nLab cartesian model category

Idea

A cartesian model category (alias cartesian closed model category) is a cartesian closed category that is equipped with the structure of a monoidal model category in a compatible way, which combines the axioms for a monoidal model category and an enriched model category.

Definition

Definition

A cartesian closed model category (following Rezk 09, 2.2 and Simpson 12) is

• a cartesian closed category

• equipped with a model category-structure

that satisfies

1. the following equivalent axioms:

   • pushout-product axiom

     For $f \colon X \to Y$ and $f' \colon X' \to Y'$ cofibrations, the induced morphism

     $$(Y \times X') \overset{X \times X'}{\coprod} (X \times Y') \longrightarrow Y \times Y'$$

     is a cofibration that is a weak equivalence if at least one of $f$ or $f'$ is;

   • pullback-power axiom

     For $f \colon X \to Y$ a cofibration and $f' \colon A \to B$ a fibration, the induced morphism

     $$[Y,A] \longrightarrow [X,A] \underset{[X,B]}{\prod} [Y,B]$$

     is a fibration, and a weak equivalence if at least one of $f$ or $f'$ is.

2. the unit axiom:

   The terminal object is cofibrant.

Examples

Cartesian monoidal model categories include:

• the standard Quillen model structure on topological spaces on compactly generated weakly Hausdorff topological spaces

  (see there)

• the fine model structure on topological G-spaces

  (see there)

• the classical model structure on simplicial sets

• the canonical model structure on categories (and that on groupoids)

  (see there)

• the model structure for complete Segal spaces

  (see there)

• the model structure for Theta-spaces

  (see there)

Related concepts

• cartesian closed category, locally cartesian closed category

• cartesian closed functor, locally cartesian closed functor

• locally cartesian closed model category

• cartesian closed enriched category

• cartesian closed (∞,1)-categorylocally cartesian closed (∞,1)-category

References

• Charles Rezk, A cartesian presentation of weak $n$-categories, Geom. Topol. 14(1): 521-571 (2010). (arXiv:0901.3602, doi:10.2140/gt.2010.14.521)

• Carlos Simpson, Chapter 10 of: Homotopy Theory of Higher Categories, Cambridge University Press 2011 (hal:00449826, ISBN:9780521516952)