[PPT] - Coherence modulo and double groupoids Benjamin Dupont Institut PowerPoint Presentation

SLIDE 1

Coherence modulo and double groupoids

Benjamin Dupont

Institut Camille Jordan, Université Lyon 1 joint work with Philippe Malbos Category Theory 2019 Edinburgh, 11 July 2019

SLIDE 2

Plan

I. Introduction and motivations
II. Double groupoids
III. Polygraphs modulo
IV. Coherence modulo

SLIDE 3

I. Introduction and motivations

SLIDE 4

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ,

SLIDE 5

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies. ,

SLIDE 6

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

,

SLIDE 7

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

,

SLIDE 8

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

,

SLIDE 9

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

,

SLIDE 10

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

◮ If a group G = X | R is presented as a monoid M = X X | R ∪ {xx− αx

→ 1, x−x

αx

→ 1},

SLIDE 11

Motivations: algebraic context

◮ Algebraic rewriting: constructive methods to study algebraic structures presented by generators and relations. ◮ Example. Computation of syzygies.

◮ Squier’s coherence theorem: basis of syzygies from a convergent presentation.

◮ If a group G = X | R is presented as a monoid M = X X | R ∪ {xx− αx

→ 1, x−x

αx

→ 1}, the confluence diagram x =

xx−1x

αx x

xαx

x

is an artefact induced by the algebraic structure and should not be considered as a syzygy.

SLIDE 12

Motivation: objectives

◮ Objective: Study diagrammatic algebras arising in representation theory using algebraic rewriting.

SLIDE 13

Motivation: objectives

◮ Objective: Study diagrammatic algebras arising in representation theory using algebraic rewriting.

◮ Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups; ◮ Temperley-Lieb algebras in statistichal mechanics; ◮ Brauer algebras and Birman-Wenzl algebras in knot theory.

SLIDE 14

Motivation: objectives

◮ Objective: Study diagrammatic algebras arising in representation theory using algebraic rewriting.

◮ Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups; ◮ Temperley-Lieb algebras in statistichal mechanics; ◮ Brauer algebras and Birman-Wenzl algebras in knot theory.

◮ Main questions:

◮ Coherence theorems; ◮ Categorification constructive results; ◮ Computation of linear bases for these algebras by rewriting.

SLIDE 15

Motivation: objectives

◮ Objective: Study diagrammatic algebras arising in representation theory using algebraic rewriting.

◮ Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups; ◮ Temperley-Lieb algebras in statistichal mechanics; ◮ Brauer algebras and Birman-Wenzl algebras in knot theory.

◮ Main questions:

◮ Coherence theorems; ◮ Categorification constructive results; ◮ Computation of linear bases for these algebras by rewriting.

◮ Structural rules of these algebras make the study of local confluence complicated.

SLIDE 16

Motivation: objectives

◮ Objective: Study diagrammatic algebras arising in representation theory using algebraic rewriting.

◮ Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups; ◮ Temperley-Lieb algebras in statistichal mechanics; ◮ Brauer algebras and Birman-Wenzl algebras in knot theory.

◮ Main questions:

◮ Coherence theorems; ◮ Categorification constructive results; ◮ Computation of linear bases for these algebras by rewriting.

◮ Structural rules of these algebras make the study of local confluence complicated. Example: Isotopy relations = =

=
=

SLIDE 17

Example: diagrammatic rewriting modulo isotopy

◮ Let P be the rewriting system on the set of diagrams composed of:

SLIDE 18

Example: diagrammatic rewriting modulo isotopy

◮ Let P be the rewriting system on the set of diagrams composed of: , , , ,

,
,

, .

SLIDE 19

Example: diagrammatic rewriting modulo isotopy

◮ Let P be the rewriting system on the set of diagrams composed of: , , , ,

,
,

, . ◮ submitted to relations:

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ for µ in {0, 1}

→
,
→
,
→
,
→
,

→ , → , → , → , → .

