Pseudofunctors and simplicial categories Overview of the problem - - PowerPoint PPT Presentation

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Pseudofunctors and simplicial categories

Nick Gurski Case Western Reserve University Category Theory Octoberfest October 26, 2019

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

This talk has two components

• 1. I want tell you about some joint work with Daniel

Sch¨

• appi. Our goal was to construct some explicit

pseudofunctors from simplicially enriched categories in the style of Gordon-Power’s Enrichment through variation.

2 / 19

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

This talk has two components

• 1. I want tell you about some joint work with Daniel

Sch¨

• appi. Our goal was to construct some explicit

pseudofunctors from simplicially enriched categories in the style of Gordon-Power’s Enrichment through variation.

• 2. The second implicit component of this talk is an

invitation for someone else to do this better.

2 / 19

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

This talk has two components

• 1. I want tell you about some joint work with Daniel

Sch¨

• appi. Our goal was to construct some explicit

pseudofunctors from simplicially enriched categories in the style of Gordon-Power’s Enrichment through variation.

• 2. The second implicit component of this talk is an

invitation for someone else to do this better.

◮ Variable base of enrichment

2 / 19

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

This talk has two components

• 1. I want tell you about some joint work with Daniel

Sch¨

• appi. Our goal was to construct some explicit

pseudofunctors from simplicially enriched categories in the style of Gordon-Power’s Enrichment through variation.

• 2. The second implicit component of this talk is an

invitation for someone else to do this better.

◮ Variable base of enrichment ◮ Variable strength of enrichment

2 / 19

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

This talk has two components

• 1. I want tell you about some joint work with Daniel

Sch¨

• appi. Our goal was to construct some explicit

pseudofunctors from simplicially enriched categories in the style of Gordon-Power’s Enrichment through variation.

• 2. The second implicit component of this talk is an

invitation for someone else to do this better.

◮ Variable base of enrichment ◮ Variable strength of enrichment ◮ But in a usable format

2 / 19

Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Outline

Overview of the problem Enrichment through variation Our higher dimensional version

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Outline

Overview of the problem Enrichment through variation Our higher dimensional version

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Setup

Let E be some nice model category whose objects we think

• f as higher categories.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Setup

Let E be some nice model category whose objects we think

• f as higher categories.

◮ Simplicial sets with Joyal model structure ◮ Cat with the canonical model structure

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Setup

Let E be some nice model category whose objects we think

• f as higher categories.

◮ Simplicial sets with Joyal model structure ◮ Cat with the canonical model structure We can study ◮ sE: simplicial objects in E, ◮ qCat(E): internal quasicategories in E, and ◮ Segal(E): internal Segal categories in E.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Application

We can write down a functor qCat(E) → Cat by A → Ho(sE/A).

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Application

We can write down a functor qCat(E) → Cat by A → Ho(sE/A). Question: Can we extend this to a functor between higher categories?

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Application

We can write down a functor qCat(E) → Cat by A → Ho(sE/A). Question: Can we extend this to a functor between higher categories? Higher dimensional versions: ◮ Cat is a 2-category* ◮ qCat(E) is a simplicially enriched category

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Application

We can write down a functor qCat(E) → Cat by A → Ho(sE/A). Question: Can we extend this to a functor between higher categories? Higher dimensional versions: ◮ Cat is a 2-category* ◮ qCat(E) is a simplicially enriched category Conclusion: we are looking for functors from a simplicially enriched category to a 2-category.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Graphs first

Recall the adjunction τ1 ⊣ N ◮ N : sSet → Cat, ◮ τ1 : Cat → sSet.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Graphs first

Recall the adjunction τ1 ⊣ N ◮ N : sSet → Cat, ◮ τ1 : Cat → sSet.

Lemma

For any adjunction F ⊣ U with F : A ⇄ B and any category D, there is an induced adjunction F∗ ⊣ U∗ with F∗ : [D, A] ⇄ [D, B] : U∗.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Graphs first

Recall the adjunction τ1 ⊣ N ◮ N : sSet → Cat, ◮ τ1 : Cat → sSet.

