# nLab enriched model category

Contents

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Enriched category theory

enriched category theory

# 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 $i \colon A \to B$ in $V_0$ and every fibration $p \colon X \to Y$ in $C_0$ the pullback powering morphism (dual to the pushout product) with respect to the powering in $V$

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

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

$\otimes : C \times V \to C$

is a Quillen bifunctor.

(…)

## References

Last revised on July 19, 2018 at 13:26:05. See the history of this page for a list of all contributions to it.