Type theoretical approaches to opetopes Journes Logique Homotopie - - PowerPoint PPT Presentation

Type theoretical approaches to opetopes

Journées Logique Homotopie Catégories

Pierre-Louis Curien1 Cédric Ho Thanh2 Samuel Mimram3 October 18th, 2018

1IRIF, Paris Diderot University 2IRIF, Paris Diderot University; this author has received funding from the European Union’s Horizon 2020 research

and innovation program under the Marie Sklodowska-Curie grant agreement №665850

3LIX, École Polytechnique

This presentation informally presents the main notions and results of [CHM18] (in preparation, draft available at chothanh.wordpress.com).

1

Contents

Opetopes The “named” approach The “unnamed” approach Conclusion

2

Opetopes

In a nutshell...

Opetopes are shapes (akin to globules, cubes, simplices, etc.) designed to represent the notion of composition in every

• dimension. As such, they were introduced in [BD98] to

describe laws and coherence if weak higher categories.

3

In a nutshell...

They are pasting diagrams where every cell is many-to-one. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell 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 arrows). We further ask those cells of dimension 1 to be 1-opetopes, i.e. pasting diagram (in a trivial way) of cells of dimension 0 (the points).

4

In a nutshell...

They are pasting diagrams where every cell is many-to-one. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell 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 arrows). We further ask those cells of dimension 1 to be 1-opetopes, i.e. pasting diagram (in a trivial way) of cells of dimension 0 (the points).

4

In a nutshell...

They are pasting diagrams where every cell is many-to-one. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell 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 arrows). We further ask those cells of dimension 1 to be 1-opetopes, i.e. pasting diagram (in a trivial way) of cells of dimension 0 (the points).

4

In a nutshell...

They are pasting diagrams where every cell is many-to-one. Here is an example of a 3-opetope: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ Every cell 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 arrows). We further ask those cells of dimension 1 to be 1-opetopes, i.e. pasting diagram (in a trivial way) of cells of dimension 0 (the points).

4

Informal definition

Definition An n-opetope is a pasting diagram of (n − 1)-opetopes i.e. a finite set of n 1 -opetopes glued along n 2 -opetopes.

5

Informal definition

Definition An n-opetope is a pasting diagram of (n − 1)-opetopes i.e. a finite set of (n − 1)-opetopes glued along (n − 2)-opetopes.

5

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, a.k.a. the

arrow

6

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, a.k.a. the

arrow

6

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, a.k.a. the

arrow ∎ . . ⇓

6

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, a.k.a. the

arrow ∎ . . . ⇓

6

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, a.k.a. the

arrow ∎ . . . . ⇓

6

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, a.k.a. the

arrow ∎ . . . . .

(1) (2) (n)

⇓

6

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, a.k.a. the

arrow ∎ . ⇓

6

Definition: dimension 3

• 3-opetopes are pasting diagrams of 2-opetopes

. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓

7

Definition: dimension 3

• 3-opetopes are pasting diagrams of 2-opetopes

. . . .

⇓ ⇓

⇛ . . . . ⇓

7

Definition: dimension 3

• 3-opetopes are pasting diagrams of 2-opetopes

.

⇓ ⇓

⇛ . ⇓

7

Definition: dimension 3

• 3-opetopes are pasting diagrams of 2-opetopes

. .

⇓

⇓ ⇛ . . ⇓

7

Definition: dimension 4

• The induction goes on: 4-opetopes are pasting diagrams
• f 3-opetopes

. . .

⇓ ⇓

⇛ . . . ⇓ . . . . . .

⇓

⇓ ⇓ ⇓ ⇛ . . . . . . ⇓ . . . . . ⇓ .

⇓

⇓ ⇓ ⇓ ⇛ . . . . . . ⇓

8

Definition: dimension 4

• The induction goes on: 4-opetopes are pasting diagrams
• f 3-opetopes

This is getting out of hand...

8

Motivation

Problem

• 1. The graphical approach is neither formal nor manageable

for dimensions ≥ 4.