Lemma

For any adjunction F ⊣ U with F : A ⇄ B and any category D, there is an induced adjunction F∗ ⊣ U∗ with F∗ : [D, A] ⇄ [D, B] : U∗. Apply with D = m ⇒ o to the adjunction above to get (τ1)∗ ⊣ N∗ with (τ1)∗ : Graph(sSet) ⇄ Graph(Cat) : N∗.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

From graphs to enriched categories

Our lemma constructing F∗ ⊣ U∗ for graphs can be extended.

Proposition

• 1. If P : A → B is a lax monoidal functor between

monoidal categories, then it induces a functor P : Mon(A) → Mon(B).

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

From graphs to enriched categories

Our lemma constructing F∗ ⊣ U∗ for graphs can be extended.

Proposition

• 1. If P : A → B is a lax monoidal functor between

monoidal categories, then it induces a functor P : Mon(A) → Mon(B).

• 2. Under the same hypotheses, P induces a functor

P∗ : A-Cat → B-Cat by applying P to the hom-objects.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

From graphs to enriched categories

Our lemma constructing F∗ ⊣ U∗ for graphs can be extended.

Proposition

• 1. If P : A → B is a lax monoidal functor between

monoidal categories, then it induces a functor P : Mon(A) → Mon(B).

• 2. Under the same hypotheses, P induces a functor

P∗ : A-Cat → B-Cat by applying P to the hom-objects.

• 3. If F ⊣ U is a monoidal adjunction, it induces an

adjunction F∗ : A-Cat ⇄ B-Cat : U∗.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Lemma

τ1 ⊣ N is a monoidal adjunction.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Lemma

τ1 ⊣ N is a monoidal adjunction.

Corollary

We have an induced adjunction between simplicially enriched categories and 2-categories.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Lemma

τ1 ⊣ N is a monoidal adjunction.

Corollary

We have an induced adjunction between simplicially enriched categories and 2-categories. Refined question: Can we extend our original functor qCat(E) → Cat to one of the equivalent kinds below? qCat(E) → N∗Cat

• (τ1)∗qCat(E) → Cat

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Outline

Overview of the problem Enrichment through variation Our higher dimensional version

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

In broad strokes

Gordon and Power compare two concepts:

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

In broad strokes

Gordon and Power compare two concepts: ◮ enriched categories (over a fixed V*) and ◮ representations of V which are pseudofunctors.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

In broad strokes

Gordon and Power compare two concepts: ◮ enriched categories (over a fixed V*) and ◮ representations of V which are pseudofunctors. Think: group action ← → group homo. G × A → A G → Aut(A) The left are the enriched categories, while the right are the representations.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

In broad strokes

Gordon and Power compare two concepts: ◮ enriched categories (over a fixed V*) and ◮ representations of V which are pseudofunctors. Think: group action ← → group homo. G × A → A G → Aut(A) The left are the enriched categories, while the right are the representations. We fix a monoidal category V over which to enrich. For any enriched category C, write C0 for the underlying category.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Tensors

Definition

A V-category C is tensored if V

• v, C(c, −)
• : C → V

is representable. Write the representing object as v ⊙ c ∈ C.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Tensors

Definition

A V-category C is tensored if V

• v, C(c, −)
• : C → V

is representable. Write the representing object as v ⊙ c ∈ C.

Our examples

◮ Any closed monoidal category (Cat) is tensored over itself

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Tensors

Definition

A V-category C is tensored if V

• v, C(c, −)
• : C → V

is representable. Write the representing object as v ⊙ c ∈ C.

Our examples

◮ Any closed monoidal category (Cat) is tensored over itself ◮ (τ1)∗qCat(E) is also tensored over Cat using nerves

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Representations

Definition

A V-representation L is a category (unenriched!) with ◮ an action ⋆ : V0 × L → L and ◮ natural iso’s I ⋆ c ∼ = c, w ⋆ (v ⋆ c) ∼ = (w ⊗ v) ⋆ c satisfying two axioms.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Representations

Definition

A V-representation L is a category (unenriched!) with ◮ an action ⋆ : V0 × L → L and ◮ natural iso’s I ⋆ c ∼ = c, w ⋆ (v ⋆ c) ∼ = (w ⊗ v) ⋆ c satisfying two axioms. There are further notions of maps of representations, and 2-cells between those, of a similar flavor.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Representations

Definition

A V-representation L is a category (unenriched!) with ◮ an action ⋆ : V0 × L → L and ◮ natural iso’s I ⋆ c ∼ = c, w ⋆ (v ⋆ c) ∼ = (w ⊗ v) ⋆ c satisfying two axioms. There are further notions of maps of representations, and 2-cells between those, of a similar flavor.

