This is a subentry of a reading guide to HTT.

Contents

A.2.1 The model category axioms

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

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

A.2.7 Simplicial sets

A.2.8 Diagram categories and homotopy colimits