SLIDE 20

Example: diagrammatic rewriting modulo isotopy

◮ Let P be the rewriting system on the set of diagrams composed of: , , , ,

,
,

, . ◮ submitted to relations:

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ for µ in {0, 1}

→
,
→
,
→
,
→
,

→ , → , → , → , → . ◮ If no rewriting modulo:

SLIDE 21

Example: diagrammatic rewriting modulo isotopy

◮ Let P be the rewriting system on the set of diagrams composed of: , , , ,

,
,

, . ◮ submitted to relations:

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ ,

µ

→ •µ for µ in {0, 1}

→
,
→
,
→
,
→
,

→ , → , → , → , → . ◮ If no rewriting modulo:

Not confluent !

SLIDE 22

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

SLIDE 23

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E.

SLIDE 24

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

SLIDE 25

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

◮ Rewriting with R modulo E, Huet ’80. u

R E

u′

R w E

v

R v ′ R w ′

SLIDE 26

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

◮ Rewriting with R modulo E, Huet ’80. u

R E

u′

R w E

v

R v ′ R w ′

◮

E RE : Rewriting with R on E-equivalence classes

SLIDE 27

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

◮ Rewriting with R modulo E, Huet ’80. u

R E

u′

R w E

v

R v ′ R w ′

◮

E RE : Rewriting with R on E-equivalence classes

u

E RE E

v

E

u′

R v ′

SLIDE 28

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

◮ Rewriting with R modulo E, Huet ’80. u

R E

u′

R w E

v

R v ′ R w ′

◮

E RE : Rewriting with R on E-equivalence classes

u

E RE E

v

E

u′

R v ′

◮ Rewriting with any system S such that R ⊆ S ⊆ E RE , Jouannaud - Kirchner ’84.

SLIDE 29

Three paradigms of rewriting modulo

◮ Rewriting system R:

◮ Coherence and confluence results in n-categories.

◮ Rewriting modulo: we consider a rewriting system R and a set of equations E. ◮ Three paradigms of rewriting modulo:

◮ Rewriting with R modulo E, Huet ’80. u

R E

u′

R w E

v

R v ′ R w ′

◮

E RE : Rewriting with R on E-equivalence classes

u

E RE E

v

E

u′

R v ′

◮ Rewriting with any system S such that R ⊆ S ⊆ E RE , Jouannaud - Kirchner ’84.

◮ Main interest and results for E R. u

E R

E

v =

u′

R v

SLIDE 30

II. Double groupoids

SLIDE 31

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.

SLIDE 32

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

SLIDE 33

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

SLIDE 34

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C0)1

(C0)0

SLIDE 35

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C0)1

(C0)0

(C0)1

(C0)0

(C0)0

SLIDE 36

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C1)0 (C0)1

(C0)0

(C0)1

(C0)0 (C1)0

(C0)0

SLIDE 37

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C1)0 (C0)1

(C0)0

(C0)1

(C0)0 (C1)0

(C0)0

(C1)1

SLIDE 38

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C1)0 (C0)1

(C0)0

(C0)1

(C0)0 (C1)0

(C0)0

(C1)1

◮ It gives four related categories

Cvo := (Cv, Co, ∂v

−,0, ∂v +,0, ◦v, iv 0 ),

Cho := (Ch, Co, ∂h

−,0, ∂h +,0, ◦h, ih 0),

Csv := (Cs, Cv, ∂v

−,1, ∂v +,1, ⋄v, iv 1 ),

Csh := (Cs, Ch, ∂h

−,1, ∂h +,1, ⋄h, ih 1),

where Csh is the category C1 and Cvo is the category C0.

SLIDE 39

Double groupoids

◮ We introduce a cubical notion of coherence, in n-categories enriched in double groupoids. ◮ A double category is an internal category (C1, C0, ∂C

−, ∂C +, ◦C, iC) in Cat, Ehresmann ’64.

