The Grothendieck construction for model categories CT2015, Protugal - - PowerPoint PPT Presentation

The Grothendieck construction for model categories

CT2015, Protugal

Matan Prasma

joint work with Y. Harpaz

Radboud University Nijmegen, The Netherlands

The Grothendieck construction Theorem A: the integral model structure Theorem B: the Grothendieck correspondence for model categories

Outline

The Grothendieck construction Theorem A: the integral model structure Theorem B: the Grothendieck correspondence for model categories

Recollection: the Grothendieck construction

Definition

Let F ∶ C → Cat be a (pseudo-)functor.

Recollection: the Grothendieck construction

Definition

Let F ∶ C → Cat be a (pseudo-)functor. The Grothendieck construction of F is the category ∫C F whose

• bjects are pairs (A,X) with A ∈ objC and X ∈ objF(A) and a

morphism (A,X) → (B,Y ) is a pair (f ,ϕ) with f ∶ A → B a morphism in C and ϕ ∶ F(f )(X) → Y a morphism in F(B).

Recollection: the Grothendieck construction

Definition

Let F ∶ C → Cat be a (pseudo-)functor. The Grothendieck construction of F is the category ∫C F whose

• bjects are pairs (A,X) with A ∈ objC and X ∈ objF(A) and a

morphism (A,X) → (B,Y ) is a pair (f ,ϕ) with f ∶ A → B a morphism in C and ϕ ∶ F(f )(X) → Y a morphism in F(B). This comes with a projection functor ∫C F → C.

Example

Let G ∈ Gp and BG the associated category.

Example

Let G ∈ Gp and BG the associated category. Consider the functor G ↦ SetBG that assigns for a group G the category of G-sets and for a group map f ∶ G → H, the functor f! ∶ SetBG → SetBH given by f!X ∶= H ×G X where the latter is the quotient of H × X under the G-action g(h,x) ∶= (hg−1,gx).

Example

Let G ∈ Gp and BG the associated category. Consider the functor G ↦ SetBG that assigns for a group G the category of G-sets and for a group map f ∶ G → H, the functor f! ∶ SetBG → SetBH given by f!X ∶= H ×G X where the latter is the quotient of H × X under the G-action g(h,x) ∶= (hg−1,gx). An object of ∫G∈sGp SBG is a pair (G,X) of a group and a G-set X.

Example

Let G ∈ Gp and BG the associated category. Consider the functor G ↦ SetBG that assigns for a group G the category of G-sets and for a group map f ∶ G → H, the functor f! ∶ SetBG → SetBH given by f!X ∶= H ×G X where the latter is the quotient of H × X under the G-action g(h,x) ∶= (hg−1,gx). An object of ∫G∈sGp SBG is a pair (G,X) of a group and a G-set X. A morphism (G,X) → (H,Y ) is a pair (f ,ϕ) where f ∶ G → H is a map of groups and ϕ ∶ f!X → Y is a map of H-sets.

The homotopical setup

Let G be now a simplicial group and let S denote the category of spaces, i.e. of simplicial sets. The (simplicial) functor category SBG can be endowed with the projective model structure (SBG)

proj.

The homotopical setup

Let G be now a simplicial group and let S denote the category of spaces, i.e. of simplicial sets. The (simplicial) functor category SBG can be endowed with the projective model structure (SBG)

proj.

Observation

The association (G

f

• → H ∈ sGp) ↦ (SBG

f!

• → SBH) defines an

adjunction SBG

f!=H×G(−) SBH ⊥ f ∗=restG

H

• (1)

The homotopical setup

Let G be now a simplicial group and let S denote the category of spaces, i.e. of simplicial sets. The (simplicial) functor category SBG can be endowed with the projective model structure (SBG)

proj.

Observation

The association (G

f

• → H ∈ sGp) ↦ (SBG

f!

• → SBH) defines an

adjunction SBG

f!=H×G(−) SBH ⊥ f ∗=restG

H

• (1)

which is a Quillen pair with respect to the projective model structure.

