Spahn HTT, A.2 model categories (Rev #9, changes)

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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.

  • (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) Let CC be a model category.

(1) Two maps f,g:X→Yf,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 (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.
Proposition Remark A.2.5.1

(characterization (descent of derived a functors) 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 (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:

In more detail we wish to extend F:C→DF : 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:C→hCQ_C : C \to hC is the universal morphism characterizing the homotopy category and similarly for Q DQ_D.

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 adjunction) 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
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 15:20:12 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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