The equivalence between many-to-one polygraphs and opetopic sets - PowerPoint PPT Presentation

The equivalence between many-to-one polygraphs and opetopic sets Cédric Ho Thanh 1 July 7 th , 2018 1 IRIF, Paris Diderot University, INSPIRE 2017 Fellow, This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 665850 1

This short talk informally presents the main notions and results of [HT, 2018] ( arXiv:1806.08645 [math.CT] ). Contents 1. Polygraphs 2. Opetopes 3. Main result and ideas of how to prove it 4. Conclusion 2

Polygraphs

In the same way, an n-polygraph (also called n-computad ) generates a free (strict) n -category, for n . Idea Given a graph G = ( G 0 − G 1 ) , s , t ← − one can generate the free category G ∗ : Objects vertices of G ; Generating morphisms edges of G ; Relations none. 3

Idea Given a graph G = ( G 0 − G 1 ) , s , t ← − one can generate the free category G ∗ : Objects vertices of G ; Generating morphisms edges of G ; Relations none. In the same way, an n-polygraph (also called n-computad ) generates a free (strict) n -category, for n ≤ ω . 3

A 1 -polygraph P is a graph s t P 1 P P 0 It generates a 1-category P which is the free category on P . Definition A 0 -polygraph P is a set. It generates a 0-category (aka a set) P ∗ = P . 4

Definition A 0 -polygraph P is a set. It generates a 0-category (aka a set) P ∗ = P . A 1 -polygraph P is a graph P = ( P 0 − P 1 ) . s , t ← − It generates a 1-category P ∗ which is the free category on P . 4

Definition An ( n + 1 ) -polygraph P is the data of an n -polygraph Q , a set P n + 1 , and two maps s , t Q ∗ − P n + 1 ← − n such that the globular identities hold: for p ∈ P n + 1 s s p = s t p , t s p = t t p . ⋅ ⋅ 5

Definition The ( n + 1 ) -category P ∗ is defined as follows: 1. its underlying n -category is Q ∗ (i.e. the n -category generated by the underlying n -polygraph Q of P ), so that P ∗ k = Q ∗ k for k ≤ n ; 2. its ( n + 1 ) -cells are the formal composites of elements of s , t P n + 1 according to Q ∗ − P n + 1 , as well as identities of cells − ← n of Q ∗ . 6

Definition Thus the ( n + 1 ) -polygraph P can be depicted as follows: P 0 P 1 P 2 P n P n + 1 ⋯ s , t s , t s , t s , t s , t P ∗ P ∗ P ∗ P ∗ P ∗ ⋯ 0 1 2 n n + 1 s , t s , t s , t s , t s , t The maps s are called source maps , and t target maps . Elements of P k are called k-generators , while elements of P ∗ k are called k-cells . The bottom row is exactly the underlying globular set of P ∗ . 7

The underlying -category P is defined as P colim P 0 P 1 Definition A ω -polygraph (or simply polygraph ) P is a sequence ( P ( n ) ∣ n < ω ) such that P ( n ) is an n -polygraph that is the underlying n -polygraph of P ( n + 1 ) . P 0 P 1 P 2 P n ⋯ ⋯ s , t s , t s , t s , t s , t P ∗ P ∗ P ∗ P ∗ ⋯ ⋯ 0 1 2 n s , t s , t s , t s , t s , t 8

Definition A ω -polygraph (or simply polygraph ) P is a sequence ( P ( n ) ∣ n < ω ) such that P ( n ) is an n -polygraph that is the underlying n -polygraph of P ( n + 1 ) . P 0 P 1 P 2 P n ⋯ ⋯ s , t s , t s , t s , t s , t P ∗ P ∗ P ∗ P ∗ ⋯ ⋯ 0 1 2 n s , t s , t s , t s , t s , t The underlying ω -category P ∗ is defined as P ∗ = colim ( P ∗ ( 1 ) ↪ ⋯ ) . ( 0 ) ↪ P ∗ 8

Definition A morphism of polygraphs f ∶ P ⟶ R is an ω -functor P ∗ ⟶ R ∗ mapping generators to generators. Let P ol be the category of polygraphs and such morphisms, and P ol n be the full subcategory of P ol spanned by n -polygraphs. 9

Proposition [Cheng, 2013] The category ol 3 is not. Thus ol n for n 3, and ol aren’t presheaf categories either. Question Which subcategories of ol are presheaf categories? Answer (sort of) A fair amount. See [Henry, 2017]. Proposition The categories P ol 0 , P ol 1 , and P ol 2 are presheaf categories. 10

Question Which subcategories of ol are presheaf categories? Answer (sort of) A fair amount. See [Henry, 2017]. Proposition The categories P ol 0 , P ol 1 , and P ol 2 are presheaf categories. Proposition [Cheng, 2013] The category P ol 3 is not. Thus P ol n for n ≥ 3, and P ol aren’t presheaf categories either. 10

