Model Category

A model category is a category with all small limits and colimits and three classes of maps, called weak equivalences, fibrations, and cofibrations, satisfying the model axioms. Weak equivalences satisfy the two-out-of-three property and all three classes are closed under retracts. Every map factors as a cofibration followed by an acyclic fibration and as an acyclic cofibration followed by a fibration. Here acyclic means also a weak equivalence. Cofibrations lift against acyclic fibrations, and acyclic cofibrations lift against fibrations. These structures define a homotopy category by formally inverting weak equivalences. The Kan-Quillen structure on simplicial sets and the Thomason structures on strict higher categories are examples.