(C0)0

(C1)0 (C0)1

(C0)0

(C0)1

(C0)0 (C1)0

(C0)0

(C1)1

◮ It gives four related categories

Cvo := (Cv, Co, ∂v

−,0, ∂v +,0, ◦v, iv 0 ),

Cho := (Ch, Co, ∂h

−,0, ∂h +,0, ◦h, ih 0),

Csv := (Cs, Cv, ∂v

−,1, ∂v +,1, ⋄v, iv 1 ),

Csh := (Cs, Ch, ∂h

−,1, ∂h +,1, ⋄h, ih 1),

where Csh is the category C1 and Cvo is the category C0. ◮ Elements of Co: point cells, elements of Ch and Cv: horizontal cells and vertical cells. x1

f

x2

x1

e

x2

SLIDE 40

Double groupoids

◮ Elements of Cs are square cells: ·

∂h

−,1(A)

∂v

−,1(A)

·

∂v

+,1(A)

·

∂h

+,1(A)

·

A

SLIDE 41

Double groupoids

◮ Elements of Cs are square cells: ·

∂h

−,1(A)

∂v

−,1(A)

·

∂v

+,1(A)

·

∂h

+,1(A)

·

A

, with identities

x1

f

iv

0(x1)

x2

iv

0(x2)

x1

f

x2

ih

1(f )

x

ih

0(x)

e

x

e

y

ih

0(y)

y

iv

1(e)

SLIDE 42

Double groupoids

◮ Elements of Cs are square cells: ·

∂h

−,1(A)

∂v

−,1(A)

·

∂v

+,1(A)

·

∂h

+,1(A)

·

A

, with identities

x1

f

iv

0(x1)

x2

iv

0(x2)

x1

f

x2

ih

1(f )

x

ih

0(x)

e

x

e

y

ih

0(y)

y

iv

1(e)

◮ Compositions

x1

f1

e1
x2

e2

f2

x3

e3

y1

g1

y2

A

g2

y3

B

x1

f1◦hf2

e1
x3

e3

y1

g1◦hg2

y3

A⋄v B

for all xi,yi,zi in Co, fi in Ch, ei,e′

i in Cv and A, A′,B in Cs.

SLIDE 43

Double groupoids

◮ Elements of Cs are square cells: ·

∂h

−,1(A)

∂v

−,1(A)

·

∂v

+,1(A)

·

∂h

+,1(A)

·

A

, with identities

x1

f

iv

0(x1)

x2

iv

0(x2)

x1

f

x2

ih

1(f )

x

ih

0(x)

e

x

e

y

ih

0(y)

y

iv

1(e)

◮ Compositions

x1

f1

e1
x2

e2

f2

x3

e3

y1

g1

y2

A

g2

y3

B

x1

f1◦hf2

e1
x3

e3

y1

g1◦hg2

y3

A⋄v B

x1

f1

e1
x2

e2

y1

f2

e′

1

y2

e′

2

A
z1

f3

z2

A′

x1

f1

e1◦v e′

1

x2

e2◦v e′

2

z1

f3

z2

A⋄hA′

for all xi,yi,zi in Co, fi in Ch, ei,e′

i in Cv and A, A′,B in Cs.

SLIDE 44

Double groupoids

◮ These compositions satisfy the middle four interchange law:

SLIDE 45

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

SLIDE 46

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

⋄h

y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

SLIDE 47

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

SLIDE 48

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

SLIDE 49

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

x1

f1

e1

x2

e2

y1

g1

y2

A

⋄v

x2

f2

e2

x3

e3

y2

g2

y3

B

SLIDE 50

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

x1

f1

e1

x2

e2

y1

g1

y2

A

⋄v

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

⋄v

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

SLIDE 51

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

x1

f1

e1

x2

e2

y1

g1

y2

A

⋄v

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

⋄v

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

◮ Double groupoid: double category in which horizontal, vertical and square cells are

invertible.

SLIDE 52

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

x1

f1

e1

x2

e2

y1

g1

y2

A

⋄v

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

⋄v

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

◮ Double groupoid: double category in which horizontal, vertical and square cells are

invertible. ◮ n-category enriched in double groupoids: n-category C such that any homset Cn(x, y) is a double groupoid.

