institut camille jordan, université lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf ·...
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![Page 1: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/1.jpg)
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
![Page 2: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/2.jpg)
Plan
I. Introduction and motivations
II. Double groupoids
III. Polygraphs modulo
IV. Coherence modulo
![Page 3: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/3.jpg)
I. Introduction and motivations
![Page 4: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/4.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 5: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/5.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 6: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/6.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 7: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/7.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
00
..
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 8: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/8.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
��
00
..
HH
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 9: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/9.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
��
00
..
HH��
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1}
,
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 10: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/10.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
��
00
..
HH��
I If a group G = 〈X | R〉 is presented as a monoid M = 〈X∐
X | R ∪ {xx− αx→ 1, x−x αx→ 1},
the confluence diagramx
=
xx−1x
αx x 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 11: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/11.jpg)
Motivations: algebraic context
I Algebraic rewriting: constructive methods to study algebraic structures presented bygenerators and relations.
I Example. Computation of syzygies.
I Squier’s coherence theorem: basis of syzygies from a convergent presentation.
��
00
..
HH��
I 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 11
xαx -- x
is an artefact induced by the algebraic structure and should not be considered as a syzygy.
![Page 12: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/12.jpg)
Motivation: objectives
I Objective: Study diagrammatic algebras arising in representation theory using algebraicrewriting.
I Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups;
I Temperley-Lieb algebras in statistichal mechanics;
I Brauer algebras and Birman-Wenzl algebras in knot theory.
I Main questions:
I Coherence theorems;
I Categorification constructive results;
I Computation of linear bases for these algebras by rewriting.
I Structural rules of these algebras make the study of local confluence complicated.
Example: Isotopy relations
= = • = • = •
![Page 13: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/13.jpg)
Motivation: objectives
I Objective: Study diagrammatic algebras arising in representation theory using algebraicrewriting.
I Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups;
I Temperley-Lieb algebras in statistichal mechanics;
I Brauer algebras and Birman-Wenzl algebras in knot theory.
I Main questions:
I Coherence theorems;
I Categorification constructive results;
I Computation of linear bases for these algebras by rewriting.
I Structural rules of these algebras make the study of local confluence complicated.
Example: Isotopy relations
= = • = • = •
![Page 14: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/14.jpg)
Motivation: objectives
I Objective: Study diagrammatic algebras arising in representation theory using algebraicrewriting.
I Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups;
I Temperley-Lieb algebras in statistichal mechanics;
I Brauer algebras and Birman-Wenzl algebras in knot theory.
I Main questions:
I Coherence theorems;
I Categorification constructive results;
I Computation of linear bases for these algebras by rewriting.
I Structural rules of these algebras make the study of local confluence complicated.
Example: Isotopy relations
= = • = • = •
![Page 15: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/15.jpg)
Motivation: objectives
I Objective: Study diagrammatic algebras arising in representation theory using algebraicrewriting.
I Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups;
I Temperley-Lieb algebras in statistichal mechanics;
I Brauer algebras and Birman-Wenzl algebras in knot theory.
I Main questions:
I Coherence theorems;
I Categorification constructive results;
I Computation of linear bases for these algebras by rewriting.
I Structural rules of these algebras make the study of local confluence complicated.
Example: Isotopy relations
= = • = • = •
![Page 16: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/16.jpg)
Motivation: objectives
I Objective: Study diagrammatic algebras arising in representation theory using algebraicrewriting.
I Khovanov-Lauda-Rouquier (KLR) algebras for categorification of quantum groups;
I Temperley-Lieb algebras in statistichal mechanics;
I Brauer algebras and Birman-Wenzl algebras in knot theory.
I Main questions:
I Coherence theorems;
I Categorification constructive results;
I Computation of linear bases for these algebras by rewriting.
I Structural rules of these algebras make the study of local confluence complicated.
Example: Isotopy relations
= = • = • = •
![Page 17: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/17.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let P be the rewriting system on the set of diagrams composed of:
, , , , • , • , , .
I submitted to relations:
•µ → •µ , •µ → •µ , •µ → •µ , •µ → •µ for µ in {0, 1}
• → • , • → • , • → • , • → • ,
→ , → , → , → , → .
