nLab
locally cartesian closed model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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 𝒞\mathcal{C} which is additionally a locally cartesian closed category is called a locally cartesian closed model category if for any fibration g:ABg\colon A\to B between fibrant objects, the dependent product adjunction

g *:𝒞/B𝒞/A:Π 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 ABA \to B as above the internal hom adjunction in the slice category over BB

()× 𝒞/ BA:𝒞/ B𝒞/ B:[A,] 𝒞/ 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 (,1)(\infty,1)-categories

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

Revised on May 10, 2012 20:24:04 by Mike Shulman (71.136.230.118)