Spahn HTT, A.2 model categories (Rev #7, 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.

  • (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.2.6 Combinatorial model categories

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 12:39:46 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.