The homotopical setup

Let G be now a simplicial group and let S denote the category of spaces, i.e. of simplicial sets. The (simplicial) functor category SBG can be endowed with the projective model structure (SBG)

proj.

Observation

The association (G

f

• → H ∈ sGp) ↦ (SBG

f!

• → SBH) defines an

adjunction SBG

f!=H×G(−) SBH ⊥ f ∗=restG

H

• (1)

which is a Quillen pair with respect to the projective model

• structure. We obtain a functor SB(−) ∶ sGp

→ ModCat where ModCat is the (2,1)-category of model categories, Quillen pairs and pseudo transformations of adjunctions.

Question

Can we obtain a model structure on ∫G∈sGp SBG from the model structure on the base category and the ones on each of the fibers?

Question

Can we obtain a model structure on ∫G∈sGp SBG from the model structure on the base category and the ones on each of the fibers? SBG SBH SBK sGp

Outline

The Grothendieck construction Theorem A: the integral model structure Theorem B: the Grothendieck correspondence for model categories

The integral model structure

Throughout, a model category will mean a bicomplete category satisfying Quillen’s axioms and admitting functorial factorizations.

The integral model structure

Throughout, a model category will mean a bicomplete category satisfying Quillen’s axioms and admitting functorial factorizations. We said that SB(−) ∶ sGp → ModCat ⊆ AdjCat where AdjCat is the (2,1)-category of adjunctions.

A diagram of adjunctions enjoys good categorical properties:

A diagram of adjunctions enjoys good categorical properties:

Proposition

If C is a bicomplete category and F ∶ C → AdjCat is such that F(A) is bicomplete for each A ∈ C then ∫C F is bicomplete.

We now wish to abstract the following

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups,

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups, the Quillen pair (SBG)

proj

f!=H×G(−) (SBH)

proj

⊥ f ∗=restG

H

• is a Quillen equivalence.

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups, the Quillen pair (SBG)

proj

f!=H×G(−) (SBH)

proj

⊥ f ∗=restG

H

• is a Quillen equivalence.

Definition

Let M be a model category.

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups, the Quillen pair (SBG)

proj

f!=H×G(−) (SBH)

proj

⊥ f ∗=restG

H

• is a Quillen equivalence.

Definition

Let M be a model category. A functor F ∶ M → ModCat is called relative if it takes weak equivalences in M to Quillen equivalences.

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups, the Quillen pair (SBG)

proj

f!=H×G(−) (SBH)

proj

⊥ f ∗=restG

H

• is a Quillen equivalence.

Definition

Let M be a model category. A functor F ∶ M → ModCat is called relative if it takes weak equivalences in M to Quillen equivalences.

Notation

For a morphism f ∶ A → B in M,

We now wish to abstract the following

Observation

If f ∶ G → H is a weak equivalence of simplicial groups, the Quillen pair (SBG)

proj

f!=H×G(−) (SBH)

proj

⊥ f ∗=restG

H

• is a Quillen equivalence.

Definition

Let M be a model category. A functor F ∶ M → ModCat is called relative if it takes weak equivalences in M to Quillen equivalences.

Notation

For a morphism f ∶ A → B in M, we denote the adjunction F(f ) by f! ∶ F(A)

F(B) ∶ f ∗.

⊥

Definition

Call a morphism (f ,ϕ) ∶ (A,X) → (B,Y ) in ∫M F a

Definition

Call a morphism (f ,ϕ) ∶ (A,X) → (B,Y ) in ∫M F a

• 1. cofibration if f ∶ A

→ B is a cofibration in M and ϕ ∶ f!X → Y is a cofibration in F(B).

Definition

Call a morphism (f ,ϕ) ∶ (A,X) → (B,Y ) in ∫M F a

• 1. cofibration if f ∶ A

→ B is a cofibration in M and ϕ ∶ f!X → Y is a cofibration in F(B).

• 2. fibration if f ∶ A

→ B is a fibration and ϕad ∶ X → f ∗Y is a fibration in F(A);

Definition

Call a morphism (f ,ϕ) ∶ (A,X) → (B,Y ) in ∫M F a

• 1. cofibration if f ∶ A

→ B is a cofibration in M and ϕ ∶ f!X → Y is a cofibration in F(B).

• 2. fibration if f ∶ A

→ B is a fibration and ϕad ∶ X → f ∗Y is a fibration in F(A);

• 3. weak equivalence if f ∶ A

→ B is a weak equivalence in M and the composite f!(X cof) → f!X → Y is a weak equivalence in F(B);

Observation

For a relative functor F ∶ M → ModCat

Observation

For a relative functor F ∶ M → ModCat, our notion of weak equivalence is symmetric in that

Observation

For a relative functor F ∶ M → ModCat, our notion of weak equivalence is symmetric in that a map (f ,ϕ) ∶ (A,X) → (B,Y ) is a weak equivalence iff f ∶ A → B is a weak equivalence and X

ϕad

• → f ∗(Y fib)

→ f ∗Y is a weak equivalence.

Observation

For a relative functor F ∶ M → ModCat, our notion of weak equivalence is symmetric in that a map (f ,ϕ) ∶ (A,X) → (B,Y ) is a weak equivalence iff f ∶ A → B is a weak equivalence and X

ϕad

• → f ∗(Y fib)

→ f ∗Y is a weak equivalence.

Definition

Let M be a model category and F ∶ M → ModCat a functor.

Observation

For a relative functor F ∶ M → ModCat, our notion of weak equivalence is symmetric in that a map (f ,ϕ) ∶ (A,X) → (B,Y ) is a weak equivalence iff f ∶ A → B is a weak equivalence and X

ϕad

• → f ∗(Y fib)

→ f ∗Y is a weak equivalence.

Definition

Let M be a model category and F ∶ M → ModCat a functor. We shall say that F is proper if whenever f ∶ A → B is a trivial cofibration in M, f!(WF(A)) ⊆ WF(B) and whenever f ∶ A → B is a trivial fibration in M, f ∗(WF(B)) ⊆ WF(A).

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

Lemma

Let F ∶ M → ModCat be a proper relative functor.

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

Lemma

Let F ∶ M → ModCat be a proper relative functor. A morphism (f ,ϕ) ∶ (A,X) → (B,Y ) is a

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

Lemma

Let F ∶ M → ModCat be a proper relative functor. A morphism (f ,ϕ) ∶ (A,X) → (B,Y ) is a

• 1. trivial cofibartion if and only if f ∶ A

→ B is a trivial cofibration and ϕ ∶ f!X → Y is a trivial cofibration in F(B);

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

Lemma

Let F ∶ M → ModCat be a proper relative functor. A morphism (f ,ϕ) ∶ (A,X) → (B,Y ) is a

• 1. trivial cofibartion if and only if f ∶ A

→ B is a trivial cofibration and ϕ ∶ f!X → Y is a trivial cofibration in F(B);

• 2. trivial fibration if and only if f ∶ A

→ B is a trivial fibration and ϕad ∶ X → f ∗Y is a trivial fibration in F(A).

The properness condition allows us to simplify the description of trivial (co)fibrations as follows.

Lemma

Let F ∶ M → ModCat be a proper relative functor. A morphism (f ,ϕ) ∶ (A,X) → (B,Y ) is a

• 1. trivial cofibartion if and only if f ∶ A

→ B is a trivial cofibration and ϕ ∶ f!X → Y is a trivial cofibration in F(B);

• 2. trivial fibration if and only if f ∶ A

→ B is a trivial fibration and ϕad ∶ X → f ∗Y is a trivial fibration in F(A). This description in turn enables us to establish the appropriate lifting properties and factorizations for these three classes. As a result, we get:

Theorem A

Let M be a model category and F ∶ M → ModCat a proper relative functor.

Theorem A

Let M be a model category and F ∶ M → ModCat a proper relative functor. The classes of weak equivalences, fibrations and cofibrations defined above endow ∫M F with the structure of a model category, called the integral model structure.

