nLab
locally cartesian closed model category

Contents

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 rational \infty-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, namely such that all right base change-adjunctions along fibrations are Quillen adjunctions, hence such that all their dependent product-functors are right Quillen functors.

Remark

(relation to cartesian closed model categories)
Beware that, despite the terminology, the axioms on a locally cartesian closed model category (Def. ) do not imply that the underlying model category (or any of its slice model categories) is a cartesian closed model category – and in most examples this is not the case. Namely, the axioms here (2) only require Quillen functors in one variable (the second variable for internal homs, with the other variable a fixed fibrant object) where those of a cartesian closed model category require Quillen bifunctors.

Definition

Definition

(locally cartesian closed model category)
A locally cartesian closed model category is

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

Equivalently this means that for all fibrations ABA \to B the internal hom adjunction in the slice category over BB

(2)()× 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

Example

Any right proper model category whose underlying category is locally cartesian closed and in which the cofibrations are the monomorphisms is a locally cartesian closed model category.

Proof

The fiber product/pullback functor g *g^\ast

  • is a left adjoint by local cartesian closure of the underlying category,

  • preserves cofibrations because these are the monomorphisms and hence are preserved by pullback (by this prop.),

  • preserves weak equivalences, and hence acyclic cofibrations by the previous item, due to right properness – using here the assumption (1) that gg is a fibration.

In summary this means that g *g^\ast is a left Quillen functor.

Example

Example subsumes the following classes of examples, in increasing generality:

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: With the assumption (1) that gg is a fibration between fibrant objects, it follows (by this Prop) that pullback along gg models the correct homotopy pullback.

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 there for the proof.

Versus homotopy type theories

Last revised on June 22, 2021 at 13:36:57. See the history of this page for a list of all contributions to it.