• 2. A formal definition either use T-operads [Lei04] or

polynomial monads and trees [KJBM10], which are both unintuitive and difficult to manipulate. Solution In this presentation, we give a rough sketch two ways to define

• petopes syntactically.

9

Motivation

Problem

• 1. The graphical approach is neither formal nor manageable

for dimensions ≥ 4.

• 2. A formal definition either use T-operads [Lei04] or

polynomial monads and trees [KJBM10], which are both unintuitive and difficult to manipulate. Solution In this presentation, we give a rough sketch two ways to define

• petopes syntactically.

9

Motivation

Problem

• 1. The graphical approach is neither formal nor manageable

for dimensions ≥ 4.

• 2. A formal definition either use T-operads [Lei04] or

polynomial monads and trees [KJBM10], which are both unintuitive and difficult to manipulate. Solution In this presentation, we give a rough sketch two ways to define

• petopes syntactically.

9

The “named” approach

Idea

• 1. Take an opetope.

. . . .

⇓ ⇓

⇛ . . . . ⇓

• 2. Give names to everything.
• 3. Write down the graftings:

A i h c g b f a

• 4. ???
• 5. Profit!

10

Idea

• 1. Take an opetope.

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A

• 2. Give names to everything.
• 3. Write down the graftings:

A i h c g b f a

• 4. ???
• 5. Profit!

10

Idea

• 1. Take an opetope.

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A

• 2. Give names to everything.
• 3. Write down the graftings:

A ∶ β(i ← α) ⊷ h(c ← g(b ← f)) ⊷ a ⊷ ∅.

• 4. ???
• 5. Profit!

10

Syntax

• We start with a set of variable V = ∐n∈N Vn, where

elements of Vn represent n-cells.

• The set of n-terms is defined as

n

::=

n n 1 n

|

n 1

Examples For a b c

0, f g h 1,

a h a g b f

1

f a f a f a f a f

1

h

2 11

Syntax

• We start with a set of variable V = ∐n∈N Vn, where

elements of Vn represent n-cells.

• The set of n-terms is defined as

Tn ::= Vn(Vn−1 ← Tn,⋯) | Vn−1 Examples For a b c

0, f g h 1,

a h a g b f

1

f a f a f a f a f

1

h

2 11

Syntax

• We start with a set of variable V = ∐n∈N Vn, where

elements of Vn represent n-cells.

• The set of n-terms is defined as

Tn ::= Vn(Vn−1 ← Tn,⋯) | Vn−1 Examples For a,b,c ∈ V0, f,g,h ∈ V1, a ∈ T0, h(a ← g,b ← f) ∈ T1, f(a ← f(a ← f),a ← f,a ← f) ∈ T1, h ∈ T2.

11

Syntax

• An n-type is a sequence of terms of the form

s1 ⊷ s2 ⊷ ⋯ ⊷ sn ⊷ sn+1 ⊷ ∅, where si ∈ Tn+1−i.

• A n-typing is an expression of the form

t T where t

n and T is an n

1 -type.

12

Syntax

• An n-type is a sequence of terms of the form

s1 ⊷ s2 ⊷ ⋯ ⊷ sn ⊷ sn+1 ⊷ ∅, where si ∈ Tn+1−i.

• A n-typing is an expression of the form

t ∶ T where t ∈ Tn and T is an (n − 1)-type.

12

Main result of the named approach

Theorem Derivable typings in system Opt! of the form α ∶ T where α ∈ Vn (as opposed to just Tn) are in bijective correspondence (up to renaming of variables) with n-opetopes.

13

System Opt!: the point rule

The first rule of Opt! states that we may create points without any prior assumption: point . point x

14

System Opt!: the point rule

The first rule of Opt! states that we may create points without any prior assumption: point . point x ∶ ∅

14

System Opt!: the degen-fill rule

This rule takes an opetope and produces a degenerate

• petope from it:

. degen-fill . ⇓ x T degen-fill x x T

15

System Opt!: the degen-fill rule

This rule takes an opetope and produces a degenerate

• petope from it:

. degen-fill . ⇓ x ∶ T degen-fill δ ∶ x ⊷ x ⊷ T

15

System Opt!: the fill rule

This rule takes a pasting diagram (that is, a term), and creates an opetope by “filling” it: . . . . .

f g h i

fill . . . . .

f g h i

⇓ µ t T fill t T

16

System Opt!: the fill rule

This rule takes a pasting diagram (that is, a term), and creates an opetope by “filling” it: . . . . .

f g h i

fill . . . . .

f g h i

⇓ µ t ∶ T fill µ ∶ t ⊷ T

16

System Opt!: the graft rule

This rules glues an opetope to a pasting diagram of the same dimension: . . .

