# nLab enriched model category

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

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.

The general definition of enriched model categories has its origin in

where it is formulated in the special case of simplicial model categories (but the generalization is immediate).

Textbook account: