[[!redirects A.2 model categories]] This is a subentry of [[a reading guide to HTT]]. # Contents * Automatic table of contents {: toc} ## A.2.1 The model category axioms +-- {: .num_defn} ###### 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. =-- +-- {: .num_remark} ###### Remark 1. The classes $(C)$ and $(F)$ is closed under retracts. (by [weak factorization systems, Lemma 2, in joyal's catlab](http://ncatlab.org/joyalscatlab/show/Weak+factorisation+systems#main_definitions_2)) 1. The class $(W)$ is closed under retracts. (by [model categories, Lemma 1, in joyal's catlab](http://ncatlab.org/joyalscatlab/show/Model+categories)) =-- ## A.2.2 The homotopy category of a model category +-- {: .num_defn} ###### Definition Let $X$ be an object in a model category. 1. 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. 1. 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 . =-- +-- {: .un_prop #propA.2.2.1} ###### 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. 1. The coproduct map $f\coprod g$ factors through every cylinder object for $X$. 1. The coproduct map $f\coprod g$ factors through some cylinder object for $X$. 1. The product map $f\times g$ factors through every path object for $Y$. 1. The product map $f\times g$ factors through some path object for $Y$. =-- +-- {: .num_defn} ###### 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](#propA.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. +-- {: .un_prop #propA.2.3.1} ###### 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. +-- {: .un_defn} ###### 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. 1. A model category is called *right proper* if the pullback of a weak equivalence along a fibration is a weak equivalence. =-- +-- {: .un_prop} ###### Proposition Any model category in which every object is cofibrant is left proper. =-- ## A.2.5 Quillen adjunctions and Quillen equivalences ## A.2.6 Combinatorial model categories +-- {: .un_defn} ###### Definition A.2.6.1 =-- +-- {: .un_prop} ###### Proposition A.2.6.13 =-- ## A.2.7 Simplicial sets ## A.2.8 Diagram categories and homotopy colimits +-- {: .un_defn} ###### Definition A.2.8.1 =-- +-- {: .un_prop} ###### Proposition A.2.8.2 =-- +-- {: .un_remark} ###### Remark A.2.8.6 =-- +-- {: .un_prop} ###### Proposition A.2.8.7 =-- +-- {: .un_remark} ###### Remark A.2.8.8 =-- +-- {: .un_prop} ###### Proposition A.2.8.9 =-- +-- {: .un_remark} ###### Remark A.2.8.11 =--