Spahn HTT, A.2 model categories (changes)

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This is a subentry of a reading guide to HTT.

A.2.1 The model category axioms
A.2.2 The homotopy category of a model category
A.2.3 A lifting criterion
A.2.4 Left properness and homotopy push out squares
A.2.5 Quillen adjunctions and Quillen equivalences
A.2.6 Combinatorial model categories
A.2.7 Simplicial sets
A.2.8 Diagram categories and homotopy colimits

A.2.1 The model category axioms

Definition

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

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

$C$ admits (small) limits and colimits.
The class of weak equivalences satisfies 2-out-of-3.
$(C\cup W,F)$ and $(C,F\cup W)$ are weak ~~factorization~~ ~~systems.~~factorization systems.

Remark

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

A.2.2 The homotopy category of a model category

HTT, A.2.2

Definition

Let $X$ be an object in a model category.

A cylinder object is defined to be a factorization of the codiagonal map $X\coprod X\to X$ for $X$ into a cofibration followed by a weak equivalence.
A path object is defined to be a factorization of the diagonal map $X\to X\times X$ for $X$ into a weak equivalence followed by a fibration .

Proposition A.2.2.1

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

The coproduct map $f\coprod g$ factors through every cylinder object for $X$ .
The coproduct map $f\coprod g$ factors through some cylinder object for $X$ .
The product map $f\times g$ factors through every path object for $Y$ .
The product map $f\times g$ factors through some path object for $Y$ .

Definition

(homotopy, homotopy category of a model category)

Let $C$ be a model category.

(1) Two maps $f,g:X\to Y$ 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 $hom_C (X,Y)$ .

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

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

A model category is called left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence.
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 $(F\dashv G):D\stackrel{G}{\to}C$ is called a Quillen adjunction if the following equivalent conditions are satisfied:

$F$ preserves cofibrations and acyclic cofibrations.
$G$ preserves fibrations and acyclic fibrations.
$F$ preserves cofibrations and $G$ preserves fibrations.
$F$ preserves acyclic cofibrations and $G$ preserves acyclic fibrations.

Remark

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

$F$ preserves weak equivalences between cofibrant objects.
$G$ preserves weak equivalences between fibrant objects.

Remark

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

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

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

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

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

$LF$ is called the (homotopy) left derived functor of $f$ .
$RG$ is called the (homotopy) right derived functor of $g$ .

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 $F : C \to D$ (for $G$ analogously) to a diagram

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

where $Q_C : C \to hC$ is the universal morphism characterizing the homotopy category and similarly for $Q_D$ .

This is accomplished by taking $hC\to hD$ to be either the left ( $LF:=Lan_{Q_C} Q_d \circ F$ ) or right ( $RF:=Ran_{Q_C} Q_d \circ F$ ) Kan extension of $Q_d \circ F$ along $Q_C$ .

Proposition A.2.5.1

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

The left derived functor $LF:hC\to hD$ is an equivalence of categories.
The right derived functor $RF:hD\to hC$ is an equivalence of categories.
For every cofibrant object $c\in C$ and every fibrant object $D\in D$ , a map $c\to G(d)$ is a weak equivalence iff the adjoint map $F(c)\to d$ is a weak equivalence.

$(F\dashv G)$ 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 $C$ be a category with all small colimits. A class $S$ of $C$ -morphisms is called a weakly saturated class if the following conditions are satisfied.

$S$ is closed under forming pushouts (along arbitrary $C$ -morphisms).
$S$ is closed under transfinite composition.
$S$ is closed under forming retracts.

)

Definition A.2.6.1

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

$A$ is presentable.
There exists a set $I$ of generating cofibrations such that the collection of all cofibrations is the smallest weakly saturated class of morphisms containing $I$ .
There exists a set $J$ of generating acyclic cofibrations such that the collection of all acyclic cofibrations is the smallest weakly saturated class of morphisms containing $J$ .

Definition A.2.6.10

(perfect class) Let $A$ be a presentable category. A class $W$ of morphisms in $C$ is called perfect class if the following conditions are satisfied:

$W$ contaings all isomorpphisms.
$W$ satisfies 2-out-of-3
$W$ is stable under poset filtered colimits.
$W$ contains a small subset which generates $W$ under filtered colimits.

Proposition A.2.6.13

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

(1) $W$ is a perfect class.

(2) For any diagram

\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, $f\in A_0$ , and $g\in W$ , the $g^\prime\in W$ .

(3) A morphism in $A$ which has the right lifting property with respect to $A_0$ belongs to $W$ .

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

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

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

A.2.7 Simplicial sets

Definition

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

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

(F) Fibrations are Kan fibrations.

A.2.8 Diagram categories and homotopy colimits

(A.2.8.1, A.2.8.2)

If $C$ is a small category and $A$ is a combinatorial model category, then

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

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

Remark A.2.8.6

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

$(F\dashv G)$ is a Quillen equivalence iff $(F^C\dashv G^C)$ is.

In other words: Forming the injective- resp. projective model structure is a functor.

