nLab
locally cartesian closed model category
Context
Model category theory
model category

Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$ -categories
Model structures
for $\infty$ -groupoids
for ∞-groupoids

for $n$ -groupoids
for $\infty$ -groups
for $\infty$ -algebras
general
specific
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 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 $\mathcal{C}$ which is additionally a locally cartesian closed category is called a locally cartesian closed model category if for any fibration $g\colon A\to B$ between fibrant objects , the dependent product adjunction

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

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 $B$

$(-) \times_{\mathcal{C}/_B} A
\;:\;
\mathcal{C}/_B
\rightleftarrows
\mathcal{C}/_B
\;:\;
[A, -]_{\mathcal{C}/_B}$

is a Quillen adjunction .

Examples
Any right proper model category which is locally cartesian closed and in which the cofibrations are the monomorphisms is a locally cartesian closed model category. This includes the classical model structure on simplicial sets , as well as the injective global model structure on simplicial presheaves . More generally, it includes any right proper Cisinski model structure .

Versus locally cartesian closed $(\infty,1)$ -categories
It is easy to see that the $(\infty,1)$ -category presented by a locally cartesian closed model category is itself locally cartesian closed . Conversely, any locally presentable locally cartesian closed $(\infty,1)$ -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.

Applications
Last revised on May 10, 2012 at 20:24:04.
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