Spahn HTT, A.2 model categories (Rev #12)

This is a subentry of a reading guide to HTT.

Contents

A.2.1 The model category axioms

Definition

(This is Joyal’s definition; it differs from A.2.1.1 in that Joyal requests CC to be finitely bicomplete.)

A model category is a category CC equipped with three distinguished classes of CC-morphisms: The classes (C)(C), (F)(F), (W)(W) of cofibrations, fibrations, and weak equivalences, respectively, satisfying the following axioms:

  • CC admits (small) limits and colimits.

  • The class of weak equivalences satisfies 2-out-of-3.

  • (CW,F)(C\cup W,F) and (C,FW)(C,F\cup W) are weak factorization systems.

Remark
  1. The classes (C)(C) and (F)(F) is closed under retracts. (by weak factorization systems, Lemma 2, in joyal’s catlab)

  2. The class (W)(W) is closed under retracts. (by model categories, Lemma 1, in joyal’s catlab)

A.2.2 The homotopy category of a model category

Definition

Let XX be an object in a model category.

  1. A cylinder object is defined to be a factorization of the codiagonal map XXXX\coprod X\to X for XX into a cofibration followed by a weak equivalence.

  2. A path object is defined to be a factorization of the diagonal map XX×XX\to X\times X for XX into a weak equivalence followed by a fibration .

Proposition A.2.2.1

Let CC be a model category. Let XX be a cofibrant object of CC. Let YY be a fibrant object of CC. Let f,g:XYf,g:X\to Y be two parallel morphisms. Then the following conditions are equivalent.

  1. The coproduct map fgf\coprod g factors through every cylinder object for XX.

  2. The coproduct map fgf\coprod g factors through some cylinder object for XX.

  3. The product map f×gf\times g factors through every path object for YY.

  4. The product map f×gf\times g factors through some path object for YY.

Definition

(homotopy, homotopy category of a model category) Let CC be a model category.

(1) Two maps f,g:XYf,g:X\to Y from a cofibrant object to a fibrant object satisfying the conditions of Proposition A.2.2.1 are called homotopic morphisms. Homotopy is an equivalence relation \simeq on hom C(X,Y)hom_C (X,Y).

(2) The homotopy category hCh C of CC is defined to have as objects the fibrant-cofibrant objects of CC. The hom objects hom hC(X,Y)hom_{hC}(X,Y) are defined to be the set of \simeq equivalence classes of hom C(X,Y)hom_C (X,Y).

A.2.3 A lifting criterion

The following proposition says that a factorization of a cofibration between cofibrant objects which exists in the homotopy category of a model category can be lifted into the model category.

Proposition A.2.3.1

A.2.4 Left properness and homotopy push out squares

In every model category the class of fibrations is stable under pullback and the class of cofibrations is stable under pushout. In general weak equivalences do not have such properties. The following definition requests such.

Definition A.2.4.1
  1. A model category is called left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence.

  2. A model category is called right proper if the pullback of a weak equivalence along a fibration is a weak equivalence.

Proposition

Any model category in which every object is cofibrant is left proper.

Lemma A.2.4.3

The push out along a cofibration of a weak equivalence between cofibrant objects is always a weak equivalence.

A.2.5 Quillen adjunctions and Quillen equivalences

A Quillen adjunction is an appropriate notion of morphism between model categories.

Proposition and Definition

An adjoint pair of functors (FG):DGC(F\dashv G):D\stackrel{G}{\to}C is called a Quillen adjunction if the following equivalent conditions are satisfied:

  1. FF preserves cofibrations and acyclic cofibrations.

  2. GG preserves fibrations and acyclic fibrations.

  3. FF preserves cofibrations and GG preserves fibrations.

  4. FF preserves acyclic cofibrations and GG preserves acyclic fibrations.

Remark

Let (FG)(F\dashv G) be a Quillen adjunction. Then

  1. FF preserves weak equivalences between cofibrant objects.

  2. GG preserves weak equivalences between fibrant objects.

Remark

(descent of a Quillen adjunction to an adjunction between the homotopy category)

Given a model category CC we obtain its homotopy category hChC be passing to its full subcategory of cofibrant objects and the formally inverting the weak equivalences.

If (FG):DGC(F\dashv G):D\stackrel{G}{\to}C is a Quillen adjunction FF induces a functor LF:hChDL F:hC\to hD since FF preserves weak equivalences between cofibrant objects.

Analogously GG preserves weak equivalences between fibrant objects and we obtain hDhD from DD by passing to the category of fibrant objects of DD and formally invert the weak equivalences and hence GG induces a functor RG:hDhCRG:hD\to hC.

In total one can show that (LFRG):hDRGhC(LF\dashv RG):hD\stackrel{RG}{\to}hC form an adjunction.

Abstracty one can obtain this result by Kan extension (this is also described at derived functor); however Quillen adjunction’s are introduced to present adjunctions between \infty-categories and to obtain such a presentation in terms of Kan extension in general requires additional assumptions:

In more detail we wish to extend F:CDF : C \to D (for GG analogously) to a diagram

C F D Q C (?) Q D hC hD, \array{ C &\stackrel{F}{\to}& D \\ \downarrow^{\mathrlap{Q_C}} &(?)& \downarrow^{\mathrlap{Q_D}} \\ hC&\to& hD } \,,

where Q C:ChCQ_C : C \to hC is the universal morphism characterizing the homotopy category and similarly for Q DQ_D.

