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

A cartesian model category (following Rezk (2010) and Simpson (2012)) is a cartesian closed category equipped with a model structure that satisfies the following additional axioms:

• (Pushout-product axiom). If $f : X \to Y$ and $f' : X' \to Y'$ are cofibrations, then the induced morphism $(Y \times X') \cup^{X \times X'} (X \times Y') \to Y \times Y'$ is a cofibration that is trivial if either $f$ or $f'$ is.

• (Unit axiom). The terminal object is cofibrant.

Examples

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

• the standard model structure on simplicial sets is cartesian closed.

Related concepts

• cartesian closed category, locally cartesian closed category

• cartesian closed functor, locally cartesian closed functor

• cartesian closed model category, locally cartesian closed model category

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

References

• Charles Rezk, A cartesian presentation of weak $n$-categories. (2010). (arXiv:0901.3602)

• Carlos Simpson, Homotopy theory of higher categories (2012)