SLIDE 1 A sequent calculus for opetopes
CT 2019
Pierre-Louis Curien1 Cédric Ho Thanh1 Samuel Mimram2 July 9th, 2019
1IRIF, Paris University 2LIX, École Polytechnique
SLIDE 2 This presentation informally presents some of the main notions and results of [Curien et al., 2019] arXiv:1903.05848, namely a “unnamed” syntax for
- petopes, and a sequent calculus Opt?.
1
SLIDE 3
Contents
Opetopes Syntax Opt?: a sequent calculus for opetopes Examples Conclusion
2
SLIDE 4
Opetopes
SLIDE 5
In a nutshell...
Opetopes are shapes (akin to globules, cubes, simplices, dendrices, etc.) designed to represent the notion of composition in every dimension. As such, they were introduced in [Baez and Dolan, 1998] to describe laws and coherence in weak higher categories. They have been actively studied over the recent years in [Hermida et al., 2002], [Cheng, 2003], [Leinster, 2004], [Kock et al., 2010] and applied to the theory of polygraphs in [Ho Thanh, 2018a]. A first syntactic account of opetopes has been tried in [Hermida et al., 2002], but does not seem usable for any computation.
3
SLIDE 6
In a nutshell...
Opetopes are shapes (akin to globules, cubes, simplices, dendrices, etc.) designed to represent the notion of composition in every dimension. As such, they were introduced in [Baez and Dolan, 1998] to describe laws and coherence in weak higher categories. They have been actively studied over the recent years in [Hermida et al., 2002], [Cheng, 2003], [Leinster, 2004], [Kock et al., 2010] and applied to the theory of polygraphs in [Ho Thanh, 2018a]. A first syntactic account of opetopes has been tried in [Hermida et al., 2002], but does not seem usable for any computation.
3
SLIDE 7
In a nutshell...
Opetopes are shapes (akin to globules, cubes, simplices, dendrices, etc.) designed to represent the notion of composition in every dimension. As such, they were introduced in [Baez and Dolan, 1998] to describe laws and coherence in weak higher categories. They have been actively studied over the recent years in [Hermida et al., 2002], [Cheng, 2003], [Leinster, 2004], [Kock et al., 2010] and applied to the theory of polygraphs in [Ho Thanh, 2018a]. A first syntactic account of opetopes has been tried in [Hermida et al., 2002], but does not seem usable for any computation.
3
SLIDE 8
Informal definition
They are pasting diagrams where every cell is many-to-one i.e. many inputs, one output. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell denoted by a above has dimension 2, so that a 3-opetope really is a pasting diagram of cells of dimension 2. We further ask those cells of dimension 2 to be 2-opetopes, i.e. pasting diagram of cells of dimension 1 (the simple arrows ). . . . . . . . . .
4
SLIDE 9
Informal definition
They are pasting diagrams where every cell is many-to-one i.e. many inputs, one output. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell denoted by a ⇓ above has dimension 2, so that a 3-opetope really is a pasting diagram of cells of dimension 2. We further ask those cells of dimension 2 to be 2-opetopes, i.e. pasting diagram of cells of dimension 1 (the simple arrows ). . . . . . . . . .
4
SLIDE 10
Informal definition
They are pasting diagrams where every cell is many-to-one i.e. many inputs, one output. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell denoted by a ⇓ above has dimension 2, so that a 3-opetope really is a pasting diagram of cells of dimension 2. We further ask those cells of dimension 2 to be 2-opetopes, i.e. pasting diagram of cells of dimension 1 (the simple arrows →). . . ⇓ . . . ⇓ . . . . ⇓
4
SLIDE 11
Informal definition
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Definition An n-dimensional opetope (or just n-opetope) is a pasting diagram of (n − 1)-opetopes, i.e. a finite set of n 1 -opetopes glued along n 2 -opetopes, in a “well-defined manner”.
5
SLIDE 12
Informal definition
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Definition An n-dimensional opetope (or just n-opetope) is a pasting diagram of (n − 1)-opetopes, i.e. a finite set of (n − 1)-opetopes glued along (n − 2)-opetopes, in a “well-defined manner”.
