Contents

model category

for ∞-groupoids

# 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.