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Identities among relations for polygraphic rewriting July 6, 2010 References: (P. M., Y. Guiraud) Identities among relations for higher-dimensional rewriting systems, arXiv:0910.4538, to appear; (P. M., Y. Guiraud) Coherence in monoidal track categories, arXiv:1004.1055, submitted; (P. M., Y. Guiraud) Higher-dimensional categories with finite derivation type, Theory and Applications of Categories, 2009;

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Page 1: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Identities among relations for polygraphic rewriting

July 6, 2010

References:

• (P. M., Y. Guiraud) Identities among relations for higher-dimensional rewriting systems, arXiv:0910.4538,

to appear;

• (P. M., Y. Guiraud) Coherence in monoidal track categories, arXiv:1004.1055, submitted;

• (P. M., Y. Guiraud) Higher-dimensional categories with finite derivation type, Theory and Applications of

Categories, 2009;

Page 2: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Contents

Two-dimensional homotopy for polygraphs

- Polygraphs and higher track categories

- Critical branchings in 3-polygraphs

Identities among relations for polygraphs

- Abelian track extensions

- Generating identities among relations

Page 3: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Two-dimensional homotopy for polygraphs

Page 4: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Track n-categories and homotopy bases

Definitions.

- A track n-category is an (n−1)-category enriched in groupoids.

- A homotopy basis of an n-category C is a cellular extension Γ such that the track n-category C/Γ

is aspherical

i.e., for every n-spheres ·f

��

g

AA · in C, there exists an (n+1)-cell ·f

��

g

AA ·�� in C(Γ),

i.e., f= g in C/Γ .

Definition. An n-polygraph Σ has finite derivation type (FDT) if

i) Σ is finite,

ii) the free track n-category Σ> = Σ∗n−1(Σn) admits a finite homotopy basis.

Proposition. The property FDT is Tietze invariant for finite polygraphs.

Page 5: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Critical branchings

• A branching is critical when it is a "minimal overlapping" of n-cells, such as:

..

_%9.

..

|t� ||||

||||

||||

||||

||||

|||| .

B�*BB

BBBB

B

BBBB

BBB

BBBB

BBB

. .

• Newman’s diamond lemma (’42) :

Termination + confluence of critical branchings ⇒ Convergence

Page 6: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

The homotopy basis of generating confluences

• A generating confluence of an n-polygraph Σ is an (n+1)-cell

uf

~~||||

|| g

!!CC

CCCC

v

f ′ @@

@@@@

⇒ w

g ′~~}}}}

}}

u ′

with (f,g) critical branching.

Theorem. Let Σ be a convergent n-polygraph.

A cellular extension of Σ∗ made of one generating confluence for each critical branching of Σ is a homotopy

basis of Σ>.

Page 7: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example I : Mac Lane’s coherence theorem

• Monoidal category (C,⊗, I,α,λ,ρ) with natural isomorphisms

αx,y,z : (x⊗y)⊗z∼−→ x⊗ (y⊗z) λx : I⊗x

∼−→ x ρx : x⊗ I∼−→ x

such that the following diagrams commute:

(x⊗(y⊗z))⊗tα

// x⊗((y⊗z)⊗t)

α

""DD

DDDD

DD

((x⊗y)⊗z)⊗t

α<<zzzzzzzz

α// (x⊗y)⊗(z⊗t)

α//

x⊗(y⊗(z⊗t))

x⊗(I⊗y)

λ

��88

8888

8

(x⊗I)⊗y

α??��������

ρ// x⊗y

Theorem. (Mac Lane’s coherence theorem ’63)

In a monoidal category, all the diagrams built from C, ⊗, I, α, λ and ρ are commutative.

Page 8: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example I : Mac Lane’s coherence theorem

• Let Mon be the 3-polygraph:

- one 0-cell, one 1-cell, two 2-cell : . , .,

- three 3-cells :

..

_%9. .

._%9 .

..

_%9 .

Lemma. Mon terminates and is locally confluent, with the following five generating confluences:

.

._%9

..

B�*BB

BBBB

B

BBBB

BBB

BBBB

BBB

.

. |4H|||||||

|||||||

|||||||

._%9

. ._%9

.��

.

.

.

9�&99

9999

99

9999

9999

9999

9999

.

. �6J�������

�������

�������

._ %9 .

.��

.

.

6�%

.

�9M.

��

.

.

<�'<<

<<<<

<<<

<<<<

<<<<

<

<<<<

<<<<

<

.

. ~5I~~~~~~~

~~~~~~~

~~~~~~~

._%9 .

��

..

<�'<<

<<<<

<<

<<<<

<<<<

<<<<

<<<<

.

. ~5I~~~~~~~

~~~~~~~

~~~~~~~

._%9 .

��

Page 9: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example I : Mac Lane’s coherence theorem

• Let Γ ={

. , .}

be the cellular extension of Mon∗ with :

.