SLIDE 53

Double groupoids

◮ These compositions satisfy the middle four interchange law: x1

f1

e1

x2

e2

y1

g1

y2

A

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

⋄v ⋄h y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

=

x1

f1

e1

x2

e2

y1

g1

y2

A

⋄v

x2

f2

e2

x3

e3

y2

g2

y3

B

⋄h

y1

g1

e′

1

y2

e′

2

z1

h1

z2

A′

⋄v

y2

g2

e′

2

y3

e′

3

z2

h2

z3

B′

◮ Double groupoid: double category in which horizontal, vertical and square cells are

invertible. ◮ n-category enriched in double groupoids: n-category C such that any homset Cn(x, y) is a double groupoid. ◮ Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category of modulo rules.

SLIDE 54

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

SLIDE 55

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

SLIDE 56

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P0 P1

SLIDE 57

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P0 P1

SLIDE 58

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P0 P1

P2

SLIDE 59

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P∗

2

P0 P1

P2

SLIDE 60

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P∗

2

(. . .) P∗

n−1

P∗

n

P0 P1

P2
(. . .)
Pn−1
Pn

SLIDE 61

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P∗

2

(. . .) P∗

n−1

P∗

n

P0 P1

P2
(. . .)
Pn−1
Pn
P⊤

n

SLIDE 62

Polygraphs

◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

◮ An n-polygraph generates a free n-category.

P∗ P∗

1

P∗

2

(. . .) P∗

n−1

P∗

n

P0 P1

P2
(. . .)
Pn−1
Pn
P⊤

n

◮ An (n − 1)-category C is presented by an n-polygraph (P0, . . . , Pn) if C ≃ P∗

n−1/ ≡Pn

SLIDE 63

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

SLIDE 64

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

◮ two (n + 1)-polygraphs Pv and Ph such that Pv

k = Ph k for k ≤ n,

Pv

n+1

P∗

n

Ph

n+1

SLIDE 65

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

◮ two (n + 1)-polygraphs Pv and Ph such that Pv

k = Ph k for k ≤ n,

(Pv

n+1)∗

(Ph

n+1)∗

Pv

n+1

P∗

n

Ph

n+1

SLIDE 66

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

◮ two (n + 1)-polygraphs Pv and Ph such that Pv

k = Ph k for k ≤ n,

◮ a 2-square extension Ps of the pair of (n + 1)-categories ((Pv)∗, (Ph)∗), that is a set equipped with four maps making Γ a 2-cubical set. Ps

(Pv

n+1)∗

(Ph

n+1)∗

Pv

n+1

P∗

n

Ph

n+1

SLIDE 67

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

◮ two (n + 1)-polygraphs Pv and Ph such that Pv

k = Ph k for k ≤ n,

◮ a 2-square extension Ps of the pair of (n + 1)-categories ((Pv)∗, (Ph)∗), that is a set equipped with four maps making Γ a 2-cubical set. Ps

(Pv

n+1)∗

(Ph

n+1)∗

Pv

n+1

P∗

n

Ph

n+1

◮ A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on

((Pv)⊤, (Ph)⊤).

SLIDE 68

Double (n + 2, n)-polygraphs

◮ A double n-polygraph is a data (Pv, Ph, Ps) made of:

◮ two (n + 1)-polygraphs Pv and Ph such that Pv

k = Ph k for k ≤ n,

◮ a 2-square extension Ps of the pair of (n + 1)-categories ((Pv)∗, (Ph)∗), that is a set equipped with four maps making Γ a 2-cubical set. Ps

(Pv

n+1)∗

(Ph

n+1)∗

Pv

n+1

P∗

n

Ph

n+1

◮ A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on

((Pv)⊤, (Ph)⊤). ◮ A double (n + 2, n)-polygraph (Pv, Ph, Ps) generates a free (n − 1)-category enriched in double groupoids, denoted by (Pv, Ph, Ps) | = .

SLIDE 69

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ ·

SLIDE 70

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ · A

there exists a square (n + 1)-cell A in (Pv, Ph, Ps)

| = such that ∂(A) = S.