Theorem A

Let M be a model category and F ∶ M → ModCat a proper relative functor. The classes of weak equivalences, fibrations and cofibrations defined above endow ∫M F with the structure of a model category, called the integral model structure. Moreover, the ∞-categorical Grothendieck construction of the associated ∞-functor F∞ ∶ M∞ → Cat∞ (given by restricting to cofibrant objects and left Quillen functors, Fcof ∶ Mcof → RelCat and then taking the marked nerve)

Theorem A

Let M be a model category and F ∶ M → ModCat a proper relative functor. The classes of weak equivalences, fibrations and cofibrations defined above endow ∫M F with the structure of a model category, called the integral model structure. Moreover, the ∞-categorical Grothendieck construction of the associated ∞-functor F∞ ∶ M∞ → Cat∞ (given by restricting to cofibrant objects and left Quillen functors, Fcof ∶ Mcof → RelCat and then taking the marked nerve) is equivalent to (∫M F)∞ over M∞.

Applications of Theorem A

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative.

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative. The resulting

integral model structure ⎛ ⎜ ⎝ ∫

G∈sGp

SBG⎞ ⎟ ⎠

int

may be referred to as a model for a global (coarse) equivariant homotopy theory.

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative. The resulting

integral model structure ⎛ ⎜ ⎝ ∫

G∈sGp

SBG⎞ ⎟ ⎠

int

may be referred to as a model for a global (coarse) equivariant homotopy theory.

▸ Let M be a right (resp. left) proper model category.

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative. The resulting

integral model structure ⎛ ⎜ ⎝ ∫

G∈sGp

SBG⎞ ⎟ ⎠

int

may be referred to as a model for a global (coarse) equivariant homotopy theory.

▸ Let M be a right (resp. left) proper model category. The slice

(resp. coslice) functor M/(−) ∶ M → ModCat (resp. M(−)/ ∶ M → ModCat) is proper and relative.

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative. The resulting

integral model structure ⎛ ⎜ ⎝ ∫

G∈sGp

SBG⎞ ⎟ ⎠

int

may be referred to as a model for a global (coarse) equivariant homotopy theory.

▸ Let M be a right (resp. left) proper model category. The slice

(resp. coslice) functor M/(−) ∶ M → ModCat (resp. M(−)/ ∶ M → ModCat) is proper and relative. The category ∫M M/(−) (resp. ∫M M(−)/) is isomorphic to the arrow category M[1] and Theorem A ensures that we get a model structure.

Applications of Theorem A

▸ The functor (SB(−))

proj is proper and relative. The resulting

integral model structure ⎛ ⎜ ⎝ ∫

G∈sGp

SBG⎞ ⎟ ⎠

int

may be referred to as a model for a global (coarse) equivariant homotopy theory.

▸ Let M be a right (resp. left) proper model category. The slice

(resp. coslice) functor M/(−) ∶ M → ModCat (resp. M(−)/ ∶ M → ModCat) is proper and relative. The category ∫M M/(−) (resp. ∫M M(−)/) is isomorphic to the arrow category M[1] and Theorem A ensures that we get a model structure. Under this identification, this is precisely the injective (resp. projective) model structure.

▸ Recall:

▸ Recall:

Theorem (Schwede-Shipley)

Let M be a combinatorial symmetric monoidal model category satisfying the Schwede-Shipley assumptions ("monoid axiom" + "flatness").

▸ Recall:

Theorem (Schwede-Shipley)

Let M be a combinatorial symmetric monoidal model category satisfying the Schwede-Shipley assumptions ("monoid axiom" + "flatness"). Then there is a (combinatorial) model structure on the category of algebra objects Alg(M)

▸ Recall:

Theorem (Schwede-Shipley)

Let M be a combinatorial symmetric monoidal model category satisfying the Schwede-Shipley assumptions ("monoid axiom" + "flatness"). Then there is a (combinatorial) model structure on the category of algebra objects Alg(M) and for each A ∈ Alg(M) there is a (combinatorial) model structure on the category of left A-modules LMod(A).

Theorem

Let M be a combinatorial symmetric monoidal model category satisfying the Schwede-Shipley assumptions.

Theorem

Let M be a combinatorial symmetric monoidal model category satisfying the Schwede-Shipley assumptions. Then the functor LMod(−) ∶ Alg(M) → ModCat is proper and relative, hence endowing ∫Alg(M) LMod(A) with an integral model structure.

