nLab
enriched model category

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Enriched category theory

enriched category theory

Background

Basic concepts

Universal constructions

Extra stuff, structure, property

Homotopical enrichment

Contents

Idea

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

Definition

Let V be a monoidal model category.

A V-enriched model category is

  • an V-enriched category C

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

  • such that

    • for every cofibration i:A→B and every fibration p:X→Y in C 0 the morphism (dual to the pushout product) in V

      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 V;

    • and is an acyclic fibration whenever i or p are acyclic.

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

⊗:C×V→C\otimes : C \times V \to C

is a Quillen bifunctor.

Properties

Change of enrichment

(…)

Examples

References

Bertrand Guillou, J.P. May, enriched model categories and diagram categories, arXiv:1110.3567v1

Revised on June 25, 2012 21:13:29 by Stephan Alexander Spahn (79.227.174.123)
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