5
SLIDE 13
Informal definition
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Definition An n-dimensional opetope (or just n-opetope) is a pasting diagram of (n − 1)-opetopes, i.e. a finite set of (n − 1)-opetopes glued along (n − 2)-opetopes, in a “well-defined manner”.
5
SLIDE 14 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
- 2-opetopes are pasting diagram of 1-opetopes:
6
SLIDE 15 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
6
SLIDE 16 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
3 = . . . . ⇓
6
SLIDE 17 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
2 = . . . ⇓
6
SLIDE 18 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
1 = . . ⇓
6
SLIDE 19 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
n = . . . . .
(n) (n − 1) (1)
⇓
6
SLIDE 20 Definition: low dimensions
- There is a unique 0-dimensional opetope, which we’ll call
the point: .
- There is a unique 1-opetope, the arrow:
. .
- 2-opetopes are pasting diagram of 1-opetopes:
= . ⇓
6
SLIDE 21 Definition: dimension 3
- 3-opetopes are pasting diagrams of 2-opetopes
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓
7
SLIDE 22 Definition: dimension 3
- 3-opetopes are pasting diagrams of 2-opetopes
. . . .
⇓ ⇓
⇛ . . . . ⇓
7
SLIDE 23 Definition: dimension 3
- 3-opetopes are pasting diagrams of 2-opetopes
.
⇓ ⇓
⇛ . ⇓
7
SLIDE 24 Definition: dimension 3
- 3-opetopes are pasting diagrams of 2-opetopes
. .
⇓
⇓ ⇛ . . ⇓
7
SLIDE 25 Definition: dimension 4
- The induction goes on: 4-opetopes are pasting diagrams
- f 3-opetopes:
. . .
⇓ ⇓
⇛ . . . ⇓ . . . . . .
⇓
⇓ ⇓ ⇓ ⇛ . . . . . . ⇓ . . . . . ⇓ .
⇓
⇓ ⇓ ⇓ ⇛ . . . . . . ⇓
This is getting out of hand...
8
SLIDE 26 Definition: dimension 4
- The induction goes on: 4-opetopes are pasting diagrams
- f 3-opetopes:
. . .
⇓ ⇓
⇛ . . . ⇓ . . . . . .
⇓
⇓ ⇓ ⇓ ⇛ . . . . . . ⇓ . . . . . ⇓ .
⇓
⇓ ⇓ ⇓ ⇛ . . . . . . ⇓
This is getting out of hand...
8
SLIDE 27 Motivation
Problem
- 1. The graphical approach is neither formal nor manageable
for dimensions ≥ 4.
- 2. A formal definition either uses T-operads [Leinster, 2004]
- r polynomial monads and trees [Kock et al., 2010], which
as is, are not suited for automated computations. Solution In this presentation, we give a way to define opetopes syntactically.
9
SLIDE 28 Motivation
Problem
- 1. The graphical approach is neither formal nor manageable
for dimensions ≥ 4.
- 2. A formal definition either uses T-operads [Leinster, 2004]
- r polynomial monads and trees [Kock et al., 2010], which
as is, are not suited for automated computations. Solution In this presentation, we give a way to define opetopes syntactically.
9
SLIDE 29 Motivation
Problem
- 1. The graphical approach is neither formal nor manageable
for dimensions ≥ 4.
- 2. A formal definition either uses T-operads [Leinster, 2004]
- r polynomial monads and trees [Kock et al., 2010], which
as is, are not suited for automated computations. Solution In this presentation, we give a way to define opetopes syntactically.
9
SLIDE 30
Syntax
SLIDE 31
Idea
Since opetopes are pasting diagrams whose cells are many-to-one, they can be represented as trees: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾
10
SLIDE 32
Idea: dimension 0 and 1
Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by the unique 1-opetope, a.k.a. the arrow: We can represent as a node of a tree as follows: Let us add address information.
11
SLIDE 33
Idea: dimension 0 and 1
Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ◾ the unique 1-opetope, a.k.a. the arrow: . . We can represent as a node of a tree as follows: Let us add address information.
11
SLIDE 34
Idea: dimension 0 and 1
Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ◾ the unique 1-opetope, a.k.a. the arrow: . . We can represent ◾ as a node of a tree as follows: ◾ ⧫ ⧫ Let us add address information.