Example

The following categories satisfy the assumptions of the last theorem:

Example

The following categories satisfy the assumptions of the last theorem:

• 1. Simplicial sets. Algebra objects in S are simplicial monoids.

Example

The following categories satisfy the assumptions of the last theorem:

• 1. Simplicial sets. Algebra objects in S are simplicial monoids.
• 2. Γ-spaces. Algebra objects are called Γ-rings and model

connective ring spectra.

Example

The following categories satisfy the assumptions of the last theorem:

• 1. Simplicial sets. Algebra objects in S are simplicial monoids.
• 2. Γ-spaces. Algebra objects are called Γ-rings and model

connective ring spectra.

• 3. All the monoidal model categories for spectra except

S-modules. The algebra objects model ring spectra.

Example

The following categories satisfy the assumptions of the last theorem:

• 1. Simplicial sets. Algebra objects in S are simplicial monoids.
• 2. Γ-spaces. Algebra objects are called Γ-rings and model

connective ring spectra.

• 3. All the monoidal model categories for spectra except

S-modules. The algebra objects model ring spectra.

• 4. Non-negatively graded chain complexes over a commutative
• ring. The algebra objects are the (non-negatively graded)

DGAs.

Example

The following categories satisfy the assumptions of the last theorem:

• 1. Simplicial sets. Algebra objects in S are simplicial monoids.
• 2. Γ-spaces. Algebra objects are called Γ-rings and model

connective ring spectra.

• 3. All the monoidal model categories for spectra except

S-modules. The algebra objects model ring spectra.

• 4. Non-negatively graded chain complexes over a commutative
• ring. The algebra objects are the (non-negatively graded)

DGAs.

• 5. Unbounded chain complexes over a commutative ring. The

algebra objects are the (unbounded) DGAs.

Outline

The Grothendieck construction Theorem A: the integral model structure Theorem B: the Grothendieck correspondence for model categories

Recall

Theorem (Grothendieck)

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

between the 2-category of (pseudo-)functors

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

between the 2-category of (pseudo-)functors and the 2-category of coCartesian fibrations over C.

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

between the 2-category of (pseudo-)functors and the 2-category of coCartesian fibrations over C.

• 2. Similarly, for every F ∶ Cop

→ Cat, the projection π ∶ ∫Cop F → C is a Cartesian fibration

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

between the 2-category of (pseudo-)functors and the 2-category of coCartesian fibrations over C.

• 2. Similarly, for every F ∶ Cop

→ Cat, the projection π ∶ ∫Cop F → C is a Cartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(Cop,Cat)

≃

• → Cart(C)

Recall

Theorem (Grothendieck)

• 1. For every F ∶ C

→ Cat, the projection π ∶ ∫C F → C is a coCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,Cat)

≃

• → coCart(C)

between the 2-category of (pseudo-)functors and the 2-category of coCartesian fibrations over C.

• 2. Similarly, for every F ∶ Cop

→ Cat, the projection π ∶ ∫Cop F → C is a Cartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(Cop,Cat)

≃

• → Cart(C)

between the 2-category of (pseudo-)functors and that of Cartesian fibrations.

Theorem (Folklore)

For every F ∶ C → AdjCat,

Theorem (Folklore)

For every F ∶ C → AdjCat, the projection π ∶ ∫C F → C is a biCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,AdjCat)

≃

• → BiFib(C)

Theorem (Folklore)

For every F ∶ C → AdjCat, the projection π ∶ ∫C F → C is a biCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,AdjCat)

≃

• → BiFib(C)

between the 2-category of (pseudo-)functors

Theorem (Folklore)

For every F ∶ C → AdjCat, the projection π ∶ ∫C F → C is a biCartesian fibration and the Grothendieck construction induces an equivalence of (2,1)-categories ∫C ∶ Fun(C,AdjCat)

≃

• → BiFib(C)

between the 2-category of (pseudo-)functors and the 2-category of biCartesian fibrations over C, adjoint functors over C whose (left) right part preserves (co)Cartesian arrows and fiberwise natural transformations of adjunctions.

