nLab cartesian model category

Contents

Context

Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-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 $f \colon X \to Y$ and $f' \colon X' \to Y'$ cofibrations, the induced morphism

$(Y \times X') \overset{X \times X'}{\coprod} (X \times Y') \longrightarrow Y \times Y'$

is a cofibration that is a weak equivalence if at least one of $f$ or $f'$ is;

• pullback-power axiom

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

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

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

2. the unit axiom:

The terminal object is cofibrant.

References

Last revised on June 14, 2021 at 08:58:12. See the history of this page for a list of all contributions to it.