I If no rewriting modulo:22
..
Not confluent !
![Page 18: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/18.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let P be the rewriting system on the set of diagrams composed of:
, , , , • , • , , .
I submitted to relations:
•µ → •µ , •µ → •µ , •µ → •µ , •µ → •µ for µ in {0, 1}
• → • , • → • , • → • , • → • ,
→ , → , → , → , → .
I If no rewriting modulo:22
..
Not confluent !
![Page 19: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/19.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let P be the rewriting system on the set of diagrams composed of:
, , , , • , • , , .
I submitted to relations:
•µ → •µ , •µ → •µ , •µ → •µ , •µ → •µ for µ in {0, 1}
• → • , • → • , • → • , • → • ,
→ , → , → , → , → .
I If no rewriting modulo:22
..
Not confluent !
![Page 20: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/20.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let P be the rewriting system on the set of diagrams composed of:
, , , , • , • , , .
I submitted to relations:
•µ → •µ , •µ → •µ , •µ → •µ , •µ → •µ for µ in {0, 1}
• → • , • → • , • → • , • → • ,
→ , → , → , → , → .
I If no rewriting modulo:22
..
Not confluent !
![Page 21: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/21.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let P be the rewriting system on the set of diagrams composed of:
, , , , • , • , , .
I submitted to relations:
•µ → •µ , •µ → •µ , •µ → •µ , •µ → •µ for µ in {0, 1}
• → • , • → • , • → • , • → • ,
→ , → , → , → , → .
I If no rewriting modulo:22
..
Not confluent !
![Page 22: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/22.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 23: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/23.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 24: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/24.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 25: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/25.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 26: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/26.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 27: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/27.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 28: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/28.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v
=��u′
R// v
![Page 29: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/29.jpg)
Three paradigms of rewriting modulo
I Rewriting system R:
I Coherence and confluence results in n-categories.
I Rewriting modulo: we consider a rewriting system R and a set of equations E .
I Three paradigms of rewriting modulo:
I Rewriting with R modulo E , Huet ’80.
uR //
E ��
u′R // w
E��v
R
// v ′R
// w ′
IERE : Rewriting with R on E -equivalence classes
uE RE //
E ��
v
E��u′
R
// v ′
I Rewriting with any system S such that R ⊆ S ⊆ ERE , Jouannaud - Kirchner ’84.
I Main interest and results for ER.
uER //
E ��
v=��
u′R// v
![Page 30: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/30.jpg)
II. Double groupoids
![Page 31: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/31.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I A double category is an internal category (C1,C0, ∂C−, ∂
C+, ◦C, iC) in Cat, Ehresmann ’64
.
(C0)0 (C0)0
(C0)0 (C0)0
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 32: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/32.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I A double category is an internal category (C1,C0, ∂C−, ∂
C+, ◦C, iC) in Cat, Ehresmann ’64.
(C0)0 (C0)0
(C0)0 (C0)0
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 33: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/33.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I A double category is an internal category (C1,C0, ∂C−, ∂
C+, ◦C, iC) in Cat, Ehresmann ’64.
(C0)0
(C0)0
(C0)0 (C0)0
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 34: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/34.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I A double category is an internal category (C1,C0, ∂C−, ∂
C+, ◦C, iC) in Cat, Ehresmann ’64.
(C0)0
(C0)1��
(C0)0
(C0)0
(C0)0
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 35: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/35.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I 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
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 36: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/36.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I 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
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 37: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/37.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I 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��
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 38: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/38.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I 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��
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 39: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/39.jpg)
Double groupoids
I We introduce a cubical notion of coherence, in n-categories enriched in double groupoids.
I 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��
I It gives four related categories
Cvo := (Cv ,Co , ∂v−,0, ∂v+,0, ◦v , iv0 ), Cho := (Ch,Co , ∂h−,0, ∂
h+,0, ◦h, ih0 ),
Csv := (Cs ,Cv , ∂v−,1, ∂v+,1, �v , iv1 ), Csh := (Cs ,Ch, ∂h−,1, ∂
h+,1, �h, ih1 ),
where Csh is the category C1 and Cvo is the category C0.
