Quillen adjunction

Summary

A Quillen adjunction is a pair of adjoint functors between model categories that respects the model structures. The left adjoint preserves cofibrations and trivial cofibrations, or equivalently the right adjoint preserves fibrations and trivial fibrations. Such a pair induces an adjunction between the homotopy categories, through the derived functors. When this derived adjunction is an equivalence, the pair is a Quillen equivalence, and the two model categories present the same homotopy theory.