SLIDE 71

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ · A

there exists a square (n + 1)-cell A in (Pv, Ph, Ps)

| = such that ∂(A) = S. ◮ A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph (Pv, Ph, Ps) such that:

◮ the (n + 1)-polygraph Pv Ph presents C; ◮ Ps is acyclic

SLIDE 72

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ · A

there exists a square (n + 1)-cell A in (Pv, Ph, Ps)

| = such that ∂(A) = S. ◮ A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph (Pv, Ph, Ps) such that:

◮ the (n + 1)-polygraph Pv Ph presents C; ◮ Ps is acyclic

◮ Example: Let E be a convergent (n + 1)-polygraph.

SLIDE 73

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ · A

there exists a square (n + 1)-cell A in (Pv, Ph, Ps)

| = such that ∂(A) = S. ◮ A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph (Pv, Ph, Ps) such that:

◮ the (n + 1)-polygraph Pv Ph presents C; ◮ Ps is acyclic

◮ Example: Let E be a convergent (n + 1)-polygraph. Cd(E) := square extension containing ·

= e1⋆n−1e′

1

·

e2⋆n−1e′

2

·

= ·

for a choice of confluence of any critical branching (e1, e2) of E.

SLIDE 74

Acyclicity

◮ A 2-square extension Ps of ((Pv)⊤, (Ph)⊤) is acyclic if for any square S = ·

(Ph)⊤ (Pv )⊤

·

(Pv )⊤

·

(Ph)⊤ · A

there exists a square (n + 1)-cell A in (Pv, Ph, Ps)

| = such that ∂(A) = S. ◮ A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph (Pv, Ph, Ps) such that:

◮ the (n + 1)-polygraph Pv Ph presents C; ◮ Ps is acyclic

◮ Example: Let E be a convergent (n + 1)-polygraph. Cd(E) := square extension containing ·

= e1⋆n−1e′

1

·

e2⋆n−1e′

2

·

= ·

for a choice of confluence of any critical branching (e1, e2) of E. ◮ From Squier’s theorem, (E, ∅, Cd(E)) is a 2-fold coherent presentation of C.

SLIDE 75

III. Polygraphs modulo

SLIDE 76

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of

SLIDE 77

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of ◮ an n-polygraph R of primary rules,

SLIDE 78

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of ◮ an n-polygraph R of primary rules, ◮ an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,

SLIDE 79

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of ◮ an n-polygraph R of primary rules, ◮ an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules, ◮ S is a cellular extension of R∗

n−1 such that R ⊆ S ⊆ E RE ,

SLIDE 80

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of ◮ an n-polygraph R of primary rules, ◮ an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules, ◮ S is a cellular extension of R∗

n−1 such that R ⊆ S ⊆ E RE , where the cellular extension E RE is defined by

γ E RE : E RE → Sphn−1(R∗

n−1)

where E RE is the set of triples (e, f , e′) in E ⊤ × R∗(1) × E ⊤ such that u

v

e

f
e′

SLIDE 81

Polygraphs modulo

A n-polygraph modulo is a data (R, E, S) made of ◮ an n-polygraph R of primary rules, ◮ an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules, ◮ S is a cellular extension of R∗

n−1 such that R ⊆ S ⊆ E RE , where the cellular extension E RE is defined by

γ E RE : E RE → Sphn−1(R∗

n−1)

where E RE is the set of triples (e, f , e′) in E ⊤ × R∗(1) × E ⊤ such that u

v

e

f
e′
and the map γ E RE is defined by γ E RE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).

SLIDE 82

Branchings and confluence modulo

◮ A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g are in S∗

n and e is in E ⊤ n , such that:

u

f

e

u′ v

g v′

SLIDE 83

Branchings and confluence modulo

◮ A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g are in S∗

n and e is in E ⊤ n , such that:

u

f

e

u′ v

g v′

◮ It is local if f is in S∗(1)

n

, g is in S∗

n and e in E ⊤ n

such that ℓ(g) + ℓ(e) = 1.

