nLab
enriched 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

Enriched category theory

Contents

Idea

An enriched model category is an enriched category CC together with the structure of a model category on the underlying category C 0C_0 such that both structures are compatible in a reasonable way.

Definition

Let VV be a monoidal model category.

A VV-enriched model category is

  • an V-enriched category CC

  • with the structure of a model category on the underlying category C 0C_0

  • such that

    • for every cofibration i:ABi : A \to B and every fibration p:XYp : X \to Y in C 0C_0 the morphism (dual to the pushout product) in VV

      C(B,X)(i *,p *)C(A,X)× C(A,Y)C(B,Y) C(B,X) \stackrel{(i^* , p_*)}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)

      is a fibration with respect to the model structure on VV;

    • and is an acyclic fibration whenever ii or pp are acyclic.

The last two conditions here are equivalent to the fact that the copower

:C×VC \otimes : C \times V \to C

is a Quillen bifunctor.

Properties

Change of enrichment

(…)

Examples

References

Revised on November 11, 2013 00:52:08 by Urs Schreiber (89.204.135.29)