# nLab enriched model category

Contents

## 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

• (pullback-power axiom) for every cofibration $i \colon A \to B$ and 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.

(…)

## Examples

###### Example

(monoidal model category is enriched model over itself)
Every monoidal model category is an enriched model category over itself, via the enrichment of its underlying closed monoidal category.

###### Proof

One just needs to see that the pullback-power axiom is implied by (in fact it is equivalent to) the pushout-product axiom. This equivalence is an instance of Joyal-Tierney calculus (see this Prop.):

Writing

• $\mathrm{F}$, $\mathrm{C}$, $\mathrm{W}$ for the classes of fibrations, cofibrations, weak equivalences, respectively;

• $\mathrm{FW}$, $\mathrm{CW}$ for the classes of acyclic fibrations and acyclic cofibrations, respectively

and

• $(-) \,⧄\, (-)$ for the lifting property,

• $(-) \Box (-)$ for the pushout product,

• $(-)^{\Box (-)}$ for the pullback power,

we have the following logical equivalences.

\begin{aligned} \mathrm{C} \Box \mathrm{C} & \,\subset\, & \mathrm{C} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{C} &\,\;⧄\;\,& \mathrm{FW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \,⧄\, & \mathrm{FW}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{FW}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{FW} \\ {\phantom{-}} \\ \mathrm{C} \Box \mathrm{CW} & \,\subset\, & \mathrm{CW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{CW} &\,\;⧄\;\,& \!\!\mathrm{F}\;\;\; & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \;⧄\; & \mathrm{F}^{\Box \mathrm{CW}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{CW}} & \,\subset\, & \mathrm{FW} \\ & & & & && & \;\;\; \Leftrightarrow \;\;\; & \mathrm{CW} & \;⧄\; & \mathrm{F}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{F} \end{aligned}

Here the outer equivalences are by definition of the lifting properties in a model category, while the middle equivalences are by Joyal-Tierney calculus. The statements on the far left constitute the pushout-product axiom, while those on the far right constiture the pullback-power axiom.

###### Example

Since the model structure on compactly-generated topological spaces as well as the classical model structure on simplicial sets are monoidal model categories, they are, by by Exp. , also enriched modal categories over themselves.

###### Example

A model category enriched over the classical model structure on simplicial sets is called a simplicial model category.

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