Skew structures in 2-category theory and homotopy theory John - - PowerPoint PPT Presentation

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Skew structures in 2-category theory and homotopy theory

John Bourke

Department of Mathematics and Statistics Masaryk University

CT2015

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Introduction

◮ A monoidal category C involves invertible maps

α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), l : I ⊗ A → A and r : A → A ⊗ I.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Introduction

◮ A monoidal category C involves invertible maps

α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), l : I ⊗ A → A and r : A → A ⊗ I.

◮ Recently skew monoidal categories have come to attention:

non-invertible maps.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Introduction

◮ A monoidal category C involves invertible maps

α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), l : I ⊗ A → A and r : A → A ⊗ I.

◮ Recently skew monoidal categories have come to attention:

non-invertible maps.

◮ If C is a category with notion of weak equivalence it is natural

to consider intermediate case in which the maps are weak equivalences.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew monoidal categories

A skew monoidal category (Szlachanyi, 2012) has

◮ A functor ⊗ : C × C → C and unit object I

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew monoidal categories

A skew monoidal category (Szlachanyi, 2012) has

◮ A functor ⊗ : C × C → C and unit object I

and natural maps

◮ α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I

satisfying five axioms.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew monoidal categories

A skew monoidal category (Szlachanyi, 2012) has

◮ A functor ⊗ : C × C → C and unit object I

and natural maps

◮ α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I

satisfying five axioms. The skew monoidal category C is called monoidal if the transformations α,l and r are invertible.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew monoidal categories

A skew monoidal category (Szlachanyi, 2012) has

◮ A functor ⊗ : C × C → C and unit object I

and natural maps

◮ α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I

satisfying five axioms. The skew monoidal category C is called monoidal if the transformations α,l and r are invertible. Then three axioms are redundant (Max Kelly).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew closed categories

A skew closed category (Street, 2013) has

◮ A functor [−, −] : C op × C → C and unit object I

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew closed categories

A skew closed category (Street, 2013) has

◮ A functor [−, −] : C op × C → C and unit object I

and natural maps

◮ L : [B, C] → [[A, B], [A, C]] ◮ j : I → [A, A] ◮ i : [I, A] → A

satisfying five axioms.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Skew closed categories

A skew closed category (Street, 2013) has

◮ A functor [−, −] : C op × C → C and unit object I

and natural maps

◮ L : [B, C] → [[A, B], [A, C]] ◮ j : I → [A, A] ◮ i : [I, A] → A

satisfying five axioms. The skew closed category C is called closed if

◮ i : [I, A] → A is invertible, ◮ the function v : C(A, B) → C(I, [A, B]) sending f : A → B to

I [A, A] [A, B]

j

• [A,f ]

is invertible.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Weak maps and skewness

◮ The iso in a genuine closed category

C(A, B) ∼ = C(I, [A, B]) says that elements of [A, B] are just maps A → B of C.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Weak maps and skewness

◮ The iso in a genuine closed category

C(A, B) ∼ = C(I, [A, B]) says that elements of [A, B] are just maps A → B of C.

◮ If we fiddle with that, we go skew.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Weak maps and skewness

◮ The iso in a genuine closed category

C(A, B) ∼ = C(I, [A, B]) says that elements of [A, B] are just maps A → B of C.

◮ If we fiddle with that, we go skew. ◮ For instance, if elements of [A, B] are weak maps.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Weak maps and skewness

◮ The iso in a genuine closed category

C(A, B) ∼ = C(I, [A, B]) says that elements of [A, B] are just maps A → B of C.

◮ If we fiddle with that, we go skew. ◮ For instance, if elements of [A, B] are weak maps. ◮ Sometimes forced to look at weak maps to get correct

enrichment.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

◮ If B is symmetric monoidal then ⊗ : B2 B is only a

pseudomap(!): (a ⊗ b) ⊗ (c ⊗ d) ∼ = (a ⊗ c) ⊗ (b ⊗ d).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

◮ If B is symmetric monoidal then ⊗ : B2 B is only a

pseudomap(!): (a ⊗ b) ⊗ (c ⊗ d) ∼ = (a ⊗ c) ⊗ (b ⊗ d).

◮ Category SMCatp(A, B) has pointwise symmetric monoidal

structure [A, B]: (f ⊗ g)(a) = fa ⊗ ga.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

◮ If B is symmetric monoidal then ⊗ : B2 B is only a

pseudomap(!): (a ⊗ b) ⊗ (c ⊗ d) ∼ = (a ⊗ c) ⊗ (b ⊗ d).

