Quillen Adjunction - Properties

Properties

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

LF: Ho(C) → Ho(D)

is a left adjoint to the total right derived functor

RG: Ho(D) → Ho(C).

This adjunction (LF, RG) is called the derived adjunction.

If (F, G) is a Quillen adjunction as above such that

F(c) → d

is a weak equivalence in D if and only if

cG(d)

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

LF(c) → d

is an isomorphism in Ho(D) if and only if

cRG(d)

is an isomorphism in Ho(C).

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