Quillen adjunction

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

(F, G): C ${\displaystyle \leftrightarrows }$ 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.

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

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

LF(c) → d

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

c → RG(d)

is an isomorphism in Ho(C).

• Goerss, Paul G. [in German]; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.
• Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages