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 $C$

  • which is powered and copowered over $V$;

• with the structure of a model category on the underlying category $C_0$

• such that for every cofibration $i:A\to B$ and every fibration $p:X\to Y$ in $C_0$ the morphism in $V$ $C(B,X)\stackrel{i^{*}\times p_{*}}{\to }C(A,X){\times }_{C(A,Y)}C(B,Y)$ is a fibration with respect to the model structure on $V$;

• and such that this fibration is an acyclic fibration whenever either $i$ or $p$ are acyclic.

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

is a Quillen bifunctor.

Remarks

• The concept can be generalized to enriched homotopical category.

Examples

• The most familiar examples of enriched model categories are simplicial model categories.