[[!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 +-- {: .un_defn} ###### Definition A.2.1.1 A *model category is a category* =-- A *model category* is a category $C$ equipped with three distinguished classes of morphisms in $C$: 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. * $(C)$,$(F)$and $(W)$ are closed under retracts. * $(C\cup W,F)$ and $(C,F\cup W)$ are weak factorization systems. ## A.2.2 The homotopy category of a model category +-- {: .un_defn} ###### Definition 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 =--