# nLab locally cartesian closed model category

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

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

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

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

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

• ### for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

• model structure for (2,1)-sheaves/for 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 $B$

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

## 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. See the history of this page for a list of all contributions to it.