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.