⇓

. . . . ⇓ graft-a . . . .

⇓

.

⇓

t s T x y U graft-a t a x s y a T

17

System Opt!: the graft rule

This rules glues an opetope to a pasting diagram of the same dimension: . . .

⇓

. . . . ⇓ graft-a . . . .

⇓

.

⇓

t ∶ s ⊷ T x ∶ y ⊷ U graft-a t(a ← x) ∶ s[y/a] ⊷ T

17

Example 1

Let’s derive

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A

18

Example 1

Let’s derive

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A Derivation of α point b ∶ ∅ fill g ∶ b ⊷ ∅ point a ∶ ∅ fill f ∶ a ⊷ ∅ graft-b g(b ← f) ∶ b[a/b] ÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

≡a

⊷ ∅ fill α ∶ g(b ← f) ⊷ a ⊷ ∅ .

18

Example 1

Let’s derive

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A Derivation of β point c ∶ ∅ fill h ∶ c ⊷ ∅ point a ∶ ∅ fill i ∶ a ⊷ ∅ graft-c h(c ← i) ∶ c[a/c] ÜÜÜÜÜÜÜÜÜÜ

≡a

⊷ ∅ fill β ∶ h(c ← i) ⊷ a ⊷ ∅

18

Example 1

Let’s derive

a b c d f g h i j ⇓α ⇓β

⇛

a b c d f g h j

⇓γ A And we assemble to get A ⋮ β ∶ h(c ← i) ⊷ a ⊷ ∅ ⋮ α ∶ g(b ← f) ⊷ a ⊷ ∅ graft-i β(i ← α) ∶ h(c ← i)[g(b ← f)/i] ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

≡h(c←g(b←f))

⊷ a ⊷ ∅ fill A ∶ β(i ← α) ⊷ h(c ← g(b ← f)) ⊷ a ⊷ ∅

18

Example 2

Let’s derive . .

⇓

⇓ ⇛ . . ⇓

19

Example 2

Let’s derive . .

⇓

⇓ ⇛ . . ⇓ Top left part point a ∶ ∅ degen-fill α ∶ a ⊷ a ⊷ ∅

19

Example 2

Let’s derive . .

⇓

⇓ ⇛ . . ⇓ Bottom part point b ∶ ∅ fill g ∶ b ⊷ ∅ point a ∶ ∅ fill f ∶ a ⊷ ∅ graft-b g(b ← f) ∶ a ⊷ ∅ fill β ∶ g(b ← f) ⊷ a ⊷ ∅

19

Example 2

Let’s derive . .

⇓

⇓ ⇛ . . ⇓ And we assemble ⋮ β ∶ g(b ← f) ⊷ a ⊷ ∅ ⋮ α ∶ a ⊷ a ⊷ ∅ graft-f a = b ⊢ β(f ← α) ∶ g(b ← f)[a/f] ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

≡g

⊷ a ⊷ ∅ fill a = b ⊢ A ∶ β(f ← α) ⊷ g ⊷ a ⊷ ∅

19

The “unnamed” approach

Idea

Since opetopes are pasting diagrams whose cells are many-to-one, they can be represented as trees: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

Then a cell in a pasting diagram no longer needs to have a name, it can be identified by its address in that tree.

20

Idea

Since opetopes are pasting diagrams whose cells are many-to-one, they can be represented as trees: . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

Then a cell in a pasting diagram no longer needs to have a name, it can be identified by its address in that tree.

20

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.

21

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.

21

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.

21

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:

∎

⧫ ⧫

∗

[ϵ] Let us add address information.

