Homotopical methods in polygraphic rewriting

Yves Guiraud and Philippe Malbos

Categorical Computer Science, Grenoble, 26/11/2009

References.

• Higher-dimensional categories with finite derivation type, Theory and Applications of Categories, 2009.

• Identities among relations for higher-dimensional rewriting systems, arXiv:0910.4538.

Part I. Two-dimensional Homotopy and String Rewriting

String Rewriting

String Rewriting System : X a set , R⊆ X∗×X∗

ulv →R urvu

oo

l

ZZ

r

¥¥ voo (r, l) ∈ R u,v ∈ X∗

→∗R : reflexive symetrique closure of →R

Terminating :

w0 →R w1 →R · · ·→R wn →R · · ·

Confluentw

∗R

||zzzz

zzzz

∗R

""DD

DDDD

DD

w1

∗R

!!CC

CCCC

CCw2

∗R

}}{{{{

{{{{

w ′

String Rewriting and word problem

Word problem

w,w ′ ∈ X∗, is w = w ′ in X∗/ ↔∗R

↔∗R : derivation.

Normal form algorithm : (X,R) : finite + convergent (terminating + confluent)

w

ÄÄÄÄÄ

""EEE

E w ′

zzvvvv

!!BB

B

ÄÄÄÄÄ ##FFF

Fyyssss

##FFF

Fzzvvv

v&&MMMMM

ÂÂ???

zzvvvv

ÂÂ???

{{xxxx

%%KKKK{{xxx

x$$HHH

Hxxqqqqq

||zzzz

zzzz

##FFF

Fyyssss

##FFF

Fzzvvv

v&&MMMMM

""EEE

EÄÄÄÄÄ !!

BBB

zzvvvv

w?

w ′

Fact. Monoids having a finite convergent presentation are decidable.

First Squier theorem

Rewriting is not universal to decidethe word problem in finite type monoids.

Theorem. (Squier ’87) There are finite type decidable monoids which do not have a finite convergent

presentation.

Proof :

• A monoid M having a finite convergent presentation (X,R) is of homological type FP3.

kerJ−→ ZM[R]J−→ ZM[X]−→ ZM−→ Z

i.e. module of homological 3-syzygies is generated by critical branchings.

• There are finite type decidable monoids which are not of type FP3.

Second Squier Theorem

Theorem. Squier ’87 (’94) The homological finiteness condition FP3 is not sufficient for a finite typedecidable monoid to admit a presentation by a finite convergent rewriting system.

Proof : • (X,R) a string rewriting system.

• S(X,R) Squier 2-dimensional combinatorial complex.

0-cells : words on X, 1-cells : derivations ↔∗R, 2-cells : Peiffer elements

ulwl ′v88

uAwl ′vxxqqqqqqqq ff

ulwA ′v&&MMMMMMMM

urwl ′vff

urwA ′v &&MMMMMMMM ulwr ′v88

uAwr ′vxxqqqqqqqq

urwr ′v

• (X,R) has finite derivation type (FDT) if

X and R are finite and S(X,R) has a finite set of homotopy trivializer.

• Property FDT is Tietze invariant for finite rewriting systems

• A monoid having a finite convergent rewriting system has FDT.

• There are finite type decidable monoids which do not have FDT and which are FP3.

Part II. Two-dimensional Homotopy for higher-dimensional rewritingsystems

Mac Lane’s coherence theorem

monoidal category is made of:

• a category C,

• functors ⊗ : C×C → C and I : ∗→ C,

• three 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

''NNNNN

((x⊗y)⊗z)⊗t

α 77pppppα

// (x⊗y)⊗(z⊗t)α

//

c©x⊗(y⊗(z⊗t))

x⊗(I⊗y)λ##

FFFF

(x⊗I)⊗y

α 99ssssρ

// x⊗y

c©

Mac Lane’s coherence theorem. "In a monoidal category (C,⊗, I,α,λ,ρ), all the diagrams built from C,

⊗, I, α, λ and ρ are commutative."

Program:

– General setting: homotopy bases of track n-categories.

– Proof method: rewriting techniques for presentations of n-categories by polygraphs.