Remark

(identity Quillen functor)

By Remark A.2.8.5 every projective cofibration is an injective cofibration and (dually) every injective fibration is a projective fibration. By definition the projective- and injective model structure have the same weak equivalences. It follows that the identity

(id_{Fun(C,A)}\dashv id_{Fun(C,A)}):Fun(C,A)_{inj}{\to}Fun(C,A)_{proj}

is a Quillen equivalence between the injective- and the projective model structure.

Proposition A.2.8.7

Let $f:C\to C^\prime$ be a functor between small categories. For a combinatorial model category $A$ let $f^*:=(-)\circ f:Fun(C^\prime,A)\to Fun(C,A)$ denote the functor given by precomposition with $f$ . By Kan extension we see that there are adjoints

(f_!\dashv f^*\dashv f_*):Fun(C,A)\stackrel{f_*}{\to} Fun(C^\prime,A)

and

$(f_!\dashv f^*):Fun(C^\prime,A)_{proj}\stackrel{f_*}{\to}Fun(C,A)_{proj}$
$(f^*\dashv f_*):Fun(C,A)_{inj}\stackrel{f_*}{\to}Fun(C^\proj,A)_{inj}$

Remark A.2.8.8

(transclusion from 5.2.4 Examples of adjoint functors:

Definition (in Proof of Corollary 5.2.4.5)

Let $C$ , $D$ be fibrant simplicially enriched categories. Let $(F\dashv G):D\stackrel{\G}{\to}C$ be a simplicially enriched adjunction. Let $M$ be the simplicially enriched category defined by

Map_M(c,d)=Map_C (c,G(d))=Map_D (F(c),d)

Map_M (d,c)=\varnothing

for every $c\in C$ , $d\in D$ .

$M$ is the correspondence associated to the adjunction $(F\dashv G)$ .

Proposition 5.2.4.6

(derived functor)

Let $A$ , $A^\prime$ be simplicially enriched model categories. Let

(F\dashv G):A^\prime\stackrel{G}{\to} A

be a simplicially enriched Quillen adjunction. Let $M$ denote the correspondence associated to the adjunction $(F\dashv G)$ . Let $M^\circ$ denote the full subcategory of $M$ consisting of those objects which are fibrant-cofibrant objects (either as objects in $A$ or as objects in $A^\prime$ ).

Then $N(M^\circ)$ determines an adjunction

(f\dashv g):N(A^{\prime\circ})\stackrel{g}{\to}N(A^\circ)

here $f$ is called left derived functor of $F$ and $g$ is called right derived functor of $G$ .

On the level of homotopy categories $f$ and $g$ reduce to the usual derived functors associated to the Quillen adjunction, see (homotopy) derived functor.

)

Definition

(homotopy right Kan extension)

Let $f:C\to C^\prime$ be a functor between small categories. The right derived functor $Rf_*$ functor of the functor $f_*=Ran_f$ (which is the right adjoint to $f^*:=(-)\circ f:Fun(C^\prime,A)\to Fun(C,A)$ ) is called homotopy right Kan extension.

Definition

(homotopy limit)

Let $1$ denote the terminal category. Let $A$ be a combinatorial model category. Let $a:1\to A$ denote a global element of $A$ . Let $C$ be a (note necessarily small) category. Let $!:C\to 1$ denote the unique functor to the terminal category. Let $F:C\to A$ be a functor.

A natural transformation $\alpha:!^* a\to F$ is called a homotopy limit of $F$ if $\alpha$ exhibits $a$ as a homotopy Kan extension of $F$ .

Note that $!^* a=\kappa_a$ is the constant functor ‘’in $a$ ’’.

Proposition A.2.8.9

Let $A$ be a combinatorial model category, let $f : C \to D$ be a functor between small categories. Let $F : C \to A$ and $G : D \to A$ be diagrams. A natural transformation $\alpha : f_* G \to F$ exhibits G as a homotopy right Kan extension of $F$ if and only if, for each object $d \in D$ , $\alpha$ exhibits $G(d)$ as a homotopy limit of the composite diagram

d/F:C\times_D d/D\to C\stackrel{F}{\to}A

The following remark defines homotopy Kan extensions which in particular model Kan extensions between $\infty$ -categories:

(…)

Remark A.2.8.11

Analog for homotopy colimits.

Last revised on December 1, 2012 at 05:26:55. See the history of this page for a list of all contributions to it.

Spahn HTT, A.2 model categories (changes)

Contents

A.2.1 The model category axioms

Definition

Remark

A.2.2 The homotopy category of a model category

Definition

Proposition A.2.2.1

Definition

A.2.3 A lifting criterion

Proposition A.2.3.1

A.2.4 Left properness and homotopy push out squares

Definition A.2.4.1

Proposition

Lemma A.2.4.3

A.2.5 Quillen adjunctions and Quillen equivalences

Proposition and Definition

Remark

Remark

Proposition A.2.5.1

A.2.6 Combinatorial model categories

Definition A.1.2.2

Definition A.2.6.1

Definition A.2.6.10

Proposition A.2.6.13

Remark 2.6.14

A.2.7 Simplicial sets

Definition

A.2.8 Diagram categories and homotopy colimits

Remark A.2.8.6

Remark

Proposition A.2.8.7

Remark A.2.8.8

Definition (in Proof of Corollary 5.2.4.5)

Proposition 5.2.4.6

Definition

Definition

Proposition A.2.8.9

Remark A.2.8.11