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Enriched algebraic weak factorisation systems Alexander Campbell Centre of Australian Category Theory Macquarie University Category Theory 2017 University of British Columbia Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 1 / 30

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SLIDE 1

Enriched algebraic weak factorisation systems

Alexander Campbell

Centre of Australian Category Theory Macquarie University

Category Theory 2017 University of British Columbia

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 1 / 30

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SLIDE 2

Enriched (co)fibrant replacement

Theorem (Folklore: Garner, Riehl, Shulman, . . . ) Let V be a monoidal model category in which every object is cofibrant. Then any cofibrantly generated model V-category has: a cofibrant replacement V-comonad, and a fibrant replacement V-monad.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 2 / 30

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Enriched (co)fibrant replacement

Theorem (Folklore: Garner, Riehl, Shulman, . . . ) Let V be a monoidal model category in which every object is cofibrant. Then any cofibrantly generated model V-category has: a cofibrant replacement V-comonad, and a fibrant replacement V-monad. Examples V = sSet, Cat.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 2 / 30

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Enriched (co)fibrant replacement

Theorem (Folklore: Garner, Riehl, Shulman, . . . ) Let V be a monoidal model category in which every object is cofibrant. Then any cofibrantly generated model V-category has: a cofibrant replacement V-comonad, and a fibrant replacement V-monad. Examples V = sSet, Cat. Theorem (Lack–Rosick´ y) Let V be a monoidal model category with cofibrant unit object. If V has a cofibrant replacement V-comonad, then every object of V is cofibrant.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 2 / 30

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The problem of enriched (co)fibrant generation

Question If V is a monoidal model category in which not every object is cofibrant, then what extra structure, if not an enrichment in the ordinary sense, is naturally possessed by the (co)fibrant replacement (co)monad of a model V-category?

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 3 / 30

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The problem of enriched (co)fibrant generation

Question If V is a monoidal model category in which not every object is cofibrant, then what extra structure, if not an enrichment in the ordinary sense, is naturally possessed by the (co)fibrant replacement (co)monad of a model V-category? An analysis of the monoidal model category V = 2-Cat suggests the decisive concept:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 3 / 30

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SLIDE 7

The problem of enriched (co)fibrant generation

Question If V is a monoidal model category in which not every object is cofibrant, then what extra structure, if not an enrichment in the ordinary sense, is naturally possessed by the (co)fibrant replacement (co)monad of a model V-category? An analysis of the monoidal model category V = 2-Cat suggests the decisive concept:

locally weak V-functor

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 3 / 30

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SLIDE 8

Outline

1

The problem of enriched (co)fibrant replacement

2

The monoidal model category of 2-categories

3

Locally weak V-functors

4

Monoidal and enriched awfs

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 4 / 30

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Outline

1

The problem of enriched (co)fibrant replacement

2

The monoidal model category of 2-categories

3

Locally weak V-functors

4

Monoidal and enriched awfs

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 5 / 30

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The monoidal category of 2-categories

Recall (Gray) The category 2-Cat of (small) 2-categories and 2-functors is a symmetric monoidal closed category with: unit object 1, tensor product A ⊗ B the (pseudo) Gray tensor product of 2-categories, internal hom Gray(A, B) the 2-category of 2-functors A − → B, pseudonatural transformations, and modifications.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 6 / 30

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The monoidal category of 2-categories

Recall (Gray) The category 2-Cat of (small) 2-categories and 2-functors is a symmetric monoidal closed category with: unit object 1, tensor product A ⊗ B the (pseudo) Gray tensor product of 2-categories, internal hom Gray(A, B) the 2-category of 2-functors A − → B, pseudonatural transformations, and modifications. Categories enriched over this monoidal category are called Gray-categories. The self-enrichment of 2-Cat is called Gray.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 6 / 30

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The monoidal model category of 2-categories

Recall (Lack) There is a model structure on 2-Cat, whose weak equivalences are the biequivalences, and which is monoidal with respect to the Gray monoidal structure.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 7 / 30

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The monoidal model category of 2-categories

Recall (Lack) There is a model structure on 2-Cat, whose weak equivalences are the biequivalences, and which is monoidal with respect to the Gray monoidal structure. A 2-category is cofibrant if and only if its underlying category is free on a

  • graph. In particular the unit 2-category 1 is cofibrant.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 7 / 30

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The monoidal model category of 2-categories

Recall (Lack) There is a model structure on 2-Cat, whose weak equivalences are the biequivalences, and which is monoidal with respect to the Gray monoidal structure. A 2-category is cofibrant if and only if its underlying category is free on a

  • graph. In particular the unit 2-category 1 is cofibrant.