– Algebraic interpretation: identities among relations.

n-categories

An n-category C is made of:

• 0-cells

• 1-cells: xu

//y with one composition

u?0 v = xu

//yv

//z

• 2-cells: x

uÃÃ

v

>>yf ¦¼ with two compositions

f?0 g = x

uÃÃ

u ′>>y

v!!

v ′== zf ¦¼

g¦¼ and f?1 g = x

u

ºº

v //

w

GGy

f ¦¼Â  ÂÂ

g¦¼ÂÂÂÂÂÂ

Exchange relation:

(f?1 g)?0 (h?1 k) = (f?0 h)?1 (g?0 k)

when · ¹¹//HH·

f ¦¼g

¦¼

¹¹//HH·

h ¦¼

k ¦¼

• 3-cells with three compositions ?0, ?1 and ?2, etc.

Track n-categories, cellular extensions and polygraphs

A track n-category is an n-category whose n-cells are invertible (for ?n−1).

A cellular extension of C is a set Γ of (n+1)-cells •f

ÃÃ

g

>>•γ¦¼ with f and g parallel n-cells in C.

C[Γ ] C(Γ) C/Γ

•

f

!!

g

==•γ

¦¼

•

f

!!

g

==•γ

µ&γ−

Rf

≈ • f

g// •

An n-polygraph is a family Σ = (Σ0, . . . ,Σn) where each Σk+1 is a cellular extension of Σ0[Σ1] · · · [Σk].

Free n-category Free track n-category Presented (n−1)-category

Σ∗ = Σ∗n−1[Σn] Σ> = Σ∗n−1(Σn) Σ = Σ∗n−1/Σn

Graphical notations for polygraphs

We draw:

• Generating 2-cells as "circuit components":

. . . . . .

• 2-cells as "circuits":

. . . .

• Generating 3-cells as "rewriting rules":

. V . .V . . V

.

• 3-cells as "rewriting paths":

.V

.V

.V

.

Example : the 2-category of monoids

Let Σ be the 3-polygraph with one 0-cell, one 1-cell, two 2-cells . and . and three 3-cells:

..

_%9. .

._%9 .

..

_%9 .

Proposition. The 2-category Σ is the theory of monoids.

i.e., there is an equivalence:

Monoids (X,×,1) in a 2-category C ↔ 2-functors M : Σ → C

M( .) = X M(. ) = × M( .) = 1

M(. ) = c© M( .) = c© M( .) = c©

Example : the track 3-category of monoidal categories

Let Γ be the cellular extension of Σ∗ with two 4-cells:

.

._%9

. .E»,EEE

EEEE

EEEE

E

.

. y2Fyyyy yyyyyyyy

.SÂ3SSSSSSSSSSSSSS

SSSSSSSSSSSSSS

SSSSSSSSSSSSSS .

..

k+?kkkkkkkkkkkkkk

kkkkkkkkkkkkkk

kkkkkkkkkkkkkk

.

ÄÂ

.

.

9µ&99

9999

99

9999

9999

9999

9999

.

. £6J£££££££

£££££££

£££££££

.

_%9 .

.

ÄÂ

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

Monoidal categories (C,⊗, I,α,λ,ρ) ↔ 3-functors M : Σ>/Γ → Cat

3-category Cat :– one 0-cell, categories as 1-cells, functors as 2-cells, natural transformations as 3-cells

– ?0 is ×, ?1 is the composition of functors, ?2 the vertical composition of natural transformations

The equivalence is given by:

M( .) = C M(. ) = ⊗ M( .) = I

M(. ) = α M( .) = λ M( .) = ρ

M(. ) = c© M(. ) = c©

Homotopy bases and finite derivation type

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

For every n-cells ·f

ÀÀ

g

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

ÀÀ

g

AA ·¦¼ in C(Γ), i.e., f = g in C/Γ .

An n-polygraph Σ has finite derivation type (FDT) if it is finite and if Σ> admits a finite homotopy basis.

Theorem. Let Σ and Υ be finite and Tietze-equivalent n-polygraphs, i.e., Σ ' Υ. Then:

Σ has FDT iff Υ has FDT.

Mac Lane’s theorem revisited. Let Σ be the 3-polygraph(∗, ., . , ., . , ., .

).

