(co)fibrant resolution
Idea
As discussed in model categories and derived functors, we can have, say, functors which are homotopical (preserve weak equivalences) only on “good” (i.e. fibrant, cofibrant) objects. Presently we discuss a method of replacing, up to weak equivalence, any object with a fibrant or cofibrant one.
Definition
A fibrant resolution of $X$ is a fibrant object $\hat{X}$ equipped with a weak equivalence into it: $$X\overset{\simeq}{\longrightarrow}\hat{X}\longrightarrow\ast$$
A cofibrant resolution of $X$ is a cofibrant object $\tilde{X}$ equipped with a weak equivalence out of it: $$\emptyset\longrightarrow\tilde{X}\overset{\simeq}{\longrightarrow} X$$