Model fibrations

Model fibrations

Let π ∶ D → C be a biCartesian fibration.

Model fibrations

Let π ∶ D → C be a biCartesian fibration. Recall that every morphism ϕ ∶ X → Y in D can factored in two ways:

Model fibrations

Let π ∶ D → C be a biCartesian fibration. Recall that every morphism ϕ ∶ X → Y in D can factored in two ways: X

ϕ

• coCart
• f!X
• f ∗Y

Cart

Y

A = π(X)

f =π(ϕ)

B = π(Y )

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor.

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor.

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M,

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

▸ If f ∶ A

→ B is a (trivial) fibration in M,

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

▸ If f ∶ A

→ B is a (trivial) fibration in M, then for every Y ∈ F(B), the Cartesian lift (A,f ∗Y ) → (B,Y ) is a (trivial) fibration in ∫M F.

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

▸ If f ∶ A

→ B is a (trivial) fibration in M, then for every Y ∈ F(B), the Cartesian lift (A,f ∗Y ) → (B,Y ) is a (trivial) fibration in ∫M F. Let us call a category M with three distinguished classes of maps W,Fib,Cof a pre-model category.

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

▸ If f ∶ A

→ B is a (trivial) fibration in M, then for every Y ∈ F(B), the Cartesian lift (A,f ∗Y ) → (B,Y ) is a (trivial) fibration in ∫M F. Let us call a category M with three distinguished classes of maps W,Fib,Cof a pre-model category. If M is a pre-model category and F ∶ M → ModCat is a (formal) proper relative functor,

In our model categorical setup there is an intimate relation between (co)Cartesian lifts and (co)fibrations:

Observation

Suppose F ∶ M → ModCat is a proper relative functor. The functor π ∶ ∫M F → M is both a biCartesian fibration and a right-left Quillen functor. Moreover,

▸ If f ∶ A

→ B is a (trivial) cofibration in M, then for every X ∈ F(A), the coCartesian lift (A,X) → (B,f!X) is a (trivial) cofibration in ∫M F.

▸ If f ∶ A

→ B is a (trivial) fibration in M, then for every Y ∈ F(B), the Cartesian lift (A,f ∗Y ) → (B,Y ) is a (trivial) fibration in ∫M F. Let us call a category M with three distinguished classes of maps W,Fib,Cof a pre-model category. If M is a pre-model category and F ∶ M → ModCat is a (formal) proper relative functor, the projection π ∶ ∫M F → M enjoys good formal properties:

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities.

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities.

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities. ▸ π(L) ⊆ L and π(R) ⊆ R.

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities. ▸ π(L) ⊆ L and π(R) ⊆ R. ▸ L (resp. R) contains any retract f of a morphism in L (resp.

R)

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities. ▸ π(L) ⊆ L and π(R) ⊆ R. ▸ L (resp. R) contains any retract f of a morphism in L (resp.

R) provided that π(f ) is contained in L (resp. R).

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities. ▸ π(L) ⊆ L and π(R) ⊆ R. ▸ L (resp. R) contains any retract f of a morphism in L (resp.

R) provided that π(f ) is contained in L (resp. R).

▸ For every morphism ϕ ∶ X

→ Y in N and every factorization of π(ϕ) as π(ϕ) = g ○ h such that h ∈ L and g ∈ R

Definition

Let π ∶ N → M be (any) functor and (L,R) be two classes of maps in M which contain all the identities. Two classes of maps (L,R) in N will be called a weak factorization systems relative to (L,R) if the following holds:

▸ L and R contain all the identities. ▸ π(L) ⊆ L and π(R) ⊆ R. ▸ L (resp. R) contains any retract f of a morphism in L (resp.

R) provided that π(f ) is contained in L (resp. R).

▸ For every morphism ϕ ∶ X

→ Y in N and every factorization of π(ϕ) as π(ϕ) = g ○ h such that h ∈ L and g ∈ R there exists a factorization of ϕ as ϕ = ψ ○ η such that η ∈ L,ψ ∈ R and such that π(ψ) = g and π(η) = h.