21

Idea: dimension 2

Then we can:

• 1. create a tree with that node representing ∎

∎ ∎ ∎

⧫ ⧫ ⧫ ⧫

∗ ∗ ∗

[ϵ] [∗] [∗∗]

• 2. consider that tree like a node, where the input edges are

the nodes of said tree

• 3. be convinced that this is a good representation of some

2-opetope!

22

Idea: dimension 2

Then we can:

• 1. create a tree with that node representing ∎

∎ ∎ ∎

⧫ ⧫ ⧫ ⧫

∗ ∗ ∗

[ϵ] [∗] [∗∗] 3

∎ ∎ ∎ ∎ [ϵ] [∗] [∗∗]

[ϵ]

• 2. consider that tree like a node, where the input edges are

the nodes of said tree

• 3. be convinced that this is a good representation of some

2-opetope!

22

Idea: dimension 2

Then we can:

• 1. create a tree with that node representing ∎

∎ ∎ ∎

⧫ ⧫ ⧫ ⧫

∗ ∗ ∗

[ϵ] [∗] [∗∗] 3

∎ ∎ ∎ ∎ [ϵ] [∗] [∗∗]

[ϵ] . . . . ⇓

• 2. consider that tree like a node, where the input edges are

the nodes of said tree

• 3. be convinced that this is a good representation of some

2-opetope!

22

Idea: dimension 2

Depending on the original tree, we obtain different 2-opetopes:

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] ⟿ 2

∎ ∎ ∎ [ϵ] [∗]

[ϵ] ⟿ . . . ⇓

23

Idea: dimension 2

Depending on the original tree, we obtain different 2-opetopes:

∎ ∎ ∎

⋮

⧫ ⧫ ⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] [∗∗⋯∗]

∗ ∗

[ϵ] ⟿ n

∎ ∎ ∎ ∎

⋯

[ϵ] [∗] [∗∗⋯∗]

[ϵ] ⟿ . . . . .

(1) (2) (n)

⇓

23

Idea: dimension 2

Depending on the original tree, we obtain different 2-opetopes:

∎

⧫ ⧫

∗

[ϵ] ⟿ 1

∎ ∎ [ϵ]

[ϵ] ⟿ . . ⇓

23

Idea: dimension 2

Depending on the original tree, we obtain different 2-opetopes:

⧫

⟿

∎

[ϵ] ⟿ . ⇓

23

Idea: dimension 3

From there, repeat the process! 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ A 3 2 2 [ϵ]

[ϵ] [[∗]]

⟿ . . . .

⇓ ⇓

⇛ . . . . ⇓

24

Idea: dimension 3

From there, repeat the process! 1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ B 1 [ϵ]

[ϵ] [[ϵ]]

⟿ .

⇓ ⇓

⇛ . ⇓

24

Idea: dimension 3

From there, repeat the process! 2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ C 2 [ϵ]

[ϵ] [[∗]]

⟿ . .

⇓

⇓ ⇛ . . ⇓

24

Idea: dimension 3

From there, repeat the process! 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ D 4 3 2 1

[ϵ] [[∗]] [[∗∗]]

[ϵ] ⟿ . . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓

24

Syntax

We now want a syntactical 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 25

Syntax

We now want a syntactical 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 25

Syntax

We now want a syntactical 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 25

Syntax

We now want a syntactical description of such trees. Solution 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ . . . .

⇓ ⇓

⇛ . . . . ⇓ 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.

25

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[ϵ]] ← 2

26

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[ϵ]] ← 2 Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗]

26

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[ϵ]] ← 2 Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

26

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [[ϵ]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

26

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [[ϵ]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ Convention

∎

= {∗ ← ⧫

26

Syntax

We encode opetopes recursively as follows: 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[ϵ]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ Convention

∎

= {∗ ← ⧫

26

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

1

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 1 [[ϵ]] ← 0

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 1 [[ϵ]] ← 0 Reminder 1 =

∎

⧫ ⧫

∗

[ϵ] = {[ϵ] ← ∎

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← ∎ [[ϵ]] ← 0 Reminder 1 =

∎

⧫ ⧫

∗

[ϵ] = {[ϵ] ← ∎

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← ∎ [[ϵ]] ← 0 Reminder

∎

= {∗ ← ⧫

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← {∗ ← ⧫ [[ϵ]] ← 0 Reminder

∎

= {∗ ← ⧫

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← {∗ ← ⧫ [[ϵ]] ← 0 Reminder + convention =