Answer (sort of) A fair amount. See [Henry, 2017]. Proposition The categories P ol 0 , P ol 1 , and P ol 2 are presheaf categories. Proposition [Cheng, 2013] The category P ol 3 is not. Thus P ol n for n ≥ 3, and P ol aren’t presheaf categories either. Question Which subcategories of P ol are presheaf categories? 10

Proposition The categories P ol 0 , P ol 1 , and P ol 2 are presheaf categories. Proposition [Cheng, 2013] The category P ol 3 is not. Thus P ol n for n ≥ 3, and P ol aren’t presheaf categories either. Question Which subcategories of P ol are presheaf categories? Answer (sort of) A fair amount. See [Henry, 2017]. 10

A polygraph P is many-to-one if for all generator p P n with n 1, we have t p P n 1 (as opposed to just P n 1 ). t t t t t P 0 P 1 P 2 P n s s s s s P 0 P 1 P 2 P n s t s t s t s t s t Teaser The category is a presheaf category. ol Many-to-one polygraphs Today we will focus on the subcategory of many-to-one polygraphs P ol ▽ . 11

Teaser The category is a presheaf category. ol Many-to-one polygraphs Today we will focus on the subcategory of many-to-one polygraphs P ol ▽ . A polygraph P is many-to-one if for all generator p ∈ P n with n ≥ 1, we have t p ∈ P n − 1 (as opposed to just P ∗ n − 1 ). t t t t t P 0 P 1 P 2 P n ⋯ ⋯ s s s s s P ∗ P ∗ P ∗ P ∗ ⋯ ⋯ 0 1 2 n s , t s , t s , t s , t s , t 11

Many-to-one polygraphs Today we will focus on the subcategory of many-to-one polygraphs P ol ▽ . A polygraph P is many-to-one if for all generator p ∈ P n with n ≥ 1, we have t p ∈ P n − 1 (as opposed to just P ∗ n − 1 ). t t t t t P 0 P 1 P 2 P n ⋯ ⋯ s s s s s P ∗ P ∗ P ∗ P ∗ ⋯ ⋯ 0 1 2 n s , t s , t s , t s , t s , t Teaser The category P ol ▽ is a presheaf category. 11

Opetopes

They have been reworked in [Kock et al., 2010] to arrive at the following moto: “An n-opetope is a tree whose nodes are n 1 -opetopes, and whose edges are n 2 -opetopes.” Idea Opetopes were originally introduced by Baez and Dolan in [Baez and Dolan, 1998] as an algebraic structure to describe compositions and coherence laws in weak higher dimensional categories. 12

Idea Opetopes were originally introduced by Baez and Dolan in [Baez and Dolan, 1998] as an algebraic structure to describe compositions and coherence laws in weak higher dimensional categories. They have been reworked in [Kock et al., 2010] to arrive at the following moto: “An n-opetope is a tree whose nodes are ( n − 1 ) -opetopes, and whose edges are ( n − 2 ) -opetopes.” 12

- there is a unique 0-opetope, the point , drawn as - there is a unique 1-opetope, the arrow , drawn as notice how both ends of the arrow are points (i.e. 0-opetopes); Definition (sketch) Here is how it goes graphically: 13

- there is a unique 1-opetope, the arrow , drawn as notice how both ends of the arrow are points (i.e. 0-opetopes); Definition (sketch) Here is how it goes graphically: - there is a unique 0-opetope, the point , drawn as . 13

Definition (sketch) Here is how it goes graphically: - there is a unique 0-opetope, the point , drawn as . - there is a unique 1-opetope, the arrow , drawn as . . notice how both ends of the arrow are points (i.e. 0-opetopes); 13

Other examples of 2-opetopes include Definition (sketch) - a 2-opetope is a shape of the form: . . ⇓ . . where the top part ( source ) is any arrangement (or pasting scheme) of 1-opetopes glued along 0-opetopes, and where the bottom part ( target ) consists in only one 1-opetope. 14

Definition (sketch) - a 2-opetope is a shape of the form: . . ⇓ . . where the top part ( source ) is any arrangement (or pasting scheme) of 1-opetopes glued along 0-opetopes, and where the bottom part ( target ) consists in only one 1-opetope. Other examples of 2-opetopes include . ⇓ ⇓ . . 14

Definition (sketch) - a 3-opetope is a shape of the form: . . ⇓ . . ⇛ . . ⇓ ⇓ ⇓ . . . . where the left part ( source ) is any pasting scheme of 2-opetopes glued along 1-opetopes, and where the right part ( target ) consists in only one 2-opetope parallel to the overall boundary of the source . 15

Here is an example of 4-opetope [Cheng and Lauda, 2004]: Definition (sketch) - and so on: an n -opetope (for n ≥ 2) is a source pasting scheme of ( n − 1 ) -opetopes glued along ( n − 2 ) -opetopes, together with a target parallel ( n − 1 ) -opetope. 16

Definition (sketch) - and so on: an n -opetope (for n ≥ 2) is a source pasting scheme of ( n − 1 ) -opetopes glued along ( n − 2 ) -opetopes, together with a target parallel ( n − 1 ) -opetope. Here is an example of 4-opetope [Cheng and Lauda, 2004]: 16