._%9

. .E�,EEE

EEEE

EEEE

E

.

. y2Fyyyy yyyyyyyy

.S�3SSSSSSSSSSSSSS

SSSSSSSSSSSSSS

SSSSSSSSSSSSSS .

..

k+?kkkkkkkkkkkkkk

kkkkkkkkkkkkkk

kkkkkkkkkkkkkk

.

��

.

.

9�&99

9999

99

9999

9999

9999

9999

.

. �6J�������

�������

�������

.

_%9 .

.

��

• The track 3-category Mon>/Γ is the theory of monoidal categories, i.e., there is an equivalence:

Monoidal categories ↔ 3-functors Mon>/Γ −→ Cat

Theorem. The 3-category Mon>/Γ is aspherical.

Corollary : Mac Lane’s coherence theorem.

Page 10: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example II : polygraph of permutations

• Perm 3-polygraph of permutations :

- one 0-cell, one 1-cell, one 2-cell :

- two 3-cells :αV and

β

V

Theorem. Perm is finite, convergent and has FDT.

• Perm has an infinite set of critical branchings.

• three regular critical branchings

• a right-indexed critical branching

k

Page 11: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example II : polygraph of permutations

• regular critical branchings are confluent

α

8�&

α

�8Lαα

α _%9

β 9�&99

9999

99

9999

9999

9999

9999

β_ %9

αβ α

�8L���������

���������

���������

β _%9

α9�&99

9999

999

9999

9999

9

9999

9999

�8L��������

��������

��������

α_%9

βα

• right-indexed have two normal instances

α _%9

α

5�$55

5555

5555

55

5555

5555

5555

5555

5555

5555

β�:N��������

��������

��������

β 6�$66

6666

66

6666

6666

6666

6666

α_%9

α

:N

ββ( )

β _%9 β _%9

β

9�&99

9999

99

9999

9999

9999

9999

β�8L��������

��������

��������

β 9�&99

9999

99

9999

9999

9999

9999

β_%9

β_%9

β

�8L��������

��������

��������

ββ( )

•{αα, αβ, βα, ββ

( ), ββ

( )}is a homotopy base.

Page 12: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Generating confluences and finite derivation type

• Generating confluences of a convergent n-polygraph Σ form a homotopy basis of Σ>.

Corollary. A finite convergent n-polygraph with a finite number of critical branchings has FDT.

Corollary. (Squier ’94) A finite convergent 2-polygraph has FDT.

- Two shapes of branching in a 2-polygraph :

Regular critical branching Inclusion critical branching

//DD//

��//

EY

��

// !!//FF

//

EY

��

Theorem. For n≥ 3, there exist finite convergent n-polygraphs without FDT.

Page 13: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Critical branchings in 3-polygraphs

• Regular critical branching

g

vv

u

sα=

g

v

u

f

h =sβ

u

vf

• Inclusion critical branching

sα = sβ

C

.

• Right-indexed critical branching

k

g

= k

f

h

g= k

f

;

Page 14: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Critical branchings in 3-polygraphs

• Regular

g

vv

u

sα=

g

v

u

f

h =sβ

u

vfsα

gu v=

f

u h

gvv =

sβu vv

f

g u

vvsα

= h

gu

v f

=fv

sβu

g

u vsα=

u vf

g

h =

v

fu

• Inclusion

sα = sβ

C

• Indexed

k

g

= k

f

h

g= k

f

; k

g

= k

g

f

h = k

f

k0 kn

g

kpk1 = knk0 h1 k1 h2

g

f

hnhp kp =f

knk0

kpk1

Page 15: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Critical branchings in 3-polygraphs

Theorem. Let Σ be a finite convergent 3-polygraph.

i) If Σ does not have indexed critical branching it has FDT.

ii) If Σ has indexed critical branchings with finitely many normal instances it has FDT.

C

tA fF _%9

Peiff

C

tA g

A ′

<�(<<

<<<<

<<<

<<<<

<<<<

<

<<<<

<<<<

<

C

sA

sBf=

A~4H~~~~~~~

~~~~~~~

~~~~~~~

B @�*@@

@@@@

@

@@@@

@@@

@@@@

@@@

F _%9

C

sBg

sA=

A~4H~~~~~~~

~~~~~~~

~~~~~~~

B @�*@@

@@@@

@

@@@@

@@@

@@@@

@@@

gen.conf.C

h

.

C

tB fF _%9

Peiff

C

tB g

B ′

�6J���������

���������

���������

Page 16: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example III : pearl necklaces

• Pearl 3-polygraph presenting the 2-category of pearl necklaces:

- one 0-cell, one 1-cell, three 2-cells :

- four 3-cells :αV

β

VδV

• Examples of 2-cells

Proposition. Pearl is finite and convergent but does not have FDT.