⧫

= { {⧫

27

Syntax: examples

1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← {∗ ← ⧫ [[ϵ]] ← { {⧫ Reminder + convention =

⧫

= { {⧫

27

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

2

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[ϵ]] ← 0

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[ϵ]] ← 0 Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [[ϵ]] ← 0 Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [[ϵ]] ← 0 Reminder

∎

= {∗ ← ⧫

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[ϵ]] ← 0 Reminder

∎

= {∗ ← ⧫

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[ϵ]] ← 0 Reminder =

⧫

= { {⧫

28

Syntax: examples

2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[ϵ]] ← { {⧫ Reminder =

⧫

= { {⧫

28

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

3 1 2

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← 3 [[ϵ]] ← 1 [[∗]] ← 2

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← 3 [[ϵ]] ← 1 [[∗]] ← 2 Reminder 3 =

∎ ∎ ∎

⧫ ⧫ ⧫ ⧫

∗ ∗ ∗

[ϵ] [∗] [∗∗] = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← 1 [[∗]] ← 2 Reminder 3 =

∎ ∎ ∎

⧫ ⧫ ⧫

∗ ∗ ∗

[ϵ] [∗] [∗∗] = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← 1 [[∗]] ← 2 Reminder 1 =

∎

⧫ ⧫

∗

[ϵ] = {[ϵ] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← {[ϵ] ← ∎ [[∗]] ← 2 Reminder 1 =

∎

⧫ ⧫

∗

[ϵ] = {[ϵ] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← {[ϵ] ← ∎ [[∗]] ← 2 Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← {[ϵ] ← ∎ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ Reminder 2 =

∎ ∎

⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ [∗∗] ← ∎ [[ϵ]] ← {[ϵ] ← ∎ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← ∎ [∗] ← ∎ Reminder

∎

= {∗ ← ⧫

29

Syntax: examples

3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⟿ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [∗∗] ← {∗ ← ⧫ [[ϵ]] ← {[ϵ] ← {∗ ← ⧫ [[∗]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ Reminder

∎

= {∗ ← ⧫

29

Syntax

Question Is this an opetope? ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [∗] ← ⧫ [∗∗] ← ⧫ [∗ ∗ ∗] ← ⧫ [∗∗] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← {[ϵ] ← ⧫ [∗ ∗ ∗] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← {[ϵ] ← ⧫ [∗] ← ⧫ [∗∗] ← ⧫ [[ϵ]] ← {[ϵ] ← {∗ ← ⧫ [[[ϵ]]] ← ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [[[∗]]] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [[∗ ∗ ∗]] ← ⧫

30

System Opt?

The set of preopetopes P is defined by the following grammar: P ::=

⧫

| ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A ← P ⋮ A ← P | { {P The Opt system aims to characterize preopetopes that actually are opetopes: Theorem Derivable preopetopes in system Opt are in bijective correspondence with opetopes.

31

System Opt?

The set of preopetopes P is defined by the following grammar: P ::=

⧫

| ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A ← P ⋮ A ← P | { {P The Opt? system aims to characterize preopetopes that actually are opetopes: Theorem Derivable preopetopes in system Opt? are in bijective correspondence with opetopes.

31

System Opt?: the point rule

The first rule of Opt? states that we may create points without any prior assumption: point . point

32

System Opt?: the point rule

The first rule of Opt? states that we may create points without any prior assumption: point . point

⧫ 32

System Opt?: the degen rule

This rule takes an opetope and produces a degenerate

• petope from it:

. degen . ⇓ p degen p

33

System Opt?: the degen rule

This rule takes an opetope and produces a degenerate

• petope from it:

. degen . ⇓ p degen { {p

33

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

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

shift A 3 2 2 [ϵ]

[ϵ] [[∗]]

p shift p

34

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

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

shift A 3 2 2 [ϵ]

[ϵ] [[∗]]

p shift {[ϵ] ← p

34

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: . . .