SLIDE 84

Branchings and confluence modulo

◮ A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g are in S∗

n and e is in E ⊤ n , such that:

u

f

e

u′ v

g v′

◮ It is local if f is in S∗(1)

n

, g is in S∗

n and e in E ⊤ n

such that ℓ(g) + ℓ(e) = 1. ◮ It is confluent modulo E if there exists f ′, g′ in S∗

n and e′ in E ⊤ n :

u

f

e

u′

f ′ w e′

v

g v′ g′ w′

SLIDE 85

Branchings and confluence modulo

◮ A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g are in S∗

n and e is in E ⊤ n , such that:

u

f

e

u′ v

g v′

◮ It is local if f is in S∗(1)

n

, g is in S∗

n and e in E ⊤ n

such that ℓ(g) + ℓ(e) = 1. ◮ It is confluent modulo E if there exists f ′, g′ in S∗

n and e′ in E ⊤ n :

u

f

e

u′

f ′ w e′

v

g v′ g′ w′

◮ Confluence modulo E (resp. local confluence modulo E): any branching (resp. local branching) of S modulo E is confluent modulo E.

SLIDE 86

IV. Coherence modulo

SLIDE 87

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗).

SLIDE 88

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗). ◮ A branching modulo E is Γ-confluent modulo E if there exist f ′, g′ in S∗

n , e′ in E ⊤ n

u

f

e

u′

f ′ w e′

v

g v′ g′ w′

SLIDE 89

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗). ◮ A branching modulo E is Γ-confluent modulo E if there exist f ′, g′ in S∗

n , e′ in E ⊤ n

and a square-cell A in (E, S, E ⋊ Γ ∪ Peiff(E, S)) | =

,v:

u

f

e

u′

f ′ A

w

e′

v

g v′ g′ w′

SLIDE 90

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗). ◮ A branching modulo E is Γ-confluent modulo E if there exist f ′, g′ in S∗

n , e′ in E ⊤ n

and a square-cell A in (E, S, E ⋊ Γ ∪ Peiff(E, S)) | =

,v:

u

f

e

u′

f ′ A

w

e′

v

g v′ g′ w′

◮ (E, S, −) | =

,v is the free n-category enriched in double categories generated by (E, S, −), in

which all vertical cells are invertible.

SLIDE 91

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗). ◮ A branching modulo E is Γ-confluent modulo E if there exist f ′, g′ in S∗

n , e′ in E ⊤ n

and a square-cell A in (E, S, E ⋊ Γ ∪ Peiff(E, S)) | =

,v:

u

f

e

u′

f ′ A

w

e′

v

g v′ g′ w′

◮ (E, S, −) | =

,v is the free n-category enriched in double categories generated by (E, S, −), in

which all vertical cells are invertible. ◮ Peiff(E, S) is the 2-square extension containing the following squares for all e,e′ ∈ E ⊤ and f ∈ S∗. u ⋆i v

f ⋆i v u⋆i e

u′ ⋆i v

u′⋆i e

u ⋆i v ′

f ⋆i v′ u′ ⋆i v ′

w ⋆i u

w⋆i f e′⋆i u

w ⋆i u′

e′⋆i u′

w ′ ⋆i u

w′⋆i f w ′ ⋆i u′

SLIDE 92

Coherent confluence modulo

◮ We consider Γ a 2-square extension of (E ⊤, S∗). ◮ A branching modulo E is Γ-confluent modulo E if there exist f ′, g′ in S∗

n , e′ in E ⊤ n

and a square-cell A in (E, S, E ⋊ Γ ∪ Peiff(E, S)) | =

,v:

u

f

e

u′

f ′ A

w

e′

v

g v′ g′ w′

◮ (E, S, −) | =

,v is the free n-category enriched in double categories generated by (E, S, −), in

which all vertical cells are invertible. ◮ Peiff(E, S) is the 2-square extension containing the following squares for all e,e′ ∈ E ⊤ and f ∈ S∗. u ⋆i v

f ⋆i v u⋆i e

u′ ⋆i v

u′⋆i e

u ⋆i v ′

f ⋆i v′ u′ ⋆i v ′

w ⋆i u

w⋆i f e′⋆i u

w ⋆i u′

e′⋆i u′

w ′ ⋆i u

w′⋆i f w ′ ⋆i u′

◮ E ⋊ Γ is to avoid "redundant" elements in Γ for different squares corresponding to the same branching of S modulo E: u

f

e

v

f ′

v ′

e′

u′

g=e1g1e2 w g′

w ′

and u

f

e⋆n−1e1

v

f ′

v ′

e′

u1

g1e2 w g′

w ′

SLIDE 93

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E.