◮ Category SMCatp(A, B) has pointwise symmetric monoidal

structure [A, B]: (f ⊗ g)(a) = fa ⊗ ga.

◮ But full subcat SMCats(A, B) of strict maps does not!

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

◮ If B is symmetric monoidal then ⊗ : B2 B is only a

pseudomap(!): (a ⊗ b) ⊗ (c ⊗ d) ∼ = (a ⊗ c) ⊗ (b ⊗ d).

◮ Category SMCatp(A, B) has pointwise symmetric monoidal

structure [A, B]: (f ⊗ g)(a) = fa ⊗ ga.

◮ But full subcat SMCats(A, B) of strict maps does not! ◮ With unit the free symmetric monoidal category F1,

(SMCatp, [−, −], F1) not even skew closed -though it is a pseudo-closed 2-category in sense of Hyland-Power.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Example of symmetric monoidal categories

◮ SMCats category of symmetric monoidal categories and strict

maps.

◮ If B is symmetric monoidal then ⊗ : B2 B is only a

pseudomap(!): (a ⊗ b) ⊗ (c ⊗ d) ∼ = (a ⊗ c) ⊗ (b ⊗ d).

◮ Category SMCatp(A, B) has pointwise symmetric monoidal

structure [A, B]: (f ⊗ g)(a) = fa ⊗ ga.

◮ But full subcat SMCats(A, B) of strict maps does not! ◮ With unit the free symmetric monoidal category F1,

(SMCatp, [−, −], F1) not even skew closed -though it is a pseudo-closed 2-category in sense of Hyland-Power.

◮ But restricts to true skew closed structure

(SMCats, [−, −], F1)! Not closed!

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The correspondence

Theorem (Street, 2013)

Consider a category C and

◮ ⊗ : C × C → C and [−.−] : C op × C → C, ◮ a natural isomorphism φ : C(A ⊗ B, C) ∼

= C(A, [B, C]),

◮ an object I ∈ C.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The correspondence

Theorem (Street, 2013)

Consider a category C and

◮ ⊗ : C × C → C and [−.−] : C op × C → C, ◮ a natural isomorphism φ : C(A ⊗ B, C) ∼

= C(A, [B, C]),

◮ an object I ∈ C.

There is a bijection between

• 1. Extensions of (C, ⊗, I) to a skew monoidal structure,
• 2. Extensions of (C, [−, −], I) to a skew closed structure.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The correspondence

Theorem (Street, 2013)

Consider a category C and

◮ ⊗ : C × C → C and [−.−] : C op × C → C, ◮ a natural isomorphism φ : C(A ⊗ B, C) ∼

= C(A, [B, C]),

◮ an object I ∈ C.

There is a bijection between

• 1. Extensions of (C, ⊗, I) to a skew monoidal structure,
• 2. Extensions of (C, [−, −], I) to a skew closed structure.

Proof.

Both skew monoidal and skew closed categories involve three transformations and five axioms. These transform into one another

• ne by one through the adjunction.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Monoidal skew closed categories

By Street’s theorem, the following are in bijection:

• 1. A skew monoidal category (C, ⊗, I) equipped with adjunctions

− ⊗ A ⊣ [A, −].

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Monoidal skew closed categories

By Street’s theorem, the following are in bijection:

• 1. A skew monoidal category (C, ⊗, I) equipped with adjunctions

− ⊗ A ⊣ [A, −].

• 2. A skew closed category (C, [−, −], I) equipped with

adjunctions − ⊗ A ⊣ [A, −].

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Monoidal skew closed categories

By Street’s theorem, the following are in bijection:

• 1. A skew monoidal category (C, ⊗, I) equipped with adjunctions

− ⊗ A ⊣ [A, −].

• 2. A skew closed category (C, [−, −], I) equipped with

adjunctions − ⊗ A ⊣ [A, −].

• 3. The totality of data (C, ⊗, [−, −], I, φ, α, l, r, L, i, j, v, t) and

relations between the monoidal and closed structure.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Monoidal skew closed categories

By Street’s theorem, the following are in bijection:

• 1. A skew monoidal category (C, ⊗, I) equipped with adjunctions

− ⊗ A ⊣ [A, −].

• 2. A skew closed category (C, [−, −], I) equipped with

adjunctions − ⊗ A ⊣ [A, −].