I Elements of Co : point cells, elements of Ch and Cv : horizontal cells and vertical cells.
x1f // x2
x1
e
��x2
![Page 40: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/40.jpg)
Double groupoids
I Elements of Cs are square cells:
·∂h−,1(A)
//
∂v−,1(A)
��
·
∂v+,1(A)
��·∂h+,1(A)
// ·
A��
, with identities
x1f //
iv0 (x1)
��
x2
iv0 (x2)
��x1
f// x2
ih1(f )��
xih0(x) //
e
��
x
e
��y
ih0(y)
// y
iv1 (e)��
I Compositionsx1
f1 //
e1
��
x2
e2
��
f2 // x3
e3
��y1 g1
// y2
A��
g2// y3
B��
x1f1◦hf2 //
e1
��
x3
e3
��y1
g1◦hg2// y3
A�vB��
x1f1 //
e1
��
x2
e2
��y1
f2//
e′1��
y2
e′2��
A��
z1f3// z2
A′��
x1f1 //
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 .
![Page 41: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/41.jpg)
Double groupoids
I Elements of Cs are square cells:
·∂h−,1(A)
//
∂v−,1(A)
��
·
∂v+,1(A)
��·∂h+,1(A)
// ·
A�� , with identities
x1f //
iv0 (x1)
��
x2
iv0 (x2)
��x1
f// x2
ih1(f )��
xih0(x) //
e
��
x
e
��y
ih0(y)
// y
iv1 (e)��
I Compositionsx1
f1 //
e1
��
x2
e2
��
f2 // x3
e3
��y1 g1
// y2
A��
g2// y3
B��
x1f1◦hf2 //
e1
��
x3
e3
��y1
g1◦hg2// y3
A�vB��
x1f1 //
e1
��
x2
e2
��y1
f2//
e′1��
y2
e′2��
A��
z1f3// z2
A′��
x1f1 //
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 .
![Page 42: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/42.jpg)
Double groupoids
I Elements of Cs are square cells:
·∂h−,1(A)
//
∂v−,1(A)
��
·
∂v+,1(A)
��·∂h+,1(A)
// ·
A�� , with identities
x1f //
iv0 (x1)
��
x2
iv0 (x2)
��x1
f// x2
ih1(f )��
xih0(x) //
e
��
x
e
��y
ih0(y)
// y
iv1 (e)��
I Compositionsx1
f1 //
e1
��
x2
e2
��
f2 // x3
e3
��y1 g1
// y2
A��
g2// y3
B��
x1f1◦hf2 //
e1
��
x3
e3
��y1
g1◦hg2// y3
A�vB��
x1f1 //
e1
��
x2
e2
��y1
f2//
e′1��
y2
e′2��
A��
z1f3// z2
A′��
x1f1 //
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 .
![Page 43: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/43.jpg)
Double groupoids
I Elements of Cs are square cells:
·∂h−,1(A)
//
∂v−,1(A)
��
·
∂v+,1(A)
��·∂h+,1(A)
// ·
A�� , with identities
x1f //
iv0 (x1)
��
x2
iv0 (x2)
��x1
f// x2
ih1(f )��
xih0(x) //
e
��
x
e
��y
ih0(y)
// y
iv1 (e)��
I Compositionsx1
f1 //
e1
��
x2
e2
��
f2 // x3
e3
��y1 g1
// y2
A��
g2// y3
B��
x1f1◦hf2 //
e1
��
x3
e3
��y1
g1◦hg2// y3
A�vB��
x1f1 //
e1
��
x2
e2
��y1
f2//
e′1��
y2
e′2��
A��
z1f3// z2
A′��
x1f1 //
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 .
![Page 44: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/44.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 45: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/45.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 46: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/46.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
�v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 47: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/47.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 48: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/48.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 49: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/49.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 50: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/50.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 51: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/51.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 52: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/52.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 53: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/53.jpg)
Double groupoids
I These compositions satisfy the middle four interchange law:
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A��
x2f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h �v �h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′��
y2g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
=
x1f1 //
e1��
x2
e2��
y1 g1 // y2
A�� �vx2
f2 //
e2��
x3
e3��
y2 g2 // y3
B��
�h
y1g1 //
e′1��
y2
e′2��
z1 h1 // z2
A′�� �vy2
g2 //
e′2��
y3
e′3��
z2 h2 // z3
B′��
I Double groupoid: double category in which horizontal, vertical and square cells areinvertible.
