related by the Dold-Kan correspondence
A 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 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 and are cofibrations, then the induced morphism is a cofibration that is trivial if either or is.
(Unit axiom). The terminal object is cofibrant.
the standard model structure on simplicial sets is cartesian closed.
cartesian closed model category, locally cartesian closed model category
Carlos Simpson, Homotopy theory of higher categories (2012)