• 3. The totality of data (C, ⊗, [−, −], I, φ, α, l, r, L, i, j, v, t) and

relations between the monoidal and closed structure. Such is called a monoidal skew closed category.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

Monoidal skew closed categories

By Street’s theorem, the following are in bijection:

• 1. A skew monoidal category (C, ⊗, I) equipped with adjunctions

− ⊗ A ⊣ [A, −].

• 2. A skew closed category (C, [−, −], I) equipped with

adjunctions − ⊗ A ⊣ [A, −].

• 3. The totality of data (C, ⊗, [−, −], I, φ, α, l, r, L, i, j, v, t) and

relations between the monoidal and closed structure. Such is called a monoidal skew closed category. The map t : [A ⊗ B, C] → [A, [B, C]] is defined by [A ⊗ B, C]

L

[[B, A ⊗ B], [B, C]]

[u,1]

[A, [B, C]]

where u : A → [B, A ⊗ B] is the unit.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The Eilenberg-Kelly theorem revisited

Immediate consequence of the proof of Street’s theorem is Eilenberg and Kelly’s theorem – reformulated using language of skew monoidal categories below. Allows us to recognise monoidal structure from closed structure, and vice versa.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The Eilenberg-Kelly theorem revisited

Immediate consequence of the proof of Street’s theorem is Eilenberg and Kelly’s theorem – reformulated using language of skew monoidal categories below. Allows us to recognise monoidal structure from closed structure, and vice versa.

Theorem (Eilenberg-Kelly, 1966)

Let (C, ⊗, [−, −], I) be monoidal skew closed.

• 1. Then (C, ⊗, I) is monoidal if and only if (C, [−, −], I) is

closed and t : [A ⊗ B, C] → [A, [B, C]] is invertible.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew monoidal categories Skew closed categories Skew monoidal versus skew closed The Eilenberg-Kelly theorem revisited

The Eilenberg-Kelly theorem revisited

Immediate consequence of the proof of Street’s theorem is Eilenberg and Kelly’s theorem – reformulated using language of skew monoidal categories below. Allows us to recognise monoidal structure from closed structure, and vice versa.

Theorem (Eilenberg-Kelly, 1966)

Let (C, ⊗, [−, −], I) be monoidal skew closed.

• 1. Then (C, ⊗, I) is monoidal if and only if (C, [−, −], I) is

closed and t : [A ⊗ B, C] → [A, [B, C]] is invertible.

• 2. (Day-LaPlaza, 1978) If (C, [−, −], I) is symmetric closed, then

t : [A ⊗ B, C] → [A, [B, C]] is automatically invertible.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Model categories

◮ A Quillen model category C with functorial factorisations.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Model categories

◮ A Quillen model category C with functorial factorisations. ◮ Cofibrant replacement functor Q and fibrant replacement

functor R.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Left deriving skew monoidal structure

C a model category with skew monoidal structure (C, ⊗, I).

Axiom M

The unit I is cofibrant. The functor ⊗ : C2 → C preserves cofibrant

• bjects and weak equivalences between them.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Left deriving skew monoidal structure

C a model category with skew monoidal structure (C, ⊗, I).

Axiom M

The unit I is cofibrant. The functor ⊗ : C2 → C preserves cofibrant

• bjects and weak equivalences between them.

Then the left derived functor ⊗l : Ho(C)2 → Ho(C) of ⊗ exists and is given by A ⊗l B = QA ⊗ QB. In particular, we get the left derived skew monoidal structure (Ho(C), ⊗l, I) on the homotopy category. (Essentially Hovey’s construction.) Unit constraints: QI ⊗ QA

p⊗1 I ⊗ QA l

QA

p

A

A

p−1 QA r QA ⊗ I (1⊗p)−1

QA ⊗ QI

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopy monoidal categories

Definition

Let (C, ⊗, I) be a skew monoidal category satisfying Axiom M. We call (C, ⊗, I) homotopy monoidal if the left derived skew monoidal structure (Ho(C), ⊗l, I) is genuine monoidal.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopy monoidal categories

Definition

Let (C, ⊗, I) be a skew monoidal category satisfying Axiom M. We call (C, ⊗, I) homotopy monoidal if the left derived skew monoidal structure (Ho(C), ⊗l, I) is genuine monoidal.

Proposition

Let (C, ⊗, I) be a skew monoidal category with a model structure satisfying Axiom M. The following are equivalent.

• 1. (C, ⊗, I) is homotopy monoidal.
• 2. When all the objects involved are cofibrant the maps

α : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), r : X → X ⊗ I and l : I ⊗ X → X are weak equivalences.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Right deriving closed structure

Similar story: Axiom C, homotopy closed categories . . . - but not required here.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Deriving monoidal skew closed structure

(C, ⊗, [−, −], I) monoidal skew closed.

Axiom MC

The unit I is cofibrant. One of the following equivalent conditions are satisfied.

• 1. For all X the functor [X, −] is right Quillen and for fibrant Y

the functor [−, Y ] preserves all weak equivalences.

• 2. For all X the functor − ⊗ X is left Quillen and for each

cofibrant Y the functor Y ⊗ − preserves all weak equivalences.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Deriving monoidal skew closed structure

(C, ⊗, [−, −], I) monoidal skew closed.

Axiom MC

The unit I is cofibrant. One of the following equivalent conditions are satisfied.

• 1. For all X the functor [X, −] is right Quillen and for fibrant Y

the functor [−, Y ] preserves all weak equivalences.

• 2. For all X the functor − ⊗ X is left Quillen and for each

cofibrant Y the functor Y ⊗ − preserves all weak equivalences.

◮ Axiom MC implies Axiom M. So the left derived skew

monoidal structure (Ho(C), ⊗l, I) exists.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopical Eilenberg-Kelly theorem

Allows us to recognise homotopy monoidal structure.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopical Eilenberg-Kelly theorem

Allows us to recognise homotopy monoidal structure.

Theorem

Let (C, ⊗, [−, −], I) be monoidal skew closed and satisfy Axiom MC.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopical Eilenberg-Kelly theorem

Allows us to recognise homotopy monoidal structure.

Theorem

Let (C, ⊗, [−, −], I) be monoidal skew closed and satisfy Axiom MC.

◮ Then (C, ⊗, I) is homotopy monoidal if and only if

• 1. For cofibrant A and fibrant B the function

v : C(A, B) → C(I, [A, B]) is a bijection on homotopy classes

• f maps.
• 2. For all fibrant A the map iA : [I, A] → A is a weak equivalence

in C.

• 3. For all cofibrant A and fibrant C the map

t : [A ⊗ B, C] → [A, [B, C]] is a weak equivalence.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Homotopical Eilenberg-Kelly theorem

Allows us to recognise homotopy monoidal structure.

Theorem

Let (C, ⊗, [−, −], I) be monoidal skew closed and satisfy Axiom MC.

◮ Then (C, ⊗, I) is homotopy monoidal if and only if

• 1. For cofibrant A and fibrant B the function

v : C(A, B) → C(I, [A, B]) is a bijection on homotopy classes

• f maps.
• 2. For all fibrant A the map iA : [I, A] → A is a weak equivalence

in C.

• 3. For all cofibrant A and fibrant C the map

t : [A ⊗ B, C] → [A, [B, C]] is a weak equivalence.

◮ If C is symmetric skew closed then the condition (3) on t is

automatic in the presence of (1) and (2).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

◮ Hyland and Power (2002) showed that if T is an accessible

pseudo-commutative 2-monad on Cat then T-Algp is a closed monoidal bicategory.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

◮ Hyland and Power (2002) showed that if T is an accessible

pseudo-commutative 2-monad on Cat then T-Algp is a closed monoidal bicategory.

• 1. Show (T-Algp, [−, −], F1) a pseudo-closed 2-category – slight

weakening of the notion of closed category.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

◮ Hyland and Power (2002) showed that if T is an accessible

pseudo-commutative 2-monad on Cat then T-Algp is a closed monoidal bicategory.

• 1. Show (T-Algp, [−, −], F1) a pseudo-closed 2-category – slight

weakening of the notion of closed category.

• 2. Prove that each [A, −] has left biadjoint − ⊗ A.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

◮ Hyland and Power (2002) showed that if T is an accessible

pseudo-commutative 2-monad on Cat then T-Algp is a closed monoidal bicategory.

• 1. Show (T-Algp, [−, −], F1) a pseudo-closed 2-category – slight

weakening of the notion of closed category.

• 2. Prove that each [A, −] has left biadjoint − ⊗ A.
• 3. Conclude that (T-Algp, ⊗, F1) is a monoidal bicategory using

bicategorical analogue of Eilenberg-Kelly theorem.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads

◮ CMon is the category of algebras for a commutative monad T

• n Set.