I n-category enriched in double groupoids: n-category C such that any homset Cn(x , y) is adouble groupoid.
I Horizontal (n + 1)-category: category of rewritings, vertical (n + 1)-category: category ofmodulo rules.
![Page 54: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/54.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1 P2 (. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 55: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/55.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1 P2 (. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 56: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/56.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0
P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb
P2 (. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 57: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/57.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1
P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2 (. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 58: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/58.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1
P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2
bb bb
(. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 59: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/59.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2
(. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2
bb bb OO
(. . .) Pn−1 Pn
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 60: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/60.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2
bb bb OO
(. . .)
cc cc OO
Pn−1
dd dd OO
Pn
cc cc OO
P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 61: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/61.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2
bb bb OO
(. . .)
cc cc OO
Pn−1
dd dd OO
Pn
cc cc OO
��P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 62: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/62.jpg)
Polygraphs
I Polygraphs are higher-dimensional generating systems of higher-dimensional globular strictcategories.
I An n-polygraph generates a free n-category.
P∗0 P∗1 P∗2 (. . .) P∗n−1 P∗n
P0 P1
bb bb OO
P2
bb bb OO
(. . .)
cc cc OO
Pn−1
dd dd OO
Pn
cc cc OO
��P>n
I An (n − 1)-category C is presented by an n-polygraph (P0, . . . ,Pn) if
C ' P∗n−1/ ≡Pn
![Page 63: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/63.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 64: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/64.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
Pvn+1
//// P∗n Phn+1oooo
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 65: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/65.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
(Pvn+1)∗
$$$$
(Phn+1)∗
zzzzPvn+1
////
OO
P∗n Phn+1oooo
OO
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 66: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/66.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
Ps
$$$$zzzz(Pv
n+1)∗
$$$$
(Phn+1)∗
zzzzPvn+1
////
OO
P∗n Phn+1oooo
OO
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 67: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/67.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
Ps
$$$$zzzz(Pv
n+1)∗
$$$$
(Phn+1)∗
zzzzPvn+1
////
OO
P∗n Phn+1oooo
OO
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 68: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/68.jpg)
Double (n + 2, n)-polygraphs
I A double n-polygraph is a data (Pv ,Ph,Ps) made of:
I two (n + 1)-polygraphs Pv and Ph such that Pvk = Ph
k for k ≤ n,
I a 2-square extension Ps of the pair of (n + 1)-categories ((Pv )∗, (Ph)∗), that is a set equippedwith four maps making Γ a 2-cubical set.
Ps
$$$$zzzz(Pv
n+1)∗
$$$$
(Phn+1)∗
zzzzPvn+1
////
OO
P∗n Phn+1oooo
OO
I A double (n + 2, n)-polygraph is a double n-polygraph in which Ps is defined on((Pv )>, (Ph)>).
I A double (n + 2, n)-polygraph (Pv ,Ph,Ps) generates a free (n − 1)-category enriched indouble groupoids, denoted by (Pv ,Ph,Ps)
|=
.
![Page 69: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/69.jpg)
Acyclicity
I A 2-square extension Ps of ((Pv )>, (Ph)>) is acyclic if for any square
S =
·(Ph)>//
(Pv )>
��
·
(Pv )>
��·
(Ph)>// ·
there exists a square (n + 1)-cell A in (Pv ,Ph,Ps)
|=
such that ∂(A) = S .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
![Page 70: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/70.jpg)
Acyclicity
I 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 .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
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Acyclicity
I 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 .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
![Page 72: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/72.jpg)
Acyclicity
I 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 .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
![Page 73: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/73.jpg)
Acyclicity
I 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 .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
![Page 74: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/74.jpg)
Acyclicity
I 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 .
