nLab
cartesian model category

Contents

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 equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-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 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

that satisfies

  1. the following equivalent axioms:

    • pushout-product axiom

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

      (Y×X)X×X(X×Y)Y×Y (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 ff or ff' is;

    • pullback-power axiom

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

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

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

  2. the unit axiom:

    The terminal object is cofibrant.

Examples

References

Last revised on June 14, 2021 at 04:58:12. See the history of this page for a list of all contributions to it.