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 codiagonal map $X\coprod X\to X$ for $X$ into a weak equivalence followed by a fibration .