Holmstrom Quillen functor

1. A functor $F: C \to D$ is a left Quillen functor if it is a left adjoint, and preserves cofibrations and trivial cofibrations.
2. A functor $U: D \to C$ is a right Quillen functor if it is a right adjoint, and preserves fibrations and trivial fibrations.
3. An adjunction $(F, U, \varphi)$ is a Quillen adjunction if $F$ is a left Quillen functor, or equivalently, if $U$is a right Quillen functor.

(Here $\varphi: D(FA,B) \to C(A, UB)$ is a natural isomorphism)

The main point of the above definition is that a left (right) Quillen functor preserves cofibrant (fibrant) objects and WEs between them.

Notation: We write $\eta: X \to UFX$ for the unit map and $\varepsilon: FUX \to X$ for the counit map.

Example: Take $F$ to be geometric realization, and $U$ to be singular complex. This gives a Quillen adjunction from $Sset$ to $Top$.

Example: Diagonal functor and product functor, or coproduct functor and diagonal functor.

Example: Disjoint basepoint functor and forgetful functor.

We can define the 2-category of model categories, using Quillen adjunctions as morphisms.

There is also a notion of Quillen adjunction of two variables, sometimes the left adjoint occurring in this definition is called a Quillen bifunctor. See Hovey, section 4.2.

In Goerss-Schemmerhorn, p. 20, there is a theorem describing how an adjoint pair can allow us to lift a model category structure from one category to another. Example: non-negatively graded cdga’s get a model structure from the category of chain complexes, when the base field has char zero but not when it has char 2.

nLab page on Quillen functor