◮ SMCatp is the 2-category of algebras and pseudomaps T-Algp

for a pseudo-commutative 2-monad T on Cat.

◮ Hyland and Power (2002) showed that if T is an accessible

pseudo-commutative 2-monad on Cat then T-Algp is a closed monoidal bicategory.

• 1. Show (T-Algp, [−, −], F1) a pseudo-closed 2-category – slight

weakening of the notion of closed category.

• 2. Prove that each [A, −] has left biadjoint − ⊗ A.
• 3. Conclude that (T-Algp, ⊗, F1) is a monoidal bicategory using

bicategorical analogue of Eilenberg-Kelly theorem.

◮ Did not give details of bicategorical EK-theorem. Expressed

dissatisfaction with this bit because of complex calculations.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

◮ Model structure on T-Algs of Lack (2007).

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

◮ Model structure on T-Algs of Lack (2007). ◮ (T-Algs, [−, −], F1, ⊗) satisfies Axiom MC and is homotopy

monoidal by homotopical EK-theorem.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

◮ Model structure on T-Algs of Lack (2007). ◮ (T-Algs, [−, −], F1, ⊗) satisfies Axiom MC and is homotopy

monoidal by homotopical EK-theorem.

◮ Restricts to skew monoidal structure on full sub 2-category

T-Algc of T-Algs consisting of cofibrant (flexible) algebras.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

◮ Model structure on T-Algs of Lack (2007). ◮ (T-Algs, [−, −], F1, ⊗) satisfies Axiom MC and is homotopy

monoidal by homotopical EK-theorem.

◮ Restricts to skew monoidal structure on full sub 2-category

T-Algc of T-Algs consisting of cofibrant (flexible) algebras.

◮ A monoidal bicategory.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Pseudo-commutative 2-monads continued

◮ We can prove this with less computations by going skew. ◮ If T is an accessible pseudocommutative 2-monad on Cat

then the 2-category T-Algs of algebras and strict maps is part

• f skew closed 2-category (T-Algs, [−, −], F1).

◮ Each [A, −] : T-Algs → T-Algs has true left adjoint − ⊗ A. So

monoidal skew closed 2-category (T-Algs, [−, −], F1, ⊗).

◮ Model structure on T-Algs of Lack (2007). ◮ (T-Algs, [−, −], F1, ⊗) satisfies Axiom MC and is homotopy

monoidal by homotopical EK-theorem.

◮ Restricts to skew monoidal structure on full sub 2-category

T-Algc of T-Algs consisting of cofibrant (flexible) algebras.

◮ A monoidal bicategory. ◮ Inclusion T-Algc → T-Algp is a biequivalence, so can

transport the monoidal bicategory structure to T-Algp.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

2-categories

◮ 2-Cats is the category of 2-categories and strict 2-functors.

This admits a Quillen model structure in which the weak equivalences are the biequivalences. (Lack 2002)

◮ Hom(A, B) is the 2-category of homomorphisms,

pseudonatural transformations and modifications. 1 the terminal 2-category.

◮ (2-Cat, Hom(−, −), 1) is part of a monoidal skew closed

category satisfying Axiom MC. It is homotopy monoidal.

◮ This is induced by a standard construction on the symmetric

monoidal closed structure associated to the Gray tensor product using by the fact that the pseudofunctor classifier Q : 2-Cat → 2-Cat is a monoidal comonad. Hom(A, B) = [QA, B].

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Bicategories

◮ Bicats is the category of bicategories and strict 2-functors.

This admits a Quillen model structure in which the weak equivalences are the biequivalences. (Lack 2004)

◮ Hom(A, B) is the bicategory of homomorphisms,

pseudonatural transformations and modifications.

◮ The unit is subtler. The problem is that if A is a bicategory

the canonical homomorphism 1 Hom(A, A) is not strict but

• nly a normal homomorphism. But it corresponds to a unique

strict map Q1 → Hom(A, A) where Q1 is the normal homomorphism classifier.

◮ (Bicats, Hom(−, −), Q1) is part of a monoidal skew closed

category satisfying Axiom MC. It is homotopy monoidal.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Future work

Theorem (Bourke-Gurski,14)

There exists no monoidal model structure on the category of Gray-categories.

John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem

Future work

Theorem (Bourke-Gurski,14)

There exists no monoidal model structure on the category of Gray-categories.

◮ Goal is to use skew framework presented here to understand

the correct setting for enrichment in Gray-categories.

John Bourke Skew structures in 2-category theory and homotopy theory