nLab
cartesian closed model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

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.

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:XYf : X \to Y and f:XYf' : X' \to Y' are cofibrations, then the induced morphism (Y×X) X×X(X×Y)Y×Y(Y \times X') \cup^{X \times X'} (X \times Y') \to Y \times Y' is a cofibration that is trivial if either ff or ff' is.

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

Examples

References

Revised on November 11, 2013 02:18:40 by Urs Schreiber (89.204.135.29)