Since not every 2-category is cofibrant, it follows from the argument of Lack and Rosick´ y that there does not exist a Gray-enriched cofibrant replacement comonad on 2-Cat.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 7 / 30

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The strictification adjunction

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 8 / 30

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The strictification adjunction

The model category 2-Cat has a canonical cofibrant replacement comonad, which is induced by the adjunction 2-Cat ⊢

Bicat

st

  • where Bicat is the category of bicategories and pseudofunctors, the right

adjoint is the inclusion, and the left adjoint st sends a bicategory to its “strictification”.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 8 / 30

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The strictification adjunction

The model category 2-Cat has a canonical cofibrant replacement comonad, which is induced by the adjunction 2-Cat ⊢

Bicat

st

  • where Bicat is the category of bicategories and pseudofunctors, the right

adjoint is the inclusion, and the left adjoint st sends a bicategory to its “strictification”. Hence this canonical cofibrant replacement stA of a 2-category A is its “pseudofunctor classifier”; i.e. it has the universal property: stA − → B 2-functors A

B

pseudofunctors

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 8 / 30

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The strictification multiadjunction

Theorem (C.) The strictification adjunction extends to an adjunction of multicategories, i.e. an adjunction in the 2-category of multicategories. 2-Cat ⊢

Bicat

st

  • Alexander Campbell (CoACT)

Enriched awfs CT2017 UBC 9 / 30

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The strictification multiadjunction

Theorem (C.) The strictification adjunction extends to an adjunction of multicategories, i.e. an adjunction in the 2-category of multicategories. 2-Cat ⊢

Bicat

st

  • The multicategory structure on 2-Cat is represented by the Gray monoidal
  • structure. Its n-ary morphisms are the “cubical functors of n variables”.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 9 / 30

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The strictification multiadjunction

Theorem (C.) The strictification adjunction extends to an adjunction of multicategories, i.e. an adjunction in the 2-category of multicategories. 2-Cat ⊢

Bicat

st

  • The multicategory structure on 2-Cat is represented by the Gray monoidal
  • structure. Its n-ary morphisms are the “cubical functors of n variables”.

The multicategory structure on Bicat, introduced in Verity’s PhD thesis, is closed but not representable. Its n-ary morphisms are the “cubical pseudofunctors of n variables”.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 9 / 30

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The strictification multiadjunction

Theorem (C.) The strictification adjunction extends to an adjunction of multicategories, i.e. an adjunction in the 2-category of multicategories. 2-Cat ⊢

Bicat

st

  • The multicategory structure on 2-Cat is represented by the Gray monoidal
  • structure. Its n-ary morphisms are the “cubical functors of n variables”.

The multicategory structure on Bicat, introduced in Verity’s PhD thesis, is closed but not representable. Its n-ary morphisms are the “cubical pseudofunctors of n variables”. Hence the comonad st on 2-Cat extends to a comonad in the 2-category

  • f multicategories. But the multicategory structure on 2-Cat is

representable, so st in fact extends to a monoidal comonad on 2-Cat.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 9 / 30

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How strict is strictification?

Corollary The strictification comonad st is a monoidal comonad on 2-Cat.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 10 / 30

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How strict is strictification?

Corollary The strictification comonad st is a monoidal comonad on 2-Cat. By adjointness, a monoidal comonad on a monoidal closed category is equally a closed comonad, so st comes equipped with 2-functors st (Gray(A, B))

Gray(stA, stB)

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 10 / 30

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How strict is strictification?

Corollary The strictification comonad st is a monoidal comonad on 2-Cat. By adjointness, a monoidal comonad on a monoidal closed category is equally a closed comonad, so st comes equipped with 2-functors st (Gray(A, B))

Gray(stA, stB)

which, by the universal property of the pseudofunctor classifier, are equally pseudofunctors Gray(A, B)

Gray(stA, stB)

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 10 / 30

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How strict is strictification?

Corollary The strictification comonad st is a monoidal comonad on 2-Cat. By adjointness, a monoidal comonad on a monoidal closed category is equally a closed comonad, so st comes equipped with 2-functors st (Gray(A, B))

Gray(stA, stB)

which, by the universal property of the pseudofunctor classifier, are equally pseudofunctors Gray(A, B)

Gray(stA, stB)

making st: Gray → Gray into a “locally weak Gray-functor”.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 10 / 30

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How strict is strictification?

Corollary The strictification comonad st is a monoidal comonad on 2-Cat. By adjointness, a monoidal comonad on a monoidal closed category is equally a closed comonad, so st comes equipped with 2-functors st (Gray(A, B))

Gray(stA, stB)

which, by the universal property of the pseudofunctor classifier, are equally pseudofunctors Gray(A, B)

Gray(stA, stB)

making st: Gray → Gray into a “locally weak Gray-functor”. Corollary (C.) The strictification comonad st is a locally weak Gray-comonad on Gray.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 10 / 30

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Outline

1

The problem of enriched (co)fibrant replacement

2

The monoidal model category of 2-categories

3

Locally weak V-functors

4

Monoidal and enriched awfs

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 11 / 30

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Locally weak V-functors

Let (Q, ϕ, ϕ0, . . .) be a monoidal comonad on a monoidal category V. We think of morphisms QX − → Y in V as “weak morphisms” X

Y in V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 12 / 30

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Locally weak V-functors

Let (Q, ϕ, ϕ0, . . .) be a monoidal comonad on a monoidal category V. We think of morphisms QX − → Y in V as “weak morphisms” X

Y in V.

Definition (C.) Let A and B be V-categories. A locally Q-weak V-functor F : A − → B consists of:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 12 / 30

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Locally weak V-functors

Let (Q, ϕ, ϕ0, . . .) be a monoidal comonad on a monoidal category V. We think of morphisms QX − → Y in V as “weak morphisms” X

Y in V.

Definition (C.) Let A and B be V-categories. A locally Q-weak V-functor F : A − → B consists of: (i) a function F : obA − → obB,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 12 / 30

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Locally weak V-functors

Let (Q, ϕ, ϕ0, . . .) be a monoidal comonad on a monoidal category V. We think of morphisms QX − → Y in V as “weak morphisms” X

Y in V.

Definition (C.) Let A and B be V-categories. A locally Q-weak V-functor F : A − → B consists of: (i) a function F : obA − → obB, (ii) for each A, B ∈ A, a morphism ψA,B : QA(A, B) − → B(FA, FB) in V, i.e. a “weak morphism” ψA,B : A(A, B)

B(FA, FB) in V,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 12 / 30

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Locally weak V-functors

Let (Q, ϕ, ϕ0, . . .) be a monoidal comonad on a monoidal category V. We think of morphisms QX − → Y in V as “weak morphisms” X

Y in V.

Definition (C.) Let A and B be V-categories. A locally Q-weak V-functor F : A − → B consists of: (i) a function F : obA − → obB, (ii) for each A, B ∈ A, a morphism ψA,B : QA(A, B) − → B(FA, FB) in V, i.e. a “weak morphism” ψA,B : A(A, B)

B(FA, FB) in V,

subject to the following two axioms.

QA(B, C) ⊗ QA(A, B)

ϕ ψ⊗ψ

  • Q (A(B, C) ⊗ A(A, B))

QK QA(A, C) ψ

  • B(FB, FC) ⊗ B(FA, FB)

K

B(FA, FC)

QI

Qj

QA(A, A)

ψ

  • I

ϕ0

  • j

B(FA, FA)

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 12 / 30

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The Kleisli 2-category of locally weak V-functors

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 13 / 30

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The Kleisli 2-category of locally weak V-functors

Let Q be a monoidal comonad on a monoidal category V. Change of base along Q defines a 2-comonad on the 2-category V-Cat.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 13 / 30

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The Kleisli 2-category of locally weak V-functors

Let Q be a monoidal comonad on a monoidal category V. Change of base along Q defines a 2-comonad on the 2-category V-Cat. The Kleisli 2-category of this 2-comonad has:

  • bjects: V-categories,

morphisms: locally Q-weak V-functors, 2-cells: locally Q-weak V-natural transformations.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 13 / 30

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The Kleisli 2-category of locally weak V-functors

Let Q be a monoidal comonad on a monoidal category V. Change of base along Q defines a 2-comonad on the 2-category V-Cat. The Kleisli 2-category of this 2-comonad has:

  • bjects: V-categories,

morphisms: locally Q-weak V-functors, 2-cells: locally Q-weak V-natural transformations. A (co)monad in this 2-category is called a locally Q-weak V-(co)monad.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 13 / 30

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Outline

1

The problem of enriched (co)fibrant replacement

2

The monoidal model category of 2-categories

3

Locally weak V-functors

4

Monoidal and enriched awfs

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 14 / 30

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Leibniz–Day constructions I

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, such that ⊗ preserves finite colimits in each variable.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 15 / 30

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Leibniz–Day constructions I

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, such that ⊗ preserves finite colimits in each variable. By Day convolution, the arrow category V2 is a monoidal category with:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 15 / 30

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Leibniz–Day constructions I

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, such that ⊗ preserves finite colimits in each variable. By Day convolution, the arrow category V2 is a monoidal category with: unit: 0 − → I,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 15 / 30

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Leibniz–Day constructions I

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, such that ⊗ preserves finite colimits in each variable. By Day convolution, the arrow category V2 is a monoidal category with: unit: 0 − → I, tensor product ( A

f

B )

⊗( C

g

D ) given by:

A ⊗ C

1⊗g f ⊗1

  • A ⊗ D
  • f ⊗1
  • B ⊗ C
  • 1⊗g
  • ·

f ⊗g

  • B ⊗ D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 15 / 30

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Leibniz–Day constructions I

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, such that ⊗ preserves finite colimits in each variable. By Day convolution, the arrow category V2 is a monoidal category with: unit: 0 − → I, tensor product ( A

f

B )

⊗( C

g

D ) given by:

A ⊗ C

1⊗g f ⊗1

  • A ⊗ D
  • f ⊗1
  • B ⊗ C
  • 1⊗g
  • ·

f ⊗g

  • B ⊗ D

definition of the associativity and unit constraints requires the above assumption on colimits.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 15 / 30

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Leibniz–Day constructions II

Let A be a V-category. Then A2 is a V2-category, with homs A (f , g) given by:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 16 / 30

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Leibniz–Day constructions II

Let A be a V-category. Then A2 is a V2-category, with homs A (f , g) given by: A(B, C)

A(1,g)

  • A

(f ,g)

  • A(f ,1)
  • Sq(f , g)
  • A(B, D)

A(f ,1)

  • A(A, C) A(1,g)

A(A, D)

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 16 / 30

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Leibniz–Day constructions II

Let A be a V-category. Then A2 is a V2-category, with homs A (f , g) given by: A(B, C)

A(1,g)

  • A

(f ,g)

  • A(f ,1)
  • Sq(f , g)
  • A(B, D)

A(f ,1)

  • A(A, C) A(1,g)

A(A, D)

Sq(f , g) is the V-object of squares f → g.

=

  • A(B, C)

A (f ,g)

  • I

(u,v)

Sq(f , g)

↔ I

= v

  • u
  • A(B, D)

A(f ,1)

  • A(A, C) A(1,g)

A(A, D)

↔ A

= u

  • f
  • C

g

  • B

v

D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 16 / 30

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SLIDE 46

Weak factorisation systems

A weak factorisation system (wfs) on a category C consists of two classes

  • f morphisms (L, R) in C subject to closure axioms, such that:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 17 / 30

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Weak factorisation systems

A weak factorisation system (wfs) on a category C consists of two classes

  • f morphisms (L, R) in C subject to closure axioms, such that:

(i) every morphism f in C has a factorisation

f

  • L∋l
  • =

r∈R

  • Alexander Campbell (CoACT)

Enriched awfs CT2017 UBC 17 / 30

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SLIDE 48

Weak factorisation systems

A weak factorisation system (wfs) on a category C consists of two classes

  • f morphisms (L, R) in C subject to closure axioms, such that:

(i) every morphism f in C has a factorisation

f

  • L∋l
  • =

r∈R

  • (ii) every square l → r has a diagonal filler:

A

  • L∋l
  • C

r∈R

  • B

∃

  • D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 17 / 30

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SLIDE 49

Weak factorisation systems

A weak factorisation system (wfs) on a category C consists of two classes

  • f morphisms (L, R) in C subject to closure axioms, such that:

(i) every morphism f in C has a factorisation

f

  • L∋l
  • =

r∈R

  • (ii) every square l → r has a diagonal filler:

A

  • L∋l
  • C

r∈R

  • B

∃

  • D

i.e. C(B, C)

C (l,r)

  • Sq(l, r)

is surjective ∀l ∈ L, r ∈ R.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 17 / 30

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SLIDE 50

Enriched weak factorisation systems

Let (L, R) be a wfs on a monoidal category V. A wfs (H, M) on a V-category A is said to be enriched over (L, R) if for each A

f

B in

H and each C

g

D in M, the morphism A(B, C)

A (f ,g) Sq(f , g) in

V belongs to R.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 18 / 30

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SLIDE 51

Enriched weak factorisation systems

Let (L, R) be a wfs on a monoidal category V. A wfs (H, M) on a V-category A is said to be enriched over (L, R) if for each A

f

B in

H and each C

g

D in M, the morphism A(B, C)

A (f ,g) Sq(f , g) in

V belongs to R. Examples (a) Every wfs is enriched over the (injective, surjective) wfs on Set.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 18 / 30

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SLIDE 52

Enriched weak factorisation systems

Let (L, R) be a wfs on a monoidal category V. A wfs (H, M) on a V-category A is said to be enriched over (L, R) if for each A

f

B in

H and each C

g

D in M, the morphism A(B, C)

A (f ,g) Sq(f , g) in

V belongs to R. Examples (a) Every wfs is enriched over the (injective, surjective) wfs on Set. (b) A wfs enriched over the (all, iso) factorisation system on Set is precisely an orthogonal factorisation system.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 18 / 30

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SLIDE 53

Enriched weak factorisation systems

Let (L, R) be a wfs on a monoidal category V. A wfs (H, M) on a V-category A is said to be enriched over (L, R) if for each A

f

B in

H and each C

g

D in M, the morphism A(B, C)

A (f ,g) Sq(f , g) in

V belongs to R. Examples (a) Every wfs is enriched over the (injective, surjective) wfs on Set. (b) A wfs enriched over the (all, iso) factorisation system on Set is precisely an orthogonal factorisation system. (c) Let V be a monoidal model category. The two defining wfs of a model V-category are enriched over the (cofibration, trivial fibration) wfs on V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 18 / 30

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Algebraic weak factorisation systems

An algebraic weak factorisation system (awfs) on a category C consists of a comonad L and a monad R on the arrow category C2, subject to various axioms,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 19 / 30

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SLIDE 55

Algebraic weak factorisation systems

An algebraic weak factorisation system (awfs) on a category C consists of a comonad L and a monad R on the arrow category C2, subject to various axioms, including that every morphism f has the canonical factorisation: A

f

  • Lf
  • =

B Ef

Rf

  • Note that E : C2 −

→ C is a functor.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 19 / 30

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SLIDE 56

Algebraic weak factorisation systems

An algebraic weak factorisation system (awfs) on a category C consists of a comonad L and a monad R on the arrow category C2, subject to various axioms, including that every morphism f has the canonical factorisation: A

f

  • Lf
  • =

B Ef

Rf

  • Note that E : C2 −

→ C is a functor. “L-map” ≡ L-coalgebra “R-map” ≡ R-algebra A

Lf

  • f
  • Ef

Rf

  • B

s

  • 1

B

C

1

  • Lg
  • C

g

  • Eg

p

  • Rg

D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 19 / 30

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SLIDE 57

Algebraic weak factorisation systems

Each square in C from an L-coalgebra (f , s) to an R-algebra (g, p) A

u

  • f
  • =

C

g

  • B

v

D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 20 / 30

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SLIDE 58

Algebraic weak factorisation systems

Each square in C from an L-coalgebra (f , s) to an R-algebra (g, p) A

u

  • f
  • =

C

g

  • B

v

D

has the canonical diagonal filler p ◦ E(u, v) ◦ s. A

1

  • 1
  • A

u

  • Lf
  • C

1

  • Lg
  • C

g

  • A

Lf

  • f
  • Ef E(u,v)
  • Rf
  • Eg

p

  • Rg
  • Rg
  • D

1

  • B

1

  • s
  • B

v

D

1

D

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 20 / 30

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SLIDE 59

(Co)fibrant replacement (co)monad

If (L, R) is an awfs on a category C with an initial object 0, then factorisation of morphisms of the form

  • =

A QA

εA

  • defines a comonad Q on C, called the cofibrant replacement comonad for

(L, R).