11
SLIDE 35 Idea: dimension 0 and 1
Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ◾ the unique 1-opetope, a.k.a. the arrow: . . We can represent ◾ as a node of a tree as follows: ◾ ⧫ ⧫
∗
[] Let us add address information.
11
SLIDE 36 Idea: dimension 2
Then we can:
- 1. create a tree with that corolla representing ◾
◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗]
- 2. consider that tree as a corolla, where the input edges are
the nodes
- 3. be convinced that this is a good representation of some
2-opetope!
12
SLIDE 37 Idea: dimension 2
Then we can:
- 1. create a tree with that corolla representing ◾
◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗] 3 ◾ ◾ ◾ ◾
[ ∗ ∗ ] [∗] [ ]
[]
- 2. consider that tree as a corolla, where the input edges are
the nodes
- 3. be convinced that this is a good representation of some
2-opetope!
12
SLIDE 38 Idea: dimension 2
Then we can:
- 1. create a tree with that corolla representing ◾
◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗] 3 ◾ ◾ ◾ ◾
[ ∗ ∗ ] [∗] [ ]
[] . . . . ⇓
- 2. consider that tree as a corolla, where the input edges are
the nodes
- 3. be convinced that this is a good representation of some
2-opetope!
12
SLIDE 39 Idea: dimension 2
Depending on the original tree, we obtain different 2-opetopes: ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] ⟿ 2 ◾ ◾ ◾
[ ∗ ] [ ]
[] ⟿ . . . ⇓
13
SLIDE 40 Idea: dimension 2
Depending on the original tree, we obtain different 2-opetopes: ◾ ◾ ◾ ⋮ ⧫ ⧫ ⧫ ⧫ ⧫
∗ ∗
[] [∗] [∗∗⋯∗]
∗ ∗
[] ⟿ n ◾ ◾ ◾ ◾ ⋯
[ ∗ ∗ ⋯ ∗ ] [∗] [ ]
[] ⟿ . . . . .
(n) (n − 1) (1)
⇓
13
SLIDE 41 Idea: dimension 2
Depending on the original tree, we obtain different 2-opetopes: ◾ ⧫ ⧫
∗
[] ⟿ 1 ◾ ◾
[]
[] ⟿ . . ⇓
13
SLIDE 42
Idea: dimension 2
Depending on the original tree, we obtain different 2-opetopes: ⧫ ⟿ ◾ [] ⟿ . ⇓
13
SLIDE 43 Idea: dimension 3
From there, repeat the process! 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ A 3 2 2 []
[ ] [ [ ∗ ] ]
⟿ . . . .
⇓ ⇓
⇛ . . . . ⇓
14
SLIDE 44 Idea: dimension 3
From there, repeat the process! 1 ◾ ◾ [] [[]]
[]
⟿ B 1 []
[ ] [ [ ] ]
⟿ .
⇓ ⇓
⇛ . ⇓
14
SLIDE 45 Idea: dimension 3
From there, repeat the process! 2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ C 2 []
[ ] [ [ ∗ ] ]
⟿ . .
⇓
⇓ ⇛ . . ⇓
14
SLIDE 46 Idea: dimension 3
From there, repeat the process! 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ D 4 3 2 1
[ ] [[∗]] [ [ ∗ ∗ ] ]
[] ⟿ . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓
14
SLIDE 47
Syntax
We now want a syntactic description of such trees. Solution In an n-opetope, every node is decorated by n 1 -opetope, but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map
n 1 15
SLIDE 48 Syntax
We now want a syntactic description of such trees. Solution 2 2 ◾ ◾ ◾ ◾ ◾ ⟿ . . . .
⇓ ⇓
⇛ . . . . ⇓ In an n-opetope, every node is decorated by (n − 1)-opetope, but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map
n 1 15
SLIDE 49 Syntax
We now want a syntactic description of such trees. Solution 2 2 ◾ ◾ ◾ ◾ ◾ ⟿ . . . .
⇓ ⇓
⇛ . . . . ⇓ In an n-opetope, every node is decorated by (n − 1)-opetope, but (n − 1)-opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map
n 1 15
SLIDE 50 Syntax
We now want a syntactic description of such trees. Solution 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ . . . .
⇓ ⇓
⇛ . . . . ⇓ In an n-opetope, every node is decorated by (n − 1)-opetope, but (n − 1)-opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map A → On−1.
