In homotopy theory, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 that induces an adjunction between the homotopy categories Ho(C) and Ho(D) 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 pair: C D
of adjoint functors with F left adjoint to G such that F preserves cofibration

Cofibration

In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...

s and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

s 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 equivalence

Weak equivalence

-Mathematics:In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same basic "shape"...

s 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

Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...

so that

LF(c) → d

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

c → RG(d)

is an isomorphism in Ho(C).