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 21 / 30

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SLIDE 60

(Co)fibrant replacement (co)monad

If (L, R) is an awfs on a category C with an initial object 0, then factorisation of morphisms of the form

  • =

A QA

εA

  • defines a comonad Q on C, called the cofibrant replacement comonad for

(L, R). Q-coalgebras are called algebraically cofibrant objects.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 21 / 30

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SLIDE 61

(Co)fibrant replacement (co)monad

If (L, R) is an awfs on a category C with an initial object 0, then factorisation of morphisms of the form

  • =

A QA

εA

  • defines a comonad Q on C, called the cofibrant replacement comonad for

(L, R). Q-coalgebras are called algebraically cofibrant objects. The Kleisli category CQ for this comonad is called the category of weak maps for (L, R).

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 21 / 30

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SLIDE 62

(Co)fibrant replacement (co)monad

If (L, R) is an awfs on a category C with an initial object 0, then factorisation of morphisms of the form

  • =

A QA

εA

  • defines a comonad Q on C, called the cofibrant replacement comonad for

(L, R). Q-coalgebras are called algebraically cofibrant objects. The Kleisli category CQ for this comonad is called the category of weak maps for (L, R). Dually, if C has a terminal object 1, then factorisation of morphisms of the form A − → 1 defines a monad on C, called the fibrant replacement monad.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 21 / 30

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SLIDE 63

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 64

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 65

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 66

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with: (i) a natural transformation ϕ: Ef ⊗ Eg − → E(f ⊗g),

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 67

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with: (i) a natural transformation ϕ: Ef ⊗ Eg − → E(f ⊗g), (ii) a morphism ϕ0 : I − → QI,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 68

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with: (i) a natural transformation ϕ: Ef ⊗ Eg − → E(f ⊗g), (ii) a morphism ϕ0 : I − → QI, making: (iii) ⊗: V × V − → V a two-variable oplax morphism of awfs,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 69

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with: (i) a natural transformation ϕ: Ef ⊗ Eg − → E(f ⊗g), (ii) a morphism ϕ0 : I − → QI, making: (iii) ⊗: V × V − → V a two-variable oplax morphism of awfs, (iv) E : V2 − → V a monoidal functor,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 70

Monoidal awfs

Let (V, ⊗, I) be a monoidal category with finite colimits and finite limits, and such that ⊗ preserves finite colimits in each variable. Recall that a wfs (L, R) on V is said to be a monoidal wfs if f , g ∈ L implies f ⊗g ∈ L. Definition (Riehl, C.) An awfs (L, R) on V is said to be a monoidal awfs when it is equipped with: (i) a natural transformation ϕ: Ef ⊗ Eg − → E(f ⊗g), (ii) a morphism ϕ0 : I − → QI, making: (iii) ⊗: V × V − → V a two-variable oplax morphism of awfs, (iv) E : V2 − → V a monoidal functor, (v) I an algebraically cofibrant object.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 22 / 30

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SLIDE 71

Two-variable oplax morphism of awfs

Axiom (iii) (⊗ is a two-variable oplax morphism of awfs) implies, inter alia, the following result. Proposition (Riehl) The tensor product ⊗ on V2 lifts to a functor

  • ⊗: L-Coalg × L-Coalg −

→ L-Coalg.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 23 / 30

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SLIDE 72

Two-variable oplax morphism of awfs

Axiom (iii) (⊗ is a two-variable oplax morphism of awfs) implies, inter alia, the following result. Proposition (Riehl) The tensor product ⊗ on V2 lifts to a functor

  • ⊗: L-Coalg × L-Coalg −

→ L-Coalg. Moreover, by the definition of two-variable oplax morphisms of awfs, ϕ defines natural transformations Lf ⊗Lg

Φ

L(f

⊗g) Lf ⊗Rg

Σ

R(f

⊗g) Rf ⊗Lg

Π

R(f

⊗g) which, together with the remaining axioms, prove the following theorem.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 23 / 30

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SLIDE 73

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.)

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 24 / 30

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SLIDE 74

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.) (i) L is a monoidal comonad on V2.