Immediate consequence

V-Rep ∼ = Lax(Ω−1V, Cat) as 2-categories. ◮ Lax(Ω−1V, Cat) is pseudofunctors, lax transformations, and modifications ◮ Ω−1V is the 1-object bicategory with V as the single endo-hom-category

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

The standard embedding

Theorem (Gordon-Power)

• 1. A tensored V-category C gives rise to a V-representation

with L = C0.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

The standard embedding

Theorem (Gordon-Power)

• 1. A tensored V-category C gives rise to a V-representation

with L = C0.

• 2. This assignment is the object part of a locally faithful

2-functor V-Cat⊙ → V-Rep.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

The standard embedding

Theorem (Gordon-Power)

• 1. A tensored V-category C gives rise to a V-representation

with L = C0.

• 2. This assignment is the object part of a locally faithful

2-functor V-Cat⊙ → V-Rep.

• 3. If V is right closed, this 2-functor is full and faithful.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

The standard embedding

Theorem (Gordon-Power)

• 1. A tensored V-category C gives rise to a V-representation

with L = C0.

• 2. This assignment is the object part of a locally faithful

2-functor V-Cat⊙ → V-Rep.

• 3. If V is right closed, this 2-functor is full and faithful.
• 4. In this case, the essential image consists of all closed

representations.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Non-standard embeddings for dense subcategories

Gordon-Power explain a further generalization of this theory.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Non-standard embeddings for dense subcategories

Gordon-Power explain a further generalization of this theory. ◮ Use dense ω ⊆ V

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Non-standard embeddings for dense subcategories

Gordon-Power explain a further generalization of this theory. ◮ Use dense ω ⊆ V ◮ Study ω-Rep instead

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Non-standard embeddings for dense subcategories

Gordon-Power explain a further generalization of this theory. ◮ Use dense ω ⊆ V ◮ Study ω-Rep instead ◮ Get locally faithful V-Cat⊙,ω → ω-Rep and conditions under which this is full and faithful

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Non-standard embeddings for dense subcategories

Gordon-Power explain a further generalization of this theory. ◮ Use dense ω ⊆ V ◮ Study ω-Rep instead ◮ Get locally faithful V-Cat⊙,ω → ω-Rep and conditions under which this is full and faithful ◮ Can use ω to detect whether C has all tensors

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Refined question: Can we extend our original functor qCat(E) → Cat to one of the equivalent kinds below? qCat(E) → N∗Cat

• (τ1)∗qCat(E) → Cat

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Refined question: Can we extend our original functor qCat(E) → Cat to one of the equivalent kinds below? qCat(E) → N∗Cat

• (τ1)∗qCat(E) → Cat

These enriched categories are all tensored, so by the above discussion a tensor-preserving map between them amounts to a map of representations.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Back to our application

Refined question: Can we extend our original functor qCat(E) → Cat to one of the equivalent kinds below? qCat(E) → N∗Cat

• (τ1)∗qCat(E) → Cat

These enriched categories are all tensored, so by the above discussion a tensor-preserving map between them amounts to a map of representations. Unfortunately, we don’t seem to have such a thing: our maps of representations are constructed from universal properties, so the axioms only hold up to isomorphism.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Outline

Overview of the problem Enrichment through variation Our higher dimensional version

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

A poorly stated theorem

Theorem [G-Sch¨ appi]

A weak map of Cat-representations induces a weakly tensor-preserving pseudofunctor between 2-categories.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

A poorly stated theorem

Theorem [G-Sch¨ appi]

A weak map of Cat-representations induces a weakly tensor-preserving pseudofunctor between 2-categories.

Consequence

We get a pseudofunctor (τ1)∗qCat(E) → Cat using the dense subcategory version of the embedding.

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Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Enrichment through variation Our higher dimensional version

Thanks!

• Σ

Σ ⇑ ⇑ =

• ⇑

⇑ Σ ⇑ ⇑

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