Holmstrom Cofibrant replacement

In a Model category, the functorial factorization $(\alpha, \beta)$ ensure that for any object $X$, the map $0 \to X$ from the initial object factors through a cofibrant object, denoted by $QX$. This construction is a functor $Q: X \to X$, called the cofibrant replacement functor.

http://mathoverflow.net/questions/54500/how-canonical-is-cofibrant-replacement

nLab page on Cofibrant replacement