15
SLIDE 51 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 2
16
SLIDE 52 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 2 Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗]
16
SLIDE 53 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 2 Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
16
SLIDE 54 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
16
SLIDE 55 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ Convention ◾ = {∗ ← ⧫
16
SLIDE 56 Syntax
We encode opetopes recursively as follows: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ Convention ◾ = {∗ ← ⧫
16
SLIDE 57 Syntax: examples
1 ◾ ◾ [] [[]]
[]
1
17
SLIDE 58 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 1 [[]] ← 0
17
SLIDE 59 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 1 [[]] ← 0 Reminder 1 = ◾ ⧫ ⧫
∗
[] = {[] ← ◾
17
SLIDE 60 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {[] ← ◾ [[]] ← 0 Reminder 1 = ◾ ⧫ ⧫
∗
[] = {[] ← ◾
17
SLIDE 61 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {[] ← ◾ [[]] ← 0 Reminder ◾ = {∗ ← ⧫
17
SLIDE 62 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {[] ← {∗ ← ⧫ [[]] ← 0 Reminder ◾ = {∗ ← ⧫
17
SLIDE 63 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {[] ← {∗ ← ⧫ [[]] ← 0 Reminder + convention = ⧫ = { {⧫
17
SLIDE 64 Syntax: examples
1 ◾ ◾ [] [[]]
[]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {[] ← {∗ ← ⧫ [[]] ← { {⧫ Reminder + convention = ⧫ = { {⧫
17
SLIDE 65 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
2
18
SLIDE 66 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 0
18
SLIDE 67 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 0 Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
18
SLIDE 68 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗]] ← 0 Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
18
SLIDE 69 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗]] ← 0 Reminder ◾ = {∗ ← ⧫
18
SLIDE 70 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗]] ← 0 Reminder ◾ = {∗ ← ⧫
18
SLIDE 71 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗]] ← 0 Reminder = ⧫ = { {⧫
18
SLIDE 72 Syntax: examples
2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗]] ← { {⧫ Reminder = ⧫ = { {⧫
18
SLIDE 73 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
3 2 1
19
SLIDE 74 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← 3 [[∗]] ← 2 [[∗∗]] ← 1
19
SLIDE 75 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← 3 [[∗]] ← 2 [[∗∗]] ← 1 Reminder 3 = ◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗] = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾
19
SLIDE 76 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← 2 [[∗∗]] ← 1 Reminder 3 = ◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗] = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾
19
SLIDE 77 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← 2 [[∗∗]] ← 1 Reminder 1 = ◾ ⧫ ⧫
∗
[] = {[] ← ◾
19
SLIDE 78 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← 2 [[∗∗]] ← {[] ← ◾ Reminder 1 = ◾ ⧫ ⧫
∗
[] = {[] ← ◾
19
SLIDE 79 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← 2 [[∗∗]] ← {[] ← ◾ Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
19
SLIDE 80 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗∗]] ← {[] ← ◾ Reminder 2 = ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾
19
SLIDE 81 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [∗∗] ← ◾ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← ◾ [∗] ← ◾ [[∗∗]] ← {[] ← ◾ Reminder ◾ = {∗ ← ⧫
19
SLIDE 82 Syntax: examples
3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [∗∗] ← {∗ ← ⧫ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗∗]] ← {[] ← {∗ ← ⧫ Reminder ◾ = {∗ ← ⧫
19
SLIDE 83
Syntax
Question Is this an opetope? ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [∗] ← ⧫ [∗∗] ← ⧫ [∗ ∗ ∗] ← ⧫ [∗∗] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← ⧫ [∗ ∗ ∗] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← {[] ← ⧫ [∗] ← ⧫ [∗∗] ← ⧫ [[]] ← {[] ← {∗ ← ⧫ [[[]]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [[[∗]]] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗ ∗ ∗]] ← ⧫
20
SLIDE 84 Opt?: a sequent calculus for
SLIDE 85
System Opt?
The set of preopetopes P is defined by the following grammar: P ::= ⧫ | ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A ← P ⋮ A ← P | { {P System Opt aims to characterize preopetopes that actually are opetopes.
21
SLIDE 86
System Opt?