▸ For every square in N of the form

X

• ψ
• Z

η

• Y

W

▸ For every square in N of the form

X

• ψ
• Z

η

• Y

W

such that ψ ∈ L and η ∈ R and for every dashed lift π(X)

• π(ψ)
• π(Z)

π(η)

• π(Y )
• u
• π(W )

▸ For every square in N of the form

X

• ψ
• Z

η

• Y

W

such that ψ ∈ L and η ∈ R and for every dashed lift π(X)

• π(ψ)
• π(Z)

π(η)

• π(Y )
• u
• π(W )

there exists a dashed lift X

• ψ
• Z

η

• Y
• γ
• W

such that π(γ) = u.

Definition

Let M,N be two pre-model categories.

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2)

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift.

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N,

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN and if the image of the third is in WM

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN and if the image of the third is in WM then the third is in WN .

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN and if the image of the third is in WM then the third is in WN .

▸ (CofN ∩ WN ,FibN ) and (CofN ,FibN ∩ WN )

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN and if the image of the third is in WM then the third is in WN .

▸ (CofN ∩ WN ,FibN ) and (CofN ,FibN ∩ WN ) are π-weak

factorization systems relative to

Definition

Let M,N be two pre-model categories. We will say that a functor π ∶ N → M is a relative model category if:

▸ π is relatively co-complete in that for every relative coCone

diagram I

δ

• N

π

• I▷

ε

M

(2) there exist an initial lift. Similarly π is relatively complete.

▸ If f ∶ X

→ Y and g ∶ Y → Z are composable morphisms in N, then if two of f ,g,g ○ f are in WN and if the image of the third is in WM then the third is in WN .

▸ (CofN ∩ WN ,FibN ) and (CofN ,FibN ∩ WN ) are π-weak

factorization systems relative to (CofM ∩ WM,FibM) and (CofM,FibM ∩ WM) respectively.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category. Since π is bicomplete we deduce that for each A ∈ M the fiber N ×M {A} is bicomplete.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category. Since π is bicomplete we deduce that for each A ∈ M the fiber N ×M {A} is

• bicomplete. We will denote by ∅A,∗A ∈ N ×M {A} the initial and

terminal objects of N ×M {A}, respectively.

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category. Since π is bicomplete we deduce that for each A ∈ M the fiber N ×M {A} is

• bicomplete. We will denote by ∅A,∗A ∈ N ×M {A} the initial and

terminal objects of N ×M {A}, respectively. We will say that an

• bject X ∈ N is π-cofibrant if the unique map ∅π(X)

→ X covering Idπ(X) is in CofN .

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category. Since π is bicomplete we deduce that for each A ∈ M the fiber N ×M {A} is

• bicomplete. We will denote by ∅A,∗A ∈ N ×M {A} the initial and

terminal objects of N ×M {A}, respectively. We will say that an

• bject X ∈ N is π-cofibrant if the unique map ∅π(X)

→ X covering Idπ(X) is in CofN . Similarly, we will say that an object X ∈ N is π-fibrant if the unique map X → ∗π(X) covering Idπ(X) is in FibN .

Remark

Let π ∶ N → M be a relative model category and assume that M is actually a model category. Then N is also a model category. Now let π ∶ N → M be a relative model category. Since π is bicomplete we deduce that for each A ∈ M the fiber N ×M {A} is

• bicomplete. We will denote by ∅A,∗A ∈ N ×M {A} the initial and

terminal objects of N ×M {A}, respectively. We will say that an

• bject X ∈ N is π-cofibrant if the unique map ∅π(X)

→ X covering Idπ(X) is in CofN . Similarly, we will say that an object X ∈ N is π-fibrant if the unique map X → ∗π(X) covering Idπ(X) is in FibN . Our main notion is defined as follows:

Definition

Let π ∶ N → M be a functor between pre-model categories.

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

• 1. π is a biCartesian fibration.

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

• 1. π is a biCartesian fibration.
• 2. π is a relative model category.

