Quillen adjunction

Formal definition

Given two closed model categories C and D, a Quillen adjunction is a pair :(F, G): C \leftrightarrows 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.

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 :**L'''F: Ho(C) → Ho(**D''') is a left adjoint to the total right derived functor :R'''G: 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 with c cofibrant and d fibrant 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 :'L'''''F(c) → d is an isomorphism in Ho(D) if and only if :c → 'R'''''G(d) is an isomorphism in Ho(C).

References

• http://www-math.mit.edu/~larsh/teaching/S2005/l12 http://www-math.mit.edu/~larsh/teaching/S2005/l13
• Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages