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
- c → G(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
- c → RG(d)
is an isomorphism in Ho(C).
Read more about this topic: Quillen Adjunction
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