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.

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.

2. 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 .

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 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 $hom_C (X,Y)$.