I A 2-fold coherent presentation of an n-category C is a double (n + 2, n)-polygraph(Pv ,Ph,Ps) such that:
I the (n + 1)-polygraph Pv ∐Ph presents C;
I Ps is acyclic
I 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 .
I From Squier’s theorem, (E , ∅,Cd(E)) is a 2-fold coherent presentation of C.
![Page 75: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/75.jpg)
III. Polygraphs modulo
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Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE , where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u v
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 77: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/77.jpg)
Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE , where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u v
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 78: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/78.jpg)
Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE , where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u v
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 79: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/79.jpg)
Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE ,
where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u v
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 80: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/80.jpg)
Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE , where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u��##;; GGv
e��
f
��
e′��
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 81: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/81.jpg)
Polygraphs modulo
A n-polygraph modulo is a data (R,E , S) made of
I an n-polygraph R of primary rules,
I an n-polygraph E such that Ek = Rk for k ≤ n − 2 and En−1 ⊆ Rn−1, of modulo rules,
I S is a cellular extension of R∗n−1 such that R ⊆ S ⊆ ERE , where the cellular extension
ERE is defined byγ ERE : ERE → Sphn−1(R∗n−1)
where ERE is the set of triples (e, f , e′) in E> × R∗(1) × E> such that
u������������������##;; GGGGGGGGGGGGGGGGGGv
e��
f
��
e′��
and the map γ ERE is defined by γ ERE (e, f , e′) = (∂−,n−1(e), ∂+,n−1(e′)).
![Page 82: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/82.jpg)
Branchings and confluence modulo
I A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g arein S∗n and e is in E>n , such that:
uf //
e��
u′
vg// v ′
I It is local if f is in S∗(1)n , g is in S∗n and e in E>n such that `(g) + `(e) = 1.
I It is confluent modulo E if there exists f ′, g ′ in S∗n and e′ in E>n :
uf //
e��
u′f ′ // w
e′��v
g// v ′
g′// w ′
I Confluence modulo E (resp. local confluence modulo E): any branching (resp. localbranching) of S modulo E is confluent modulo E .
![Page 83: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/83.jpg)
Branchings and confluence modulo
I A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g arein S∗n and e is in E>n , such that:
uf //
e��
u′
vg// v ′
I It is local if f is in S∗(1)n , g is in S∗n and e in E>n such that `(g) + `(e) = 1.
I It is confluent modulo E if there exists f ′, g ′ in S∗n and e′ in E>n :
uf //
e��
u′f ′ // w
e′��v
g// v ′
g′// w ′
I Confluence modulo E (resp. local confluence modulo E): any branching (resp. localbranching) of S modulo E is confluent modulo E .
![Page 84: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/84.jpg)
Branchings and confluence modulo
I A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g arein S∗n and e is in E>n , such that:
uf //
e��
u′
vg// v ′
I It is local if f is in S∗(1)n , g is in S∗n and e in E>n such that `(g) + `(e) = 1.
I It is confluent modulo E if there exists f ′, g ′ in S∗n and e′ in E>n :
uf //
e��
u′f ′ // w
e′��v
g// v ′
g′// w ′
I Confluence modulo E (resp. local confluence modulo E): any branching (resp. localbranching) of S modulo E is confluent modulo E .
![Page 85: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/85.jpg)
Branchings and confluence modulo
I A branching modulo E of the n-polygraph modulo S is a triple (f , e, g) where f and g arein S∗n and e is in E>n , such that:
uf //
e��
u′
vg// v ′
I It is local if f is in S∗(1)n , g is in S∗n and e in E>n such that `(g) + `(e) = 1.
I It is confluent modulo E if there exists f ′, g ′ in S∗n and e′ in E>n :
uf //
e��
u′f ′ // w
e′��v
g// v ′
g′// w ′
I Confluence modulo E (resp. local confluence modulo E): any branching (resp. localbranching) of S modulo E is confluent modulo E .
