nLab enriched 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

• (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.

References

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