Then the cellular extension{

. , .}

of Σ∗ is a homotopy basis of Σ>.

Part III. Computation of homotopy bases

Rewriting properties of an n-polygraph Σ: termination and confluence

A reduction of Σ is a non-identity n-cell uf

//v of Σ∗.

A normal form is an (n−1)-cell u of Σ∗ such that no reduction uf

//v exists.

The polygraph Σ terminates when it has no infinite sequence of reductions u1f1

// u2f2

// u3f3

// (· · ·)Termination ⇒ Existence of normal forms

A branching of Σ is a diagramu

f

ÄÄÄÄÄÄ

ÄÄ g

ÂÂ??

???

wv

in Σ∗.

– It is local when f and g contain exactly one generating n-cell of Σn.

– It is confluent when there exists a diagramv

f ′ ÂÂ??

????

u ′

w

g ′ÄÄÄÄÄÄ

Ä

The polygraph Σ is (locally) confluent when every (local) branching is confluent.

Confluence ⇒ Unicity of normal forms

Rewriting properties of an n-polygraph Σ: convergence

The polygraph Σ is convergent if it terminates and it is confluent.

Theorem [Newman’s lemma]. Termination + local confluence ⇒ Convergence.

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

..

|t© ||||

||||

||||

||||

||||

|||| .

B¹*BB

BBBB

B

BBBB

BBB

BBBB

BBB

. .

..

£v­ ££££

£££

££££

£££

££££

£££

.

9µ&99

9999

99

9999

9999

9999

9999

..

Theorem. Termination + confluence of critical branchings ⇒ Convergence.

The homotopy basis of generating confluences

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

uf}}{{{

{ g

""EEE

E

v

f ′ ÃÃBB

B %9 w

g ′}}{{{{

u ′

with (f,g) critical.

Theorem. Let Σ be a convergent n-polygraph. Let Γ be a cellular extension of Σ∗ made of one generating

confluence for each critical branching of Σ. Then Γ is a homotopy basis of Σ>.

Corollary. If Σ is a finite convergent n-polygraph with a finite number of critical branchings, then it has FDT.

Generating confluences : pear necklaces

Theorem, Squier ’94. If a monoid admits a presentation by a finite convergent word rewriting system, then

it has FDT.

Theorem There exists a 2-category that lacks FDT, even though it admits a presentation by a finite

convergent 3-polygraph.

• A 3-polygraph presenting the 2-category of pear necklaces.

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

four 3-cells :αV

β

Vγ

VδV

• Σ is finite and convergent but does not have FDT.

Generating confluences : pear necklaces

• Four regular critical branching

γ

QÁ2

δ

m,@γδ

δQÁ2

γ

m,@δγ

β _%9

γ

@)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

αy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

γ_%9

αγ

α _%9

δ

@)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

βy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

δ_%9

βδ

• One right-indexed critical branching

k k ∈{

, , ,n}

n

Generating confluences : pear necklaces

β _%9

Peiffγ

)­Á

Peiff

δ

Ä5IÄÄÄÄÄÄÄÄ

ÄÄÄÄÄÄÄÄ

ÄÄÄÄÄÄÄÄ

β _%9

δ~~~ ~~~~~~

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

γ _%9

δ

DDDD

DDD

DDDD

DDD

DDDD

DDD

Dº+DD

DDDD

DD

DDDD

DDDD

DDDD

DDDDα

ÂEY

β¦¼

γhhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhh

h*>hhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhhhhαγ

δVVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVV

VÃ4VVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVVVVβδ

Peiff

α _%9

γ?)??

????

??

????

????

????

????

δ _%9

γ@@

@@@

@ @@@

@)@@

@@@@

@@ @@@@Peiff

γzzzzzzz

zzzzzzz

zzzzzzz

z3Gzzzzzzzz

zzzzzzzz

zzzzzzzz

α_%9

δ

@T

Peiff

Generating confluences : pear necklaces

α _%9

β D»,DDDDDDD

DDDDDDD

DDDDDDD

δVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVV

VVVVVVVVVVVVVVV

VÃ4VVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVV

δ _%9

Peiff

α_%9

βδ

δ_%9

α

¦8L¦¦¦¦¦¦¦¦¦

¦¦¦¦¦¦¦¦¦

¦¦¦¦¦¦¦¦¦

β _%9 γ _%9

β

9µ&99

9999

999

9999

9999

9

9999

9999

9

αz3Gzzzzzzz

zzzzzzz

zzzzzzz

β_%9

γhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhh

hhhhhhhhhhhhhhh

h*>hhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhh

hhhhhhhhhhhhhhhhhαγ

γ_%9

Peiff... n

α

SÂ3

β

k+? ... n+1

.

