialent" free n-category Σ Presentation of C: 2-polygraph Σ Graph Σ 1 Celllar extension Σ 2 of Σ 1 st C Σ 1 /Σ 2 = pasting, sch as φ ψ ξ with φ, ψ, ξ in Σ 2 α Problem: Another pasting c from to β φ Soltion: Fill the 2-sphere with a 3-cell ξ ψ ω α c β

34 32 Higher presentations 2-fold presentation of C: 3-polygraph Σ 2-polygraph Σ 2 Free 2-category Σ 2 Free track 2-category Σ 2 Presented 1-category Σ f f f

35 32 Higher presentations 2-fold presentation of C: 3-polygraph Σ 2-polygraph Σ 2 Free 2-category Σ 2 Free track 2-category Σ 2 Presented 1-category Σ f f Homotopy basis Σ 3 of Σ 2 = Celllar extension of Σ 2 st f f g in Σ 2 f W g in Σ 3

36 32 Higher presentations 2-fold presentation of C: 3-polygraph Σ 2-polygraph Σ 2 Free 2-category Σ 2 Free track 2-category Σ 2 Presented 1-category Σ f f Homotopy basis Σ 3 of Σ 2 = Celllar extension of Σ 2 st f f g in Σ 2 f W g in Σ 3 n-fold presentation of C: (n + 1)-polygraph Σ (n 1)-fold presentation Σ n Homotopy basis Σ n of Σ n

37 33 The main reslt Theorem [Girad-Malbos 10] If a category C admits a conergent presentation, then C admits an n-fold presentation, for eery n N

38 33 The main reslt Theorem [Girad-Malbos 10] If a category C admits a conergent presentation, then C admits an n-fold presentation, for eery n N Class FDT n : Categories admitting a finite (n + 1)-fold presentation FDT 1 : finite generating graph FDT 0 : finite presentation FDT 1 : finite presentation + homotopy basis = finite deriation type (FDT [Sqier 94])

39 33 The main reslt Theorem [Girad-Malbos 10] If a category C admits a conergent presentation, then C admits an n-fold presentation, for eery n N Class FDT n : Categories admitting a finite (n + 1)-fold presentation FDT 1 : finite generating graph FDT 0 : finite presentation FDT 1 : finite presentation + homotopy basis = finite deriation type (FDT [Sqier 94]) Corollary FCP = FDT n Corollary [Sqier 94] For monoids FCP = FDT

40 34 Key element of the proof: normalisation strategies Normalisation strategy: coherent choice of normal forms in eery dimension In eery class Σ, a representatie 1-cell û Σ 1 For eery 1-cell in Σ 1, a 2-cell σ û in Σ 2

41 34 Key element of the proof: normalisation strategies Normalisation strategy: coherent choice of normal forms in eery dimension In eery class Σ, a representatie 1-cell û Σ 1 For eery 1-cell in Σ 1, a 2-cell σ û in Σ 2 For eery 2-cell f : in Σ 2, a 3-cell in Σ 3 f σ σ f σ û

42 34 Key element of the proof: normalisation strategies Normalisation strategy: coherent choice of normal forms in eery dimension In eery class Σ, a representatie 1-cell û Σ 1 For eery 1-cell in Σ 1, a 2-cell σ û in Σ 2 For eery 2-cell f : in Σ 2, a 3-cell in Σ 3 f σ σ f σ û For eery 3-cell A : f g in Σ 3, a 4-cell in Σ 4 f W g σ σ g σ û f σ σ f σ W σ û Proposition: Normalisation strategies Homotopy bases

43 35 Indction start: the 2-fold presentation of critical branchings Theorem [Girad-Malbos 09, 10]: If Σ is a conergent 2-polygraph then its critical branchings generate a homotopy basis Σ 3 of Σ

44 35 Indction start: the 2-fold presentation of critical branchings Theorem [Girad-Malbos 09, 10]: If Σ is a conergent 2-polygraph then its critical branchings generate a homotopy basis Σ 3 of Σ Example: Let As 2 = ( ) a, aa φ a = (, ) Define σ as the rightmost normalisation strategy: σ a = 1 a σ aa = φ = σ aaa = aφ 1 φ =

45 35 Indction start: the 2-fold presentation of critical branchings Theorem [Girad-Malbos 09, 10]: If Σ is a conergent 2-polygraph then its critical branchings generate a homotopy basis Σ 3 of Σ Example: Let As 2 = ( ) a, aa φ a = (, ) Define σ as the rightmost normalisation strategy: σ a = 1 a σ aa = φ = σ aaa = aφ 1 φ = Define Σ 3 : one critical branching (φa, aφ) (** Conflence **) aaa φa aφ aa σ aa ω a or aa σ aa or Define σ φa = and extend σ to eery 2-cell f (**Termination**)

46 36 Indction: the n-fold presentation of n-fold critical branchings Theorem [Girad-Malbos 10]: If Σ is a conergent 2-polygraph then its n-fold critical branchings form a homotopy basis Σ n+1 of Σ n

47 36 Indction: the n-fold presentation of n-fold critical branchings Theorem [Girad-Malbos 10]: If Σ is a conergent 2-polygraph then its n-fold critical branchings form a homotopy basis Σ n+1 of Σ n Example: As 2 = ( ) [ ] [ ], As 3 = As 2 As 4 = As 3 aaaa aaa aaa aaa

48 36 Indction: the n-fold presentation of n-fold critical branchings Theorem [Girad-Malbos 10]: If Σ is a conergent 2-polygraph then its n-fold critical branchings form a homotopy basis Σ n+1 of Σ n Example: As 2 = ( ) [ ] [ ], As 3 = As 2 As 4 = As 3 aaaa aaa aaa aaa Corollary: If Σ has no n-fold critical branchings then Σ n is "aspherical" ie Σ n is a cofibrant replacement for Σ Corollary: If Σ has n-fold critical branchings for eery n then Σ is a cofibrant replacement for Σ

49 37 Link with homological finiteness condition FP n Class FP n : Categories C sch that the triial C-modle Z admits a projectie resoltion M n M n 1 ( ) M 1 M 0 Z 0

50 37 Link with homological finiteness condition FP n Class FP n : Categories C sch that the triial C-modle Z admits a projectie resoltion M n M n 1 ( ) M 1 M 0 Z 0 Theorem [Girad-Malbos 10]: FDT n = FP n+2 Proof: C[Σ n+2 ] C[Σ n+1 ] C[Σ n ] C[Σ 1 ] C[Σ 0 ] Z 0 Differentials: sorce target Contracting homotopies: normalisation strategy Corollary [Sqier 87]: For monoids FCP = FP 3