Quillen adjunction

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {D} }$ is a special kind of adjunction between categories that induces an adjunction between the homotopy categories ${\displaystyle {\text{Ho}}(\mathbf {C} )}$ and ${\displaystyle {\text{Ho}}(\mathbf {D} )}$ via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

Formal definition

Given two closed model categories ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {D} }$, a Quillen adjunction is a pair

${\displaystyle (F,G):\mathbf {C} \rightleftarrows \mathbf {D} }$

of adjoint functors with ${\displaystyle F}$ left adjoint to ${\displaystyle G}$ such that ${\displaystyle F}$ preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that ${\displaystyle G}$ preserves fibrations and trivial fibrations. In such an adjunction, ${\displaystyle F}$ is called the left Quillen functor and ${\displaystyle G}$ 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

${\displaystyle \mathbf {L} F:\operatorname {Ho} (\mathbf {C} )\rightarrow \operatorname {Ho} (\mathbf {D} )}$

is a left adjoint to the total right derived functor

${\displaystyle \mathbf {R} G:{\text{Ho}}(\mathbf {D} )\rightarrow {\text{Ho}}(\mathbf {C} )}$

This adjunction ${\displaystyle (\mathbf {L} F,\mathbf {R} G)}$ is called the derived adjunction.

If ${\displaystyle (F,G)}$ is a Quillen adjunction as above such that

${\displaystyle F(c)\rightarrow d}$

with ${\displaystyle c}$ cofibrant and ${\displaystyle d}$ fibrant is a weak equivalence in ${\displaystyle \mathbf {D} }$ if and only if

${\displaystyle c\rightarrow G(d)}$

is a weak equivalence in ${\displaystyle \mathbf {C} }$ then it is called a Quillen equivalence of the closed model categories ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {D} }$. In this case the derived adjunction is an adjoint equivalence of categories so that

${\displaystyle \mathbf {L} F(c)\rightarrow d}$

is an isomorphism in ${\displaystyle {\text{Ho}}(\mathbf {D} )}$ if and only if

${\displaystyle c\rightarrow \mathbf {R} G(d)}$

is an isomorphism in ${\displaystyle {\text{Ho}}(\mathbf {C} )}$.

References

• 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