⇓

. . . . ⇓ graft-[b] . . . .

⇓

.

⇓

a1 r1 ak rk q graft- b a1 r1 ak rk b q (we omitted some technical assumptions that ensure this

• peration is geometrically meaningful)

35

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: . . .

⇓

. . . . ⇓ graft-[b] . . . .

⇓

.

⇓

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [a1] ← r1 ⋮ [ak] ← rk q graft-[b] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [a1] ← r1 ⋮ [ak] ← rk [b] ← q (we omitted some technical assumptions that ensure this

• peration is geometrically meaningful)

35

Example

The proof tree of . is:

point

⧫ 36

Example

The proof tree of . . is:

point

⧫

shift {[ϵ] ← ⧫

36

Example

The proof tree of . . is:

point

⧫

shift {∗ ← ⧫

36

Example

The proof tree of . . ⇓ is:

point

⧫

shift {∗ ← ⧫ shift {[ϵ] ← {∗ ← ⧫

36

Example

The proof tree of . . . ⇓ is:

point

⧫

shift {∗ ← ⧫ shift {[ϵ] ← {∗ ← ⧫ . . graft-[∗] ?

36

Example

The proof tree of . . . ⇓ is:

point

⧫

shift {∗ ← ⧫ shift {[ϵ] ← {∗ ← ⧫ point

⧫

shift {∗ ← ⧫ graft-[∗] ?

36

Example

The proof tree of . . . ⇓ is:

point

⧫

shift {∗ ← ⧫ shift {[ϵ] ← {∗ ← ⧫ point

⧫

shift {[ϵ] ← ⧫ graft-[∗] {[ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫

36

Example

The proof tree of . . . . ⇓ is:

point

⧫

shift {∗ ← ⧫ shift {[ϵ] ← {∗ ← ⧫ point

⧫

shift {[ϵ] ← ⧫ graft-[∗] {[ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ point

⧫

shift {[ϵ] ← ⧫ graft-[∗∗] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [∗∗] ← {∗ ← ⧫

36

Examples

Write n = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← {∗ ← ⧫ [∗] ← {∗ ← ⧫ [∗∗] ← {∗ ← ⧫ ⋮ [∗n−1] ← {∗ ← ⧫ =

∎ ∎ ∎

⋮

⧫ ⧫ ⧫ ⧫ ⧫

∗ ∗

[ϵ] [∗] [∗∗⋯∗]

∗ ∗

[ϵ]

37

Examples

The proof tree of . . . .

⇓ ⇓

⇛ . . . . ⇓ ⟿ 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

is: ⋮ 2

38

Examples

The proof tree of . . . .

⇓ ⇓

⇛ . . . . ⇓ ⟿ 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

is: ⋮ 2 shift {[ϵ] ← 2

38

Examples

The proof tree of . . . .

⇓ ⇓

⇛ . . . . ⇓ ⟿ 2 2

∎ ∎ ∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ] [ϵ] [∗]

is: ⋮ 2 shift {[ϵ] ← 2 ⋮ 2 graft-[[∗]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[∗]] ← 2

38

Example

The proof tree of .

⇓ ⇓

⇛ . ⇓ ⟿ 1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

is ⋮ 1

39

Example

The proof tree of .

⇓ ⇓

⇛ . ⇓ ⟿ 1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

is ⋮ 1 shift {[ϵ] ← 1

39

Example

The proof tree of .

⇓ ⇓

⇛ . ⇓ ⟿ 1

∎ ∎

[ϵ] [[ϵ]]

[ϵ]

is ⋮ 1 shift {[ϵ] ← 1 ⋮ graft-[[ϵ]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 1 [[ϵ]] ← 0

39

Example

The proof tree of . .

⇓

⇓ ⇛ . . ⇓ ⟿ 2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

is ⋮ 2

40

Example

The proof tree of . .

⇓

⇓ ⇛ . . ⇓ ⟿ 2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

is ⋮ 2 shift {[ϵ] ← 2

40

Example

The proof tree of . .