SLIDE 94

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

SLIDE 95

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E;

SLIDE 96

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E;

SLIDE 97

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

SLIDE 98

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

◮ S satisfies properties a) and b) for any critical branching of S modulo E.

SLIDE 99

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

◮ S satisfies properties a) and b) for any critical branching of S modulo E.

◮ For S = E R, property b) is trivially satisfied.

SLIDE 100

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

◮ S satisfies properties a) and b) for any critical branching of S modulo E.

◮ For S = E R, property b) is trivially satisfied. u

f e

v v′

SLIDE 101

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

◮ S satisfies properties a) and b) for any critical branching of S modulo E.

◮ For S = E R, property b) is trivially satisfied. u

f e

v v′

e−·f

v

SLIDE 102

Coherent Newman and critical pair lemmas

◮ S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branching modulo E (resp. local branching modulo E) is Γ-confluent modulo E. ◮ Theorem. [D.-Malbos ’18] If E RE is terminating, the following assertions are equivalent:

◮ S is Γ-confluent modulo E; ◮ S is locally Γ-confluent modulo E; ◮ S satisfies properties a) and b): a): u

S∗(1)

=

v

S∗ v ′ E⊤

u

R∗(1) w S∗ w ′ A

b):

u

S∗(1) E⊤(1)

v

S∗ v ′ E⊤

u′

S∗

w

B

for any local branching of S modulo E.

◮ S satisfies properties a) and b) for any critical branching of S modulo E.

◮ For S = E R, property b) is trivially satisfied. u

f e

v =

v′

e−·f

v

SLIDE 103

Coherence modulo

◮ A set X of (n − 1)-cells in R∗

n−1 is E-normalizing with respect to S if for any u in X,

NF(S, u) ∩ Irr(E) = ∅.

SLIDE 104

Coherence modulo

◮ A set X of (n − 1)-cells in R∗

n−1 is E-normalizing with respect to S if for any u in X,

NF(S, u) ∩ Irr(E) = ∅. ◮ Theorem. [D.-Malbos ’18] Let (R, E, S) be an n-polygraph modulo, and Γ be a square extension of (E ⊤, S∗) such that

◮ E is convergent, ◮ S is Γ-confluent modulo E, ◮ Irr(E) is E-normalizing with respect to S, ◮

E RE is terminating,

then E ⋊ Γ ∪ Peiff(E, S) ∪ Cd(E) is acyclic.

SLIDE 105

Coherent completion

◮ Coherent completion modulo E of S: square extension of (E ⊤, S⊤) containing square cells Af ,g and Bf ,e: u

f

=
u′

f ′ Af ,g

w

e′

u

g v g′ w′

u

f

e

u′

f ′ Bf ,e

w

e′

v

g′

w′

for any critical branchings (f , g) and (f , e) of S modulo E.

SLIDE 106

Coherent completion

◮ Coherent completion modulo E of S: square extension of (E ⊤, S⊤) containing square cells Af ,g and Bf ,e: u

f

=
u′

f ′ Af ,g

w

e′

u

g v g′ w′

u

f

e

u′

f ′ Bf ,e

w

e′

v

g′

w′

for any critical branchings (f , g) and (f , e) of S modulo E. ◮ Corollary. [D.-Malbos ’18] Let (R, E, S) be an n-polygraph modulo such that

◮ E is convergent, ◮ S is confluent modulo E, ◮ Irr(E) is E-normalizing with respect to S, ◮

E RE is terminating,

For any coherent completion Γ of S modulo E, E ⋊ Γ ∪ Peiff(E, S) ∪ Cd(E) is acyclic.