This is accomplished by taking hChDhC\to hD to be either the left (LF:=Lan Q CQ dFLF:=Lan_{Q_C} Q_d \circ F) or right (RF:=Ran Q CQ dFRF:=Ran_{Q_C} Q_d \circ F) Kan extension of Q dFQ_d \circ F along Q CQ_C.

Proposition A.2.5.1

(characterization of derived functors, Quilen adjunction) Let (FG):DGC(F\dashv G):D\stackrel{G}{\to}C be a Quillen adjunction of model categories. Then the following are equivalent:

  1. The left derived functor LF:hChDLF:hC\to hD is an equivalence of categories.

  2. The right derived functor RF:hDhCRF:hD\to hC is an equivalence of categories.

  3. For every cofibrant object cCc\in C and every fibrant object DDD\in D, a map cG(d)c\to G(d) is a weak equivalence iff the adjoint map F(c)dF(c)\to d is a weak equivalence.

(FG)(F\dashv G) is called Quillen equivalence if these conditions are satisfied.

A.2.6 Combinatorial model categories

(transclusion:

Definition A.1.2.2

(weakly saturated class of morphisms)

Let CC be a category with all small colimits. A class SS of CC-morphisms is called a weakly saturated class if the following conditions are satisfied.

  1. SS is closed under forming pushouts (along arbitrary CC-morphisms).

  2. SS is closed under transfinite composition.

  3. SS is closed under forming retracts.

)

Definition A.2.6.1

A model category AA is called combinatorial model category if the following conditions are satisfied:

  1. AA is presentable.

  2. There exists a set II of generating cofibrations such that the collection of all cofibrations is the smallest weakly saturated class of morphisms containing II.

  3. There exists a set JJ of generating acyclic cofibrations such that the collection of all acyclic cofibrations is the smallest weakly saturated class of morphisms containing JJ.

Definition A.2.6.10

(perfect class) Let AA be a presentable category. A class WW of morphisms in CC is called perfect class if the following conditions are satisfied:

  1. WW contaings all isomorpphisms.

  2. WW satisfies 2-out-of-3

  3. WW is stable under poset filtered colimits.

  4. WW contains a small subset which generates WW under filtered colimits.

Proposition A.2.6.13

Let AA be a presentable category. Let WW be a class of AA-morphisms called called weak equivalences. Let A 0A_0 be a small set of morphisms of AA called generating cofibrations satisfying:

(1) WW is a perfect class.

(2) For any diagram

X f Y X Y g g X Y \array{ X&\stackrel{f}{\to}& Y \\ \downarrow&&\downarrow \\ X^{\prime}&\to&Y^{\prime} \\ \downarrow^g&&\downarrow^{g^\prime} \\ X^{\prime\prime}&\to& Y^{\prime\prime} }

where both sub squares are cocartesian, fA 0f\in A_0, and gWg\in W, the g Wg^\prime\in W.

(3) A morphism in AA which has the right lifting property with respect to A 0A_0 belongs to WW.

Then there exists a left proper, combinatorial model structure on CC defined by:

(C) A morphism is a cofibration if it belongs to the smallest weakly saturated class of morphisms generated by A 0A_0.

(W) A morphism is a weak equivalence if it belongs to WW.

(F) A morphism is a fibration if it has the right lifting property with respect to the class of acyclic cofibrations.

Remark 2.6.14

Let AA be a model category. Then AA arises via the construction of Proposition A.2.6.13 iff it is left proper, combinatorial and the class of weak equivalences in AA is stable under filtered colimits.

A.2.7 Simplicial sets

Definition

The standard model structure on the category sSetsSet of simplicial sets is defined by:

(W) A morphism is a weak equivalence if its geometric realization is a weak homotopy equivalence.

(C) Cofibrations are the monomorphisms.

(F) Fibrations are Kan fibrations.

A.2.8 Diagram categories and homotopy colimits

If CC is a small category and AA is a combinatorial model category, then

  1. The injective model structure on Fun(C,A)Fun (C,A) is a combinatorial model structure, determined by the strong cofibrations, weak equivalences, and projective fibrations.

  2. The projective model structure on Fun(C,A)Fun (C,A) is a combinatorial model structure, determined by the weak cofibrations, weak equivalences, and injective fibrations.

If AA is moreover right proper resp. left proper, then Fun(C,A)Fun(C,A) is right proper resp. left proper.

A Quillen adjunction (FG):BGA(F\dashv G):B\stackrel{G}{\to}A induces for every small category CC a Quillen adjunction (F CG C):Fun(C,B)G CFun(C,A)(F^C\dashv G^C):Fun(C,B)\stackrel{G^C}{\to}Fun(C,A) with respect to either the injective- or the projective model structure.

Remark

(identity Quillen functor)

Definition A.2.8.1
Proposition A.2.8.2
Remark A.2.8.6
Proposition A.2.8.7
Remark A.2.8.8
Proposition A.2.8.9
Remark A.2.8.11

=–

Proposition A.2.6.13

A.2.7 Simplicial sets

A.2.8 Diagram categories and homotopy colimits

Definition A.2.8.1
Proposition A.2.8.2
Remark A.2.8.6
Proposition A.2.8.7
Remark A.2.8.8
Proposition A.2.8.9
Remark A.2.8.11

Revision on June 24, 2012 at 19:54:06 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.