A complete and cocomplete category with designated model weak equivalences, model fibrations and model cofibrations, closed under retracts. Weak equivalences satisfy two-out-of-three; maps admit the two cofibration/fibration factorizations, and either acyclic class has the required lifting property against the opposite class. These axioms organize homotopy-theoretic calculations abstractly.
A **model category** is a concept from category theory, which is a branch of mathematics that deals with abstract structures and relationships between them. Specifically, a model category provides a framework for doing homotopy theory in a categorical setting. It allows mathematicians to work with "homotopical" concepts such as homotopy equivalences, fibrations, and cofibrations in a systematic way.
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