αβ(

n)

Generating confluences : pear necklaces

• An infinite homotopy base :

γ

QÁ2

δ

m,@γδ

δQÁ2

γ

m,@δγ

β _%9

γ

@)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

αy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

γ_%9

αγ

α _%9

δ

@)@@

@@@@

@@

@@@@

@@@@

@@@@

@@@@

βy2Fyyyyyyyy

yyyyyyyy

yyyyyyyy

δ_%9

βδ

... n

α

SÂ3

β

k+? ... n+1

.

αβ(

n)

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

Generating confluences : Mac Lane’s coherence theorem

• Let Σ be the finite 3-polygraph(∗, ., . , ., . , ., .

).

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

.

._%9

..

B¹*BB

BBBB

B

BBBB

BBB

BBBB

BBB

.

. |4H|||||||

|||||||

|||||||

._%9

. ._%9

.ÄÂ

.

.

.

9µ&99

9999

99

9999

9999

9999

9999

.

. £6J£££££££

£££££££

£££££££

._%9 .

.ÄÂ

.

.

6%

.

©9M.

ÄÂ

.

.

<¶'<<

<<<<

<<<

<<<<

<<<<

<

<<<<

<<<<

<

.

. ~5I~~~~~~~

~~~~~~~

~~~~~~~

._%9 .

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..

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<<<<

<<

<<<<

<<<<

<<<<

<<<<

.

. ~5I~~~~~~~

~~~~~~~

~~~~~~~

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Theorem. The cellular extension{

. , .}

is a homotopy basis of Σ>.

Corollary (Mac Lane’s coherence theorem). "In a monoidal category (C,⊗, I,α,λ,ρ), all the diagrams

built from C, ⊗, I, α, λ and ρ are commutative."

Part IV. Identities among relations

Defining identities among relations

The contexts of an n-category C are the partial maps C : Cn → Cn generated by:

x 7→ f ?i x and x 7→ x ?i f

The category of contexts of C is the category CC with:

– Objects: n-cells of C.

– Morphisms from f to g: contexts C of C such that C[f] = g.

The natural system of identities among relations of an n-polygraph Σ is the functor Π(Σ) : CΣ → Abdefined as follows:

• If u is an (n−1)-cell of Σ, then Π(Σ)u is the quotient of

Z{bfc

∣∣ v fbb in Σ>, v = u}

by (with ? denoting ?n−1):

– bf?gc = bfc+ bgc for every vf""

gbb with v = u.

– bf?gc = bg? fc for every vf &&

wg

ee with v = w = u.

• If C is a context of Σ from u to v, then Cbfc= bB[f]c, with B = C.

Generating identities among relations

A generating set of Π(Σ) is a part X⊆ Π(Σ) such that, for every bfc:

bfc =

k∑

i=1

±Ci[xi], with xi ∈ X, Ci ∈ CΣ.

Proposition. Let Σ and Υ be finite Tietze-equivalent n-polygraphs. Then

Π(Σ) is finitely generated iff Π(Υ) is finitely generated.

Proposition. Let Γ be a homotopy basis of Σ> and Γ ={

γ = f?g−∣∣γ : f → g in Γ

}.

Then⌊Γ⌋

is a generating set for Π(Σ).

3.2. Generating identities among relations

Theorem. If a n-polygraph Σ has FDT, then Π(Σ) is finitely generated.

Proposition. If Σ is a convergent n-polygraph, then Π(Σ) is generated by the generating confluences of Σ.

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

).

It is a finite convergent presentation of the monoid {1,a} with aa = a.

It has one generating confluence:

..

_%9.

Hence the following element generates Π(Σ):

⌊.

⌋=

⌊. ?1

(.

)−⌋

=

.

=⌊

.⌋

=⌊

. .⌋