Quillen adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed [model category|closed model categories] C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho and Ho via the total [derived functor] construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.
Formal definition
Given two closed model categories C and D, a Quillen adjunction is a pairof 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.
Properties
It is a consequence of the axioms that a left Quillen functor preserves weak [equivalence (homotopy theory)|weak equivalence]s between cofibrant objects. The total [derived functor theorem] of Quillen says that the total left derived functoris a left adjoint to the total right derived functor
This adjunction is called the derived adjunction.
If 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 if and only if
is an isomorphism in Ho.