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

• 1. π is a biCartesian fibration.
• 2. π is a relative model category.
• 3. If f ∶ X

→ Y ∈ N is a π-coCartesian arrow such that X is π-cofibrant and if π(f ) ∶ A = π(X) → π(Y ) = B is a weak equivalence in M then f is a weak equivalence in N.

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

• 1. π is a biCartesian fibration.
• 2. π is a relative model category.
• 3. If f ∶ X

→ Y ∈ N is a π-coCartesian arrow such that X is π-cofibrant and if π(f ) ∶ A = π(X) → π(Y ) = B is a weak equivalence in M then f is a weak equivalence in N.

• 4. If f ∶ X

→ Y ∈ N is a π-Cartesian arrow such that Y is π-fibrant and if π(f ) ∶ A = π(X) → π(Y ) = B is a weak equivalence in M then f is a weak equivalence in N.

Definition

Let π ∶ N → M be a functor between pre-model categories. We will say that π is a model fibration if:

• 1. π is a biCartesian fibration.
• 2. π is a relative model category.
• 3. If f ∶ X

→ Y ∈ N is a π-coCartesian arrow such that X is π-cofibrant and if π(f ) ∶ A = π(X) → π(Y ) = B is a weak equivalence in M then f is a weak equivalence in N.

• 4. If f ∶ X

→ Y ∈ N is a π-Cartesian arrow such that Y is π-fibrant and if π(f ) ∶ A = π(X) → π(Y ) = B is a weak equivalence in M then f is a weak equivalence in N.

A morphism of model fibrations over M is a Quillen adjunction

A morphism of model fibrations over M is a Quillen adjunction N

π

• Φ

N ′

π′

• Ψ

⊥

• M

A morphism of model fibrations over M is a Quillen adjunction N

π

• Φ

N ′

π′

• Ψ

⊥

• M

such that Ψ (resp. Φ) preserves Cartesian (resp. coCartesian) morphisms.

A morphism of model fibrations over M is a Quillen adjunction N

π

• Φ

N ′

π′

• Ψ

⊥

• M

such that Ψ (resp. Φ) preserves Cartesian (resp. coCartesian)

• morphisms. The resulting (2,1)-category is denoted ModFib(M).

Let M be a pre-model category.

Let M be a pre-model category. The association F ↦ ∫M F determines a functor of (2,1)-categories ∫M ∶ FunPR(M,ModCat) → ModFib(M). (3)

Let M be a pre-model category. The association F ↦ ∫M F determines a functor of (2,1)-categories ∫M ∶ FunPR(M,ModCat) → ModFib(M). (3) We can now state a model categorical analogue of Grothendieck’s classical correspondence:

Let M be a pre-model category. The association F ↦ ∫M F determines a functor of (2,1)-categories ∫M ∶ FunPR(M,ModCat) → ModFib(M). (3) We can now state a model categorical analogue of Grothendieck’s classical correspondence:

Theorem B

Let M be a pre-model category.

Let M be a pre-model category. The association F ↦ ∫M F determines a functor of (2,1)-categories ∫M ∶ FunPR(M,ModCat) → ModFib(M). (3) We can now state a model categorical analogue of Grothendieck’s classical correspondence:

Theorem B

Let M be a pre-model category. The functor ∫M above is an equivalence of (2,1)-categories FunPR(M,ModCat) ≃ ModFib(M).

Further perspectives

Further perspectives

▸ What conditions on the indexing category M and the functor

F ∶ M → ModCat ensure that the integral model structure ∫M F is left proper? simplicial? monoidal?

Further perspectives

▸ What conditions on the indexing category M and the functor

F ∶ M → ModCat ensure that the integral model structure ∫M F is left proper? simplicial? monoidal?

▸ What is the interaction between model fibrations and known

constructions of model categories. For example, can model fibrations be used to compute homotopy limits of model categories?

Further perspectives

▸ What conditions on the indexing category M and the functor

F ∶ M → ModCat ensure that the integral model structure ∫M F is left proper? simplicial? monoidal?

▸ What is the interaction between model fibrations and known

constructions of model categories. For example, can model fibrations be used to compute homotopy limits of model categories? Thank you!