In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction.
Given two closed model categories C and D, a Quillen adjunction is a pair of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations.
It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects.
The total derived functor theorem of Quillen says that the total left derived functor is a left adjoint to the total right derived functor This adjunction (LF, RG) is called the derived adjunction.
If (F, G) is a Quillen adjunction as above such that with c cofibrant and d fibrant is a weak equivalence in D if and only if is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint equivalence of categories so that is an isomorphism in Ho(D) if and only if is an isomorphism in Ho(C).