[[!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 codiagonal map $X\coprod X\to X$ for $X$ into a weak equivalence followed by a fibration . =-- +-- {: .un_prop} ###### Proposition A.2.2.1 =-- ## 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 +-- {: .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 =--