model category

for ∞-groupoids

# Contents

## Idea

A locally cartesian closed model category is a locally cartesian closed category which is equipped with the structure of a model category in a compatible way.

## Definition

A model category $𝒞$ which is additionally a locally cartesian closed category is called a locally cartesian closed model category if for any fibration $g:A\to B$ between fibrant objects, the dependent product adjunction

${g}^{*}:𝒞/B⇄𝒞/A:{\Pi }_{g}$g^* : \mathcal{C}/B \rightleftarrows \mathcal{C}/A : \Pi_g

is a Quillen adjunction between the corresponding slice model structures.

Concretely, this means that both cofibrations and trivial cofibrations are stable under pullback along fibrations between fibrant objects.

Equivalently this means that for all $A\to B$ as above the internal hom adjunction in the slice category over $B$

$\left(-\right){×}_{𝒞{/}_{B}}A\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}𝒞{/}_{B}⇄𝒞{/}_{B}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left[A,-{\right]}_{𝒞{/}_{B}}$(-) \times_{\mathcal{C}/_B} A \;:\; \mathcal{C}/_B \rightleftarrows \mathcal{C}/_B \;:\; [A, -]_{\mathcal{C}/_B}

## Versus locally cartesian closed $\left(\infty ,1\right)$-categories
It is easy to see that the $\left(\infty ,1\right)$-category presented by a locally cartesian closed model category is itself locally cartesian closed. Conversely, any locally presentable locally cartesian closed $\left(\infty ,1\right)$-category can be presented by some right proper Cisinski model category, which is therefore a locally cartesian closed model category; see locally cartesian closed (infinity,1)-category for the proof.