model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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.
A cartesian closed model category (following Rezk 09, 2.2 and Simpson 12) is
equipped with a model category-structure
that satisfies
the following equivalent axioms:
For $f \colon X \to Y$ and $f' \colon X' \to Y'$ cofibrations, the induced morphism
is a cofibration that is a weak equivalence if at least one of $f$ or $f'$ is;
For $f \colon X \to Y$ a cofibration and $f' \colon A \to B$ a fibration, the induced morphism
is a fibration, and a weak equivalence if at least one of $f$ or $f'$ is.
the unit axiom:
The terminal object is cofibrant.
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.
cartesian closed category, locally cartesian closed category
cartesian closed model category, locally cartesian closed model category
cartesian closed (∞,1)-categorylocally cartesian closed (∞,1)-category
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)