The set of preopetopes P is defined by the following grammar: P ::= ⧫ | ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A ← P ⋮ A ← P | { {P System Opt? aims to characterize preopetopes that actually are opetopes.
21
SLIDE 87
System Opt?: the point rule
The first rule of Opt? states that we may create points without any prior assumption: point . point ⧫
22
SLIDE 88 System Opt?: the shift rule
This rule takes an opetope p and produces a new opetope having a unique node, decorated in p: 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
shift A 3 2 2 []
[ ] [ [ ∗ ] ]
p shift {[] ← p
23
SLIDE 89 System Opt?: the degen rule
This rule takes an opetope and produces a degenerate
. degen . ⇓ p degen { {p
24
SLIDE 90 System Opt?: the graft rule
This rule glues an n-opetope q to an (n + 1)-opetope p, the latter really just being a pasting diagram of n-opetopes, and “glues” them together: . . .
⇓
. . . . ⇓
⇓
. . . .
⇓ ⇓
.
⇓
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [a1] ← r1 ⋮ [ak] ← rk q graft-[b] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [a1] ← r1 ⋮ [ak] ← rk [b] ← q
25
SLIDE 91
Main result
Theorem Derivable preopetopes in system Opt? are in bijective correspondence with opetopes.
26
SLIDE 92
Examples
SLIDE 93
Examples
The proof tree of ⧫ = . is
point ⧫
27
SLIDE 94
Examples
The proof tree of ◾ = . . is
point ⧫ shift {[] ← ⧫
27
SLIDE 95
Examples
The proof tree of ◾ = . . is
point ⧫ shift {∗ ← ⧫
27
SLIDE 96 Examples
The proof tree of 1 = . . ⇓ ⟿ ◾ ⧫ ⧫
∗
[] is
point ⧫ shift {∗ ← ⧫ shift {[] ← {∗ ← ⧫
27
SLIDE 97 Examples
The proof tree of 2 = . . . ⇓ ⟿ ◾ ◾ ⧫ ⧫ ⧫
∗ ∗
[] [∗] is
point ⧫ shift {∗ ← ⧫ shift {[] ← {∗ ← ⧫ point ⧫ shift {[] ← ⧫ graft-[∗] {[] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫
27
SLIDE 98 Examples
The proof tree of 3 = . . . . ⇓ ⟿ ◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫
∗ ∗ ∗
[] [∗] [∗∗] is
point ⧫ shift {∗ ← ⧫ shift {[] ← {∗ ← ⧫ point ⧫ shift {[] ← ⧫ graft-[∗] {[] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ point ⧫ shift {[] ← ⧫ graft-[∗∗] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [∗∗] ← {∗ ← ⧫
27
SLIDE 99 Examples
The proof tree of . . . .
⇓ ⇓
⇛ . . . . ⇓ ⟿ 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
is: ⋮ 2
28
SLIDE 100 Examples
The proof tree of . . . .
⇓ ⇓
⇛ . . . . ⇓ ⟿ 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
is: ⋮ 2 shift {[] ← 2
28
SLIDE 101 Examples
The proof tree of . . . .
⇓ ⇓
⇛ . . . . ⇓ ⟿ 2 2 ◾ ◾ ◾ ◾ ◾ [] [[∗]]
[∗] [ ] [ ] [ ∗ ]
is: ⋮ 2 shift {[] ← 2 ⋮ 2 graft-[[∗]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 2
28
SLIDE 102 Example
The proof tree of .
⇓ ⇓
⇛ . ⇓ ⟿ 1 ◾ ◾ [] [[]]
[]
is ⋮ 1
29
SLIDE 103 Example
The proof tree of .
⇓ ⇓
⇛ . ⇓ ⟿ 1 ◾ ◾ [] [[]]
[]
is ⋮ 1 shift {[] ← 1
29
SLIDE 104 Example
The proof tree of .
⇓ ⇓
⇛ . ⇓ ⟿ 1 ◾ ◾ [] [[]]
[]
is ⋮ 1 shift {[] ← 1 ⋮ graft-[[]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 1 [[]] ← 0
29
SLIDE 105 Examples
The proof tree of . .
⇓
⇓ ⇛ . . ⇓ ⟿ 2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
is ⋮ 2
30
SLIDE 106 Examples
The proof tree of . .
⇓
⇓ ⇛ . . ⇓ ⟿ 2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
is ⋮ 2 shift {[] ← 2
30
SLIDE 107 Examples
The proof tree of . .
⇓
⇓ ⇛ . . ⇓ ⟿ 2 ◾ ◾ ◾ [] [[∗]]
[∗] [ ]
is ⋮ 2 shift {[] ← 2 ⋮ graft-[[∗]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [] ← 2 [[∗]] ← 0
30
SLIDE 108 Examples
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⋮ 3
31
SLIDE 109 Examples
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⋮ 3 shift {[] ← 3
31
SLIDE 110 Examples
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⋮ 3 shift {[] ← 3 ⋮ 2 graft-[[∗]] {[] ← 3 [[∗]] ← 2
31
SLIDE 111 Examples
. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2 ◾ ◾ ◾ ◾ ◾ ◾ ◾ [] [[∗∗]] [[∗]]
[∗∗] [∗] [ ] [] [ ] [ ∗ ]
⋮ 3 shift {[] ← 3 ⋮ 2 graft-[[∗]] {[] ← 3 [[∗]] ← 2 ⋮ 1 graft-[[∗∗]] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [] ← 3 [[∗]] ← 2 [[∗∗]] ← 1
31
SLIDE 112
Conclusion
SLIDE 113 Conclusion
- In this presentation, we gave a “unnamed” way to decribe
- petopes using terms and system Opt?.
- In [Curien et al., 2019] arXiv:1903.05848 we also
present variants of this system for opetopic sets.
- We are experimenting with those new tools to
automatically check coherence laws for an appropriate definition of opetopic
32
SLIDE 114 Conclusion
- In this presentation, we gave a “unnamed” way to decribe
- petopes using terms and system Opt?.
- In [Curien et al., 2019] arXiv:1903.05848 we also
present variants of this system for opetopic sets.
- We are experimenting with those new tools to
automatically check coherence laws for an appropriate definition of opetopic
32
SLIDE 115 Conclusion
- In this presentation, we gave a “unnamed” way to decribe
- petopes using terms and system Opt?.
- In [Curien et al., 2019] arXiv:1903.05848 we also
present variants of this system for opetopic sets.
- We are experimenting with those new tools to
automatically check coherence laws for an appropriate definition of opetopic ∞-groupoid.
32
SLIDE 116
The various constructs and algorithms can be easilyTM implemented, and opetopes amount to valid proof trees. An example implementation in Python 3 is available at github.com/altaris/
- petopy, where valid proof
trees are represented by certain expressions that evaluate without throwing any
p shift {[] ← p
def shift(seq: Sequent) -> Sequent: n = seq.source.dimension ctx = Context(n + 1) for a in seq.source.nodeAddresses(): ctx += (a.shift(), a) return Sequent( ctx, Preopetope.fromDictOfPreopetopes( {Address.epsilon(n): seq.source} ↪ ), seq.source )
33
SLIDE 117
Thank you for your attention!
33
SLIDE 118
References i
Baez, J. C. and Dolan, J. (1998). Higher-dimensional algebra. III. n-categories and the algebra of opetopes. Advances in Mathematics, 135(2):145–206. Cheng, E. (2003). The category of opetopes and the category of opetopic sets. Theory and Applications of Categories, 11:No. 16, 353–374. Curien, P.-L., Ho Thanh, C., and Mimram, S. (2019). Syntactic approaches for opetopes. arXiv:1903.05848 [math.CT].
34
SLIDE 119 References ii
Hermida, C., Makkai, M., and Power, J. (2002). On weak higher-dimensional categories. I. 3. Journal of Pure and Applied Algebra, 166(1-2):83–104. Ho Thanh, C. (2018a). The equivalence between opetopic sets and many-to-one polygraphs. arXiv:1806.08645 [math.CT]. Ho Thanh, C. (2018b).
https://github.com/altaris/opetopy.
35
SLIDE 120
References iii
Kock, J., Joyal, A., Batanin, M., and Mascari, J.-F. (2010). Polynomial functors and opetopes. Advances in Mathematics, 224(6):2690–2737. Leinster, T. (2004). Higher Operads, Higher Categories. Cambridge University Press.
36