Page 17: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example III : pearl necklaces

• Four regular critical branching

γQ�2

δ

m,@γδ

δQ�2

γ

m,@δγ

β _ %9

γ

@�)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

αy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

γ_%9

αγ

α _ %9

δ

@�)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@β y2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

δ_%9

βδ

• One right-indexed critical branching

k k ∈{

, , ,n}

n

Page 18: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example III : pearl necklaces

β _%9

Peiffγ

! ��

Peiff

δ�5I��������

��������

��������β _%9

δ~5I~~~~~~~~

~~~~~~~~

~~~~~~~~γ _%9

δ

D�+DD

DDDD

DDDD

DDDD

D

DDDD

DDDD

DDDD

DDD

DDDD

DDDD

DDDD

DDD

α

�EY

β ���

γ

h*>hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

αγ

δ

V 4VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

βδ

Peiff

α _%9

γ?�)??

????

??

????

????

????

????

δ _%9

γ

@�)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

Peiff

γ

z3Gzzzzzzzzzzzzzzz

zzzzzzzzzzzzzzz

zzzzzzzzzzzzzzz

α_%9

δ

� DX

Peiff

Page 19: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example III : pearl necklaces

α _%9

β D�,DDDDDDD

DDDDDDD

DDDDDDDδ

V 4VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVδ _ %9

Peiff

α_%9

βδ

δ_%9

α

�8L���������

���������

���������

β _%9 γ _%9

β

9�&99

9999

999

9999

9999

9

9999

9999

9

αz3Gzzzzzzz

zzzzzzz

zzzzzzz

β_%9

γ

h*>hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

αγ

γ_ %9

Peiff... n

αS�3

β

k+?

... n+1

.

αβ(

n)

Page 20: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Example III : pearl necklaces

γQ�2

δ

m,@γδ

δQ�2

γ

m,@δγ

β _ %9

γ

@�)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

αy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

γ_%9

αγ

α _ %9

δ

@�)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@β y2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

δ_%9

βδ

... n

αS�3

β

k+?

... n+1

.

αβ(

n)

• An infinite homotopy base :{γδ, δγ, αγ, βδ, αβ

(n), n ∈ N

}.

• The 3-polygraph Pearl is finite and convergent but does not have finite derivation type

Page 21: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Identities among relations for polygraphs

Page 22: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

The special case of groups

• (Peiffer-Reidemester-Whitehead ’49) the free crossed module

∂ : C(P )−→ F(X)

on a presentation P = (X,R) of a group G.

• Whitehead’s lemma (’49) : the following diagram commutes

ker∂ // C(P )∂

//

( )ab��

F(X) //

[ ]��

G

kerJ // 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 generated.

Corollary. A finitely presented group is of type FP3 if and only if it has FDT.

Page 23: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Identities among relations

• Let C be a n-category presented by a (n+1)-polygraph Σ.

• Let Σ>ab be the Abelianised free (n+1)-track category on Σ.

Σ>abn+1

s//

t// Σ

∗n

// // C

Theorem A track category is Abelian if and only if it is linear.

• There exists a C-module Π(Σ) of identities among relations of Σ and an isomorphism of

Σ∗n−1-modules

Π̃(Σ)' AutΣ>ab

• Π(Σ) is defined as follows:

- for an n-cell u of C, then Π(Σ)(u) is the quotient of

Z{bfc

∣∣ v fbb in Σ>ab , v= u}

by :

– bf?n gc = bfc+ bgc for every vf"" gbb with v= u

– bf?n gc = bg?n fc for every vf ''

wg

ff with v=w= u

Page 24: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Generating identities among relations

Theorem. Any homotopy basis of the free track category Σ> forms a generating set for the module Π(Σ).

Corollary. If a polygraph Σ has FDT, then the module Π(Σ) is finitely generated.

Corollary. If Σ is convergent, the module Π(Σ) is generated by the generating confluences of Σ.

Example. Let As be the 2-polygraph(∗, ., .

).

- As is a finite convergent presentation of the monoid 〈a | aa= a〉.

- As has one generating confluence:

..

_ %9.

- Hence the following element generates Π(Σ):

⌊.

⌋=

⌊. ?1

(.

)−1⌋=

.

=⌊

.⌋=

⌊.

Page 25: Identities among relations for polygraphic rewritingmath.univ-lyon1.fr/homes-www/malbos/Pres/identamongrelpolygraph_10.pdfIdentities among relations for polygraphic rewriting ... DD

Finite Abelian derivation type

Definition. An n-polygraph Σ has finite Abelian derivation type (TDFab) if

i) Σ is finite,

ii) Σ>ab admits a finite homotopy basis.

• TDF implies TDFab,

Proposition. Let C be an n-category presented a polygraph Σ, then Σ has TDFab if and only if the

C-module Π(Σ) is finitely generated.

Corollary. A 1-category has TDFab if and only if it is of homological type FP3.