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

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 ϕ ad � → f ∗ ( Y fib ) � → f ∗ Y is a weak equivalence. X 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 ϕ ad � → f ∗ ( Y fib ) � → f ∗ Y is a weak equivalence. X 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 ! (W F( A ) ) ⊆ W F( B ) and whenever f ∶ A � → B is a trivial fibration in M , f ∗ (W F( B ) ) ⊆ W F( 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 ϕ ad ∶ 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 ϕ ad ∶ 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, F cof ∶ M cof � → 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, F cof ∶ M cof � → RelCat and then taking the marked nerve) is equivalent to (∫ M F) ∞ over M ∞ .

Applications of Theorem A

Applications of Theorem A ▸ The functor (S B ( − ) ) proj is proper and relative.

Applications of Theorem A ▸ The functor (S B ( − ) ) proj is proper and relative. The resulting integral model structure ⎛ S B G ⎞ ⎝ ∫ ⎜ ⎟ ⎠ G ∈ sGp int may be referred to as a model for a global (coarse) equivariant homotopy theory .

Applications of Theorem A ▸ The functor (S B ( − ) ) proj is proper and relative. The resulting integral model structure ⎛ S B G ⎞ ⎝ ∫ ⎜ ⎟ ⎠ G ∈ sGp 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 (S B ( − ) ) proj is proper and relative. The resulting integral model structure ⎛ S B G ⎞ ⎝ ∫ ⎜ ⎟ ⎠ G ∈ sGp 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 (S B ( − ) ) proj is proper and relative. The resulting integral model structure ⎛ S B G ⎞ ⎝ ∫ ⎜ ⎟ ⎠ G ∈ sGp 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 (S B ( − ) ) proj is proper and relative. The resulting integral model structure ⎛ S B G ⎞ ⎝ ∫ ⎜ ⎟ ⎠ G ∈ sGp 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 ∶ C op � → Cat , the projection π ∶ ∫ C op 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 ∶ C op � → Cat , the projection π ∶ ∫ C op F � → C is a Cartesian fibration and the Grothendieck construction induces an equivalence of ( 2 , 1 ) -categories ∫ C ∶ Fun (C op , 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 ∶ C op � → Cat , the projection π ∶ ∫ C op F � → C is a Cartesian fibration and the Grothendieck construction induces an equivalence of ( 2 , 1 ) -categories ∫ C ∶ Fun (C op , 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: coCart X f ! X ϕ f ∗ Y � Y Cart A = π ( X ) � B = π ( Y ) f = π ( ϕ )

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 , F ib , C of 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 , F ib , C of 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 , F ib , C of 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 ).