SLIDE 107

Coherent completion

◮ Coherent completion modulo E of S: square extension of (E ⊤, S⊤) containing square cells Af ,g and Bf ,e: u

f

=
u′

f ′ Af ,g

w

e′

u

g v g′ w′

u

f

e

u′

f ′ Bf ,e

w

e′

v

g′

w′

for any critical branchings (f , g) and (f , e) of S modulo E. ◮ Corollary. [D.-Malbos ’18] Let (R, E, S) be an n-polygraph modulo such that

◮ E is convergent, ◮ S is confluent modulo E, ◮ Irr(E) is E-normalizing with respect to S, ◮

E RE is terminating,

For any coherent completion Γ of S modulo E, E ⋊ Γ ∪ Peiff(E, S) ∪ Cd(E) is acyclic. ◮ Corollary: Usual Squier’s theorem. (E = ∅)

SLIDE 108

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

SLIDE 109

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗},

SLIDE 110

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨},

SLIDE 111

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨}, ◮ E2 =      , , , ,

,


   

SLIDE 112

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨}, ◮ E2 =      , , , ,

,


    ◮ E3 =     

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ for µ in {0, 1} ,

⇛
,
⇛
,
⇛
,
⇛

SLIDE 113

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨}, ◮ E2 =      , , , ,

,


    R2 = E2



    ,      ◮ E3 =     

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ for µ in {0, 1} ,

⇛
,
⇛
,
⇛
,
⇛

SLIDE 114

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨}, ◮ E2 =      , , , ,

,


    R2 = E2



    ,      ◮ E3 =     

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ for µ in {0, 1} ,

⇛
,
⇛
,
⇛
,
⇛
◮ R3 =

     ⇛ , ⇛ , ⇛ , ⇛ , ⇛     

SLIDE 115

Example: diagrammatic rewriting modulo isotopy

◮ Let E and R be two 3-polygraphs defined by:

◮ E0 = R0 = {∗}, ◮ E1 = R1 = {∧, ∨}, ◮ E2 =      , , , ,

,


    R2 = E2



    ,      ◮ E3 =     

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ ,

µ

⇛ •µ for µ in {0, 1} ,

⇛
,
⇛
,
⇛
,
⇛
◮ R3 =

     ⇛ , ⇛ , ⇛ , ⇛ , ⇛     

◮ Facts:

◮ E is convergent. ◮

E RE is terminating.

◮

E R is confluent modulo E.

SLIDE 116

Conclusion

◮ We proved a coherence result for polygraphs modulo.

SLIDE 117

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ?

SLIDE 118

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ? ◮ Is any polygraph modulo Tietze-equivalent to an E-normalizing polygraph modulo ?

SLIDE 119

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ? ◮ Is any polygraph modulo Tietze-equivalent to an E-normalizing polygraph modulo ?

◮ Explicit a quotient of a square extension by all modulo rules. ◮ Constructions extended to the linear setting.

◮ Linear bases from termination (or quasi-termination) or E RE and confluence of R modulo E.

SLIDE 120

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ? ◮ Is any polygraph modulo Tietze-equivalent to an E-normalizing polygraph modulo ?

◮ Explicit a quotient of a square extension by all modulo rules. ◮ Constructions extended to the linear setting.

◮ Linear bases from termination (or quasi-termination) or E RE and confluence of R modulo E.

◮ Work in progress:

SLIDE 121

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ? ◮ Is any polygraph modulo Tietze-equivalent to an E-normalizing polygraph modulo ?

◮ Explicit a quotient of a square extension by all modulo rules. ◮ Constructions extended to the linear setting.

◮ Linear bases from termination (or quasi-termination) or E RE and confluence of R modulo E.

◮ Work in progress:

◮ Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.

SLIDE 122

Conclusion

◮ We proved a coherence result for polygraphs modulo.

◮ How to weaken E-normalization assumption ? ◮ Is any polygraph modulo Tietze-equivalent to an E-normalizing polygraph modulo ?

◮ Explicit a quotient of a square extension by all modulo rules. ◮ Constructions extended to the linear setting.

◮ Linear bases from termination (or quasi-termination) or E RE and confluence of R modulo E.

◮ Work in progress:

◮ Rise this construction in dimensions, in n-categories enriched in p-fold groupoids. ◮ Formalize these constructions with rewriting modulo all the algebraic axioms.

SLIDE 123