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SLIDE 75

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.) (i) L is a monoidal comonad on V2. (ii) R is a L-bistrong monad on V2.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 24 / 30

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SLIDE 76

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.) (i) L is a monoidal comonad on V2. (ii) R is a L-bistrong monad on V2. (iii) The cofibrant replacement comonad Q is a monoidal comonad on V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 24 / 30

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SLIDE 77

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.) (i) L is a monoidal comonad on V2. (ii) R is a L-bistrong monad on V2. (iii) The cofibrant replacement comonad Q is a monoidal comonad on V. (iv) The fibrant replacement monad P is a Q-bistrong monad on V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 24 / 30

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SLIDE 78

Cofibrant replacement is a monoidal comonad

Let (L, R) be a monoidal awfs on V. Theorem (C.) (i) L is a monoidal comonad on V2. (ii) R is a L-bistrong monad on V2. (iii) The cofibrant replacement comonad Q is a monoidal comonad on V. (iv) The fibrant replacement monad P is a Q-bistrong monad on V. Corollary (i) The monoidal structure on V2 lifts to a monoidal structure on L-Coalg. (ii) R-Kl is a two-sided (L-Coalg)-actegory. (iii) The monoidal structure on V lifts to a monoidal structure on Q-Coalg. (iv) P-Kl is a two-sided (Q-Coalg)-actegory.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 24 / 30

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SLIDE 79

The multicategory of weak maps

Let (L, R) be a monoidal awfs on V with cofibrant replacement comonad

  • Q. Recall that the Kleisli category VQ for Q is called the category of weak

maps for (L, R).

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 25 / 30

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SLIDE 80

The multicategory of weak maps

Let (L, R) be a monoidal awfs on V with cofibrant replacement comonad

  • Q. Recall that the Kleisli category VQ for Q is called the category of weak

maps for (L, R). Corollary The Kleisli adjunction for Q extends to an adjunction of multicategories. V ⊢

VQ

Q

  • n-ary morphisms (X1, . . . , Xn) −

→ Y in the multicategory structure on VQ are morphisms QX1 ⊗ · · · ⊗ QXn − → Y in V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 25 / 30

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SLIDE 81

Enriched awfs

Let (L, E, R) be a monoidal awfs on V.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 82

Enriched awfs

Let (L, E, R) be a monoidal awfs on V. Definition (Riehl, C.) An awfs (H, N, M) on a V-category A is said to be enriched over (L, R) when it is equipped with:

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 83

Enriched awfs

Let (L, E, R) be a monoidal awfs on V. Definition (Riehl, C.) An awfs (H, N, M) on a V-category A is said to be enriched over (L, R) when it is equipped with: (i) a natural transformation ψ: EA (f , g) − → A(Nf , Ng),

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 84

Enriched awfs

Let (L, E, R) be a monoidal awfs on V. Definition (Riehl, C.) An awfs (H, N, M) on a V-category A is said to be enriched over (L, R) when it is equipped with: (i) a natural transformation ψ: EA (f , g) − → A(Nf , Ng), making: (ii) A(−, −): Aop × A − → V a two-variable lax morphism of awfs,

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 85

Enriched awfs

Let (L, E, R) be a monoidal awfs on V. Definition (Riehl, C.) An awfs (H, N, M) on a V-category A is said to be enriched over (L, R) when it is equipped with: (i) a natural transformation ψ: EA (f , g) − → A(Nf , Ng), making: (ii) A(−, −): Aop × A − → V a two-variable lax morphism of awfs, (iii) (N, E): (A2, V2) − → (A, V) a morphism of enriched categories.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 86

Enriched awfs

Let (L, E, R) be a monoidal awfs on V. Definition (Riehl, C.) An awfs (H, N, M) on a V-category A is said to be enriched over (L, R) when it is equipped with: (i) a natural transformation ψ: EA (f , g) − → A(Nf , Ng), making: (ii) A(−, −): Aop × A − → V a two-variable lax morphism of awfs, (iii) (N, E): (A2, V2) − → (A, V) a morphism of enriched categories.