⇓

⇓ ⇛ . . ⇓ ⟿ 2

∎ ∎ ∎

[ϵ] [[∗]]

[ ∗ ] [ϵ]

is ⋮ 2 shift {[ϵ] ← 2 ⋮ graft-[[∗]] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [ϵ] ← 2 [[∗]] ← 0

40

Example

. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⋮ 3

41

Example

. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⋮ 3 shift {[ϵ] ← 3

41

Example

. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⋮ 3 shift {[ϵ] ← 3 ⋮ 2 graft-[[∗]] {[ϵ] ← 3 [[∗]] ← 2

41

Example

. . . . . ⇓ ⇓ ⇓ ⇛ . . . . . ⇓ ⟿ 3 1 2

∎ ∎ ∎ ∎ ∎ ∎ ∎

[ϵ] [[∗∗]] [[∗]]

[∗∗] [∗] [ϵ] [ϵ] [ϵ] [∗]

⋮ 3 shift {[ϵ] ← 3 ⋮ 2 graft-[[∗]] {[ϵ] ← 3 [[∗]] ← 2 ⋮ 1 graft-[[∗∗]] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [ϵ] ← 3 [[∗]] ← 2 [[∗∗]] ← 1

41

Conclusion

Conclusion

• In this presentation, we gave two ways to define opetopes

syntactically:

• 1. in a “named” way, using terms and system Opt!;
• 2. in an “unnamed” way, using preopetopes and system Opt?;
• 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 [Ho 18], where valid proof trees are represented by certain expressions that evaluate without throwing any exception.

• In [CHM18] (see link on the first slide for a draft), we also

present variants of those systems for opetopic sets.

• We are experimenting with those new tools to

automatically check coherence laws for an appropriate definition of opetopic

• groupoid.

42

Conclusion

• In this presentation, we gave two ways to define opetopes

syntactically:

• 1. in a “named” way, using terms and system Opt!;
• 2. in an “unnamed” way, using preopetopes and system Opt?;
• 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 [Ho 18], where valid proof trees are represented by certain expressions that evaluate without throwing any exception.

• In [CHM18] (see link on the first slide for a draft), we also

present variants of those systems for opetopic sets.

• We are experimenting with those new tools to

automatically check coherence laws for an appropriate definition of opetopic

• groupoid.

42

Conclusion

• In this presentation, we gave two ways to define opetopes

syntactically:

• 1. in a “named” way, using terms and system Opt!;
• 2. in an “unnamed” way, using preopetopes and system Opt?;
• 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 [Ho 18], where valid proof trees are represented by certain expressions that evaluate without throwing any exception.

• In [CHM18] (see link on the first slide for a draft), we also

present variants of those systems for opetopic sets.

• We are experimenting with those new tools to

automatically check coherence laws for an appropriate definition of opetopic

• groupoid.

42

Conclusion

• In this presentation, we gave two ways to define opetopes

syntactically:

• 1. in a “named” way, using terms and system Opt!;
• 2. in an “unnamed” way, using preopetopes and system Opt?;
• 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 [Ho 18], where valid proof trees are represented by certain expressions that evaluate without throwing any exception.

• In [CHM18] (see link on the first slide for a draft), we also

present variants of those systems for opetopic sets.

• We are experimenting with those new tools to

automatically check coherence laws for an appropriate definition of opetopic ω-groupoid.

42

Thank you for your attention!

42

References i

John C. Baez and James Dolan. Higher-dimensional algebra. III. n-categories and the algebra of opetopes. Advances in Mathematics, 135(2):145–206, 1998. Pierre-Louis Curien, Cédric Ho Thanh, and Samuel Mimram. Type theoretical approaches for opetopes. In preparation, 2018. Cédric Ho Thanh.

• petopy.

https://github.com/altaris/opetopy, April 2018.

43

References ii

Joachim Kock, André Joyal, Michael Batanin, and Jean-François Mascari. Polynomial functors and opetopes. Advances in Mathematics, 224(6):2690–2737, 2010. Tom Leinster. Higher Operads, Higher Categories. Cambridge University Press, 2004.

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