![Page 86: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/86.jpg)
IV. Coherence modulo
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Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 88: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/88.jpg)
Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n
and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
uf //
e��
u′f ′ // w
e′��v
g// v ′
g′// w ′
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 89: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/89.jpg)
Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
uf //
e��
u′f ′ //
A��
w
e′��v
g// v ′
g′// w ′
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 90: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/90.jpg)
Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
uf //
e��
u′f ′ //
A��
w
e′��v
g// v ′
g′// w ′
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 91: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/91.jpg)
Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
uf //
e��
u′f ′ //
A��
w
e′��v
g// v ′
g′// w ′
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 92: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/92.jpg)
Coherent confluence modulo
I We consider Γ a 2-square extension of (E>, S∗).
I A branching modulo E is Γ-confluent modulo E if there exist f ′, g ′ in S∗n , e′ in E>n and asquare-cell A in (E ,S ,E o Γ ∪ Peiff(E ,S))
|=
,v :
uf //
e��
u′f ′ //
A��
w
e′��v
g// v ′
g′// w ′
I (E , S,−)
|=
,v is the free n-category enriched in double categories generated by (E , S,−), inwhich all vertical cells are invertible.
I Peiff(E , S) is the 2-square extension containing the following squares for all e,e′ ∈ E> andf ∈ S∗.
u ?i vf ?i v //
u?i e ��
u′ ?i v
u′?i e��u ?i v
′
f ?i v′// u′ ?i v ′
w ?i uw?i f //
e′?i u ��
w ?i u′
e′?i u′
��w ′ ?i u
w′?i f// w ′ ?i u′
I Eo Γ is to avoid "redundant" elements in Γ for different squares corresponding to the samebranching of S modulo E :
uf //
e ��
vf ′ // v ′
e′��u′
g=e1g1e2// w
g′// w ′
and
uf //
e?n−1e1��
vf ′ // v ′
e′��u1 g1e2
// wg′// w ′
![Page 93: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/93.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 94: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/94.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 95: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/95.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 96: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/96.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 97: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/97.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 98: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/98.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 99: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/99.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied.
![Page 100: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/100.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied. uf //
e ��
v
v ′
![Page 101: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/101.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied. uf //
e ��
v
v ′e−·f// v
![Page 102: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/102.jpg)
Coherent Newman and critical pair lemmas
I S is Γ-confluent modulo E (resp. locally Γ-confluent modulo E) if any of its branchingmodulo E (resp. local branching modulo E) is Γ-confluent modulo E .
I Theorem. [D.-Malbos ’18] If ERE is terminating, the following assertions are equivalent:
I S is Γ-confluent modulo E ;
I S is locally Γ-confluent modulo E ;
I S satisfies properties a) and b):
a):
uS∗(1) //
=
��
vS∗ // v ′
E>��u
R∗(1)
// wS∗// w ′
A�� b):
uS∗(1) //
E>(1)
��
vS∗ // v ′
E>��
u′S∗
// w
B��
for any local branching of S modulo E .
I S satisfies properties a) and b) for any critical branching of S modulo E .
I For S = ER, property b) is trivially satisfied. uf //
e ��
v
=��v ′
e−·f// v
![Page 103: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/103.jpg)
Coherence modulo
I 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) 6= ∅.
I Theorem. [D.-Malbos ’18] Let (R,E , S) be an n-polygraph modulo, and Γ be a squareextension of (E>, S∗) such that
I E is convergent,
I S is Γ-confluent modulo E ,
I Irr(E) is E -normalizing with respect to S,
IERE is terminating,
then E o Γ ∪ Peiff(E ,S) ∪ Cd(E) is acyclic.
![Page 104: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/104.jpg)
Coherence modulo
I 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) 6= ∅.
I Theorem. [D.-Malbos ’18] Let (R,E , S) be an n-polygraph modulo, and Γ be a squareextension of (E>, S∗) such that
I E is convergent,
I S is Γ-confluent modulo E ,
I Irr(E) is E -normalizing with respect to S,
IERE is terminating,
then E o Γ ∪ Peiff(E ,S) ∪ Cd(E) is acyclic.
![Page 105: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/105.jpg)
Coherent completion
I Coherent completion modulo E of S : square extension of (E>, S>) containing square cellsAf ,g and Bf ,e :
uf //
= ��
u′f ′ //
Af ,g��
w
e′��u
g// v
g′// w ′
uf //
e��
u′f ′ //
Bf ,e��
w
e′��v
g′// w ′
for any critical branchings (f , g) and (f , e) of S modulo E .