EA (g, h) ⊗ EA (f , g)

ϕ ψ⊗ψ

  • E
  • A

(g, h) ⊗A (f , g)

  • E

K EA

(f , h)

ψ

  • A(Ng, Nh) ⊗ A(Nf , Ng)

K

A(Nf , Nh)

QI

E j

EA

(f , f )

ψ

  • I

ϕ0

  • j

A(Nf , Nf )

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 26 / 30

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SLIDE 87

Two-variable lax morphism of awfs

Axiom (ii) (A(−, −) is a two-variable lax morphism of awfs) implies, inter alia, the following result. Proposition (Riehl) The V2-valued hom A (−, −) on A2 lifts to a functor A (−, −): H-Coalg × M-Alg − → R-Alg.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 27 / 30

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SLIDE 88

Two-variable lax morphism of awfs

Axiom (ii) (A(−, −) is a two-variable lax morphism of awfs) implies, inter alia, the following result. Proposition (Riehl) The V2-valued hom A (−, −) on A2 lifts to a functor A (−, −): H-Coalg × M-Alg − → R-Alg. Moreover, by the definition of two-variable lax morphisms of awfs, ψ defines natural transformations RA (f , g)

Θ

A

  • (Hf , Mg)

LA (f , g)

Ψ

A

(Hf , Hg) LA (f , g)

Ω

A

  • (Mf , Mg)

which, together with the remaining axioms, prove the following theorem.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 27 / 30

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SLIDE 89

(Co)fibrant replacement is a locally weak (co)monad

Theorem (C.) Let (H, M) be an (L, R)-enriched awfs on A. Then the following are true.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 28 / 30

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SLIDE 90

(Co)fibrant replacement is a locally weak (co)monad

Theorem (C.) Let (H, M) be an (L, R)-enriched awfs on A. Then the following are true. (i) H is a locally L-weak V2-comonad on A2.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 28 / 30

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SLIDE 91

(Co)fibrant replacement is a locally weak (co)monad

Theorem (C.) Let (H, M) be an (L, R)-enriched awfs on A. Then the following are true. (i) H is a locally L-weak V2-comonad on A2. (ii) M is a locally L-weak V2-monad on A2.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 28 / 30

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SLIDE 92

(Co)fibrant replacement is a locally weak (co)monad

Theorem (C.) Let (H, M) be an (L, R)-enriched awfs on A. Then the following are true. (i) H is a locally L-weak V2-comonad on A2. (ii) M is a locally L-weak V2-monad on A2. (iii) The cofibrant replacement comonad for (H, M) is a locally Q-weak V-comonad on A.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 28 / 30

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SLIDE 93

(Co)fibrant replacement is a locally weak (co)monad

Theorem (C.) Let (H, M) be an (L, R)-enriched awfs on A. Then the following are true. (i) H is a locally L-weak V2-comonad on A2. (ii) M is a locally L-weak V2-monad on A2. (iii) The cofibrant replacement comonad for (H, M) is a locally Q-weak V-comonad on A. (iv) The fibrant replacement monad for (H, M) is a locally Q-weak V-monad on A.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 28 / 30

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SLIDE 94

The enriched category of weak maps

Let (L, R) be a monoidal awfs on V with cofibrant replacement comonad

  • Q. Let (H, M) be a (L, R)-enriched awfs on a V-category A with

cofibrant replacement comonad S.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 29 / 30

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SLIDE 95

The enriched category of weak maps

Let (L, R) be a monoidal awfs on V with cofibrant replacement comonad

  • Q. Let (H, M) be a (L, R)-enriched awfs on a V-category A with

cofibrant replacement comonad S. Corollary (C.) The Kleisli adjunction for S extends to a VQ-enriched adjunction, i.e. an adjunction in the 2-category VQ-Cat of categories enriched over the multicategory of weak maps for (L, R). A ⊢

AS

S

  • The hom-objects in the VQ-category AS are AS(A, B) = A(SA, B).

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 29 / 30

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SLIDE 96

Examples of monoidal and enriched awfs

Examples (a) Every monoidal awfs on a monoidal closed category is enriched over itself. (b) The (all,iso) factorisation system on a monoidal category V is a monoidal awfs (with canonical factorisation f = 1 ◦ f ) . An awfs on a V-category A enriched over this monoidal awfs is precisely a V-enriched orthogonal factorisation system on A. (c) The “split epi” awfs on Set (in which f : X → Y factors through X + Y ) is monoidal with respect to cartesian product. Every awfs is canonically enriched over this monoidal awfs. (d) Let V be a monoidally cocomplete category, so that U = V(I, −): V → Set has a left adjoint F. The “U-split epi” awfs on V (in which f : X → Y factors through X + FUY ) is monoidal. Every awfs on a V-category is canonically enriched over this monoidal awfs.

Alexander Campbell (CoACT) Enriched awfs CT2017 UBC 30 / 30