Spahn HTT, A.2 model categories (Rev #16, changes)

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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.

  • (C∪W,F)(C\cup W,F) and (C,F∪W)(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 X∐X→XX\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 X→X×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:X→Yf,g:X\to Y be two parallel morphisms. Then the following conditions are equivalent.

  1. The coproduct map f∐gf\coprod g factors through every cylinder object for XX.

  2. The coproduct map f∐gf\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) LetCC be a model category.

(1) Let Two maps f C,g:X→Y f,g:X\to C Y from be a cofibrant model object category. to a fibrant object satisfying the conditions ofProposition A.2.2.1 are called homotopic morphisms. Homotopy is an equivalence relation ≃\simeq on hom C(X,Y)hom_C (X,Y).

(2) (1) The Two mapshomotopy category hCh C of CCf,g:X→Yf,g:X\to Y is from defined a cofibrant object to have a as fibrant objects object satisfying the fibrant-cofibrant conditions objects ofCCProposition A.2.2.1 . The are hom called objectshom hC(X,Y)hom_{hC}(X,Y)homotopic morphisms . are Homotopy defined is to an be equivalence the relation set of≃\simeq equivalence on classes ofhom 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 (F⊣G):D→GC(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 (F⊣G)(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) category, derived functor)

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 (F⊣G):D→GC(F\dashv G):D\stackrel{G}{\to}C is a Quillen adjunction FF induces a functor LF:hC→hDL 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:hD→hCRG:hD\to hC.

In total one can show that (LF⊣RG):hD→RGhC(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:

  1. LFLF is called the (homotopy) left derived functor of ff.

  2. RGRG is called the (homotopy) right derived functor of gg.

In Abstracty more one detail can we obtain wish this to result extend byF:C→DF : C \to DKan extension (for (this is also described atGGderived functor ); analogously) however Quillen adjunction’s are introduced to a present diagram adjunctions between∞\infty-categories and to obtain such a presentation in terms of Kan extension in general requires additional assumptions:

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 } \,,

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

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

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 } \,,

This where is accomplished by takinghCQ C:C→ hD hC hC\to Q_C hD : C \to hC to is be either the left universal ( morphism characterizing theLF:=Lan Q CQ d∘FLF:=Lan_{Q_C} Q_d \circ Fhomotopy category ) or and right similarly ( forRFQ D:=Ran Q CQ d∘F RF:=Ran_{Q_C} Q_D Q_d \circ F) Kan extension of Q d∘FQ_d \circ F along Q CQ_C.

This is accomplished by taking hC→hDhC\to hD to be either the left (LF:=Lan Q CQ d∘FLF:=Lan_{Q_C} Q_d \circ F) or right (RF:=Ran Q CQ d∘FRF:=Ran_{Q_C} Q_d \circ F) Kan extension of Q d∘FQ_d \circ F along Q CQ_C.
Proposition A.2.5.1

(characterization of derived functors, Quilen equivalence) Let (F⊣G):D→GC(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:hC→hDLF:hC\to hD is an equivalence of categories.

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

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

(F⊣G)(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, f∈A 0f\in A_0, and g∈Wg\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

(A.2.8.1, A.2.8.2)

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.

Remark A.2.8.6

A Quillen adjunction (F⊣G):B→GA(F\dashv G):B\stackrel{G}{\to}A induces for every small category CC a Quillen adjunction (F C⊣G 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.

(F⊣G)(F\dashv G) is a Quillen equivalence iff (F C⊣G C)(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)⊣id Fun(C,A)):Fun(C,A) inj→Fun(C,A) proj(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→C ′f:C\to C^\prime be a functor between small categories. For a combinatorial model category AA let f *:=(−)∘f:Fun(C ′,A)→Fun(C,A)f^*:=(-)\circ f:Fun(C^\prime,A)\to Fun(C,A) denote the functor given by precomposition with ff. By Kan extension we see that there are adjoints

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

and

  1. (f !⊣f *):Fun(C ′,A) proj→f *Fun(C,A) proj(f_!\dashv f^*):Fun(C^\prime,A)_{proj}\stackrel{f_*}{\to}Fun(C,A)_{proj}

  2. (f *⊣f *):Fun(C,A) inj→f *Fun(C proj,A) inj(f^*\dashv f_*):Fun(C,A)_{inj}\stackrel{f_*}{\to}Fun(C^\proj,A)_{inj}

Remark A.2.8.8
Definition

(homotopy right Kan extension)

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

(transclusion from 5.2.4 Examples of adjoint functors:
Definition (in Proof of Corollary 5.2.4.5)

(homotopy limit) Let1C 1 C , denote the terminal category. Let A D A D be a fibrant combinatorial simplicially model enriched category. categories. Leta(F⊣G):1D→→G A C a:1\to (F\dashv A G):D\stackrel{\G}{\to}C denote be a global simplicially element enriched of adjunction. Let A M A M . Let be the simplicially enriched category defined byCC be a (note necessarily small) category. Let !:C→1!:C\to 1 denote the unique functor to the terminal category. Let F:C→AF:C\to A be a functor.

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

Map M(c,d)=Map C(c,G(d))=Map D(F(c),d)Map_M(c,d)=Map_C (c,G(d))=Map_D (F(c),d)
Map M(d,c)=∅Map_M (d,c)=\varnothing

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

for every c∈Cc\in C, d∈Dd\in D.

MM is the correspondence associated to the adjunction (F⊣G)(F\dashv G).
Proposition A.2.8.9 5.2.4.6

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

d/F:C× Dd/D→C→FAd/F:C\times_D d/D\to C\stackrel{F}{\to}A

Let AA, A ′A^\prime be simplicially enriched model categories. Let
(F⊣G):A ′→GA(F\dashv G):A^\prime\stackrel{G}{\to} A

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

Then N(M ∘)N(M^\circ) determines an adjunction
(f⊣g):N(A ′∘)→gN(A ∘)(f\dashv g):N(A^{\prime\circ})\stackrel{g}{\to}N(A^\circ)

here ff is called left derived functor of FF and gg is called right derived functor of GG.

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

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

Definition

(homotopy right Kan extension)

Let f:C→C ′f:C\to C^\prime be a functor between small categories. The right derived functor Rf *Rf_* functor of the functor f *=Ran ff_*=Ran_f (which is the right adjoint to f *:=(−)∘f:Fun(C ′,A)→Fun(C,A)f^*:=(-)\circ f:Fun(C^\prime,A)\to Fun(C,A)) is called homotopy right Kan extension.
Definition

(homotopy limit)

Let 11 denote the terminal category. Let AA be a combinatorial model category. Let a:1→Aa:1\to A denote a global element of AA. Let CC be a (note necessarily small) category. Let !:C→1!:C\to 1 denote the unique functor to the terminal category. Let F:C→AF:C\to A be a functor.

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

Note that ! *a=κ a!^* a=\kappa_a is the constant functor ‘’in aa’’.
Proposition A.2.8.9

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

d/F:C× Dd/D→C→FAd/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.

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