Quillen Adjunction - Formal Definition

Formal Definition

Given two closed model categories C and D, a Quillen adjunction is a pair

(F, G): C D

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. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.