I Corollary. [D.-Malbos ’18] Let (R,E ,S) be an n-polygraph modulo such that
I E is convergent,
I S is confluent modulo E ,
I Irr(E) is E -normalizing with respect to S,
IERE is terminating,
For any coherent completion Γ of S modulo E , E o Γ ∪ Peiff(E , S) ∪ Cd(E) is acyclic.
I Corollary: Usual Squier’s theorem. (E = ∅)
![Page 106: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/106.jpg)
Coherent completion
I Coherent completion modulo E of S : square extension of (E>, S>) containing square cellsAf ,g and Bf ,e :
uf //
= ��
u′f ′ //
Af ,g��
w
e′��u
g// v
g′// w ′
uf //
e��
u′f ′ //
Bf ,e��
w
e′��v
g′// w ′
for any critical branchings (f , g) and (f , e) of S modulo E .
I Corollary. [D.-Malbos ’18] Let (R,E ,S) be an n-polygraph modulo such that
I E is convergent,
I S is confluent modulo E ,
I Irr(E) is E -normalizing with respect to S,
IERE is terminating,
For any coherent completion Γ of S modulo E , E o Γ ∪ Peiff(E , S) ∪ Cd(E) is acyclic.
I Corollary: Usual Squier’s theorem. (E = ∅)
![Page 107: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/107.jpg)
Coherent completion
I Coherent completion modulo E of S : square extension of (E>, S>) containing square cellsAf ,g and Bf ,e :
uf //
= ��
u′f ′ //
Af ,g��
w
e′��u
g// v
g′// w ′
uf //
e��
u′f ′ //
Bf ,e��
w
e′��v
g′// w ′
for any critical branchings (f , g) and (f , e) of S modulo E .
I Corollary. [D.-Malbos ’18] Let (R,E ,S) be an n-polygraph modulo such that
I E is convergent,
I S is confluent modulo E ,
I Irr(E) is E -normalizing with respect to S,
IERE is terminating,
For any coherent completion Γ of S modulo E , E o Γ ∪ Peiff(E , S) ∪ Cd(E) is acyclic.
I Corollary: Usual Squier’s theorem. (E = ∅)
![Page 108: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/108.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 109: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/109.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 110: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/110.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 111: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/111.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 112: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/112.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 113: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/113.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 114: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/114.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 115: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/115.jpg)
Example: diagrammatic rewriting modulo isotopy
I Let E and R be two 3-polygraphs defined by:
I E0 = R0 = {∗},
I E1 = R1 = {∧,∨},
I E2 =
, , , , • , •
R2 = E2∐ ,
I E3 =
•µ V •µ , •µ V •µ , •µ V •µ , •µ V •µ for µ in {0, 1} ,
• V • ,•
V•, • V • ,
•V
•}
I R3 = V , V , V , V , V
I Facts:
I E is convergent.
IERE is terminating.
IER is confluent modulo E .
![Page 116: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/116.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 117: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/117.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 118: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/118.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 119: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/119.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 120: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/120.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 121: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/121.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 122: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/122.jpg)
Conclusion
I We proved a coherence result for polygraphs modulo.
I How to weaken E -normalization assumption ?
I Is any polygraph modulo Tietze-equivalent to an E -normalizing polygraph modulo ?
I Explicit a quotient of a square extension by all modulo rules.
I Constructions extended to the linear setting.
I Linear bases from termination (or quasi-termination) or ERE and confluence of R modulo E .
I Work in progress:
I Rise this construction in dimensions, in n-categories enriched in p-fold groupoids.
I Formalize these constructions with rewriting modulo all the algebraic axioms.
![Page 123: Institut Camille Jordan, Université Lyon 1conferences.inf.ed.ac.uk/ct2019/slides/55.pdf · Coherence modulo and double groupoids BenjaminDupont Institut Camille Jordan, Université](https://reader035.vdocuments.us/reader035/viewer/2022071213/6041b3d7ae6dbb4bae4e09b4/html5/thumbnails/123.jpg)
Thank you !