nLab module over an enriched category

Idea

The action on a module over a monoid $A$ in a closed monoidal category $V$ may be equivalently encoded in terms of a $V$-enriched functor

$$\rho : \mathbf{B}A^{op} \to V$$

from the delooping one-object $V$-enriched category $\mathbf{B}A$, corresponding to $A$, to $V$ itself.

More generally it makes sense to replace $\mathbf{B}A$ by any $V$-enriched category $C$ – regarded as the horizontal categorification of a monoid, a “monoid-oid” – and think of a $V$-enriched presheaf $\rho : C \to V$ as a module over the category $C$.

From this perspective a $C$-$D$-bimodule is a $V$-enriched functor $C^{op}\times D \to V$, which is in this context known as a profunctor from $C$ to $D$. The notion of the bicategory $V Mod$ of $V$-enriched categories, $V$-profunctors between these and transformations between those is then a generalization of the category of monoids in $V$ and bimodules between them.

Definition

Let $V$ be a closed monoidal category. Let $C$ be a $V$-enriched category.

Definition

A left module over $V$ is a $V$-presheaf on $C$, i.e. a functor of $V$-enriched categories

$$M : C^{op} \longrightarrow V.$$

Dually a right module is a $V$-enriched functor $M : C \to V$.

Let $C$ and $D$ be $V$-enriched categories.

Definition

A $C$-$D$-bimodule is a $V$-enriched functor

$$C^{op} \otimes D \longrightarrow V,$$

i.e. a left $C \otimes D^{op}$-module.

$C$-$D$-bimodules are also known as profunctors or distributors from $C$ to $D$.

Examples

Modules over a ring

For $R$ a ring, write $\mathbf{B}R$ for the Ab-enriched category with a single object and hom-object $R = \mathbf{B}R(\bullet, \bullet)$.

Then a left $R$-module $N$ is equivalently an Ab-enriched functor

$$N : \mathbf{B}R^{op} \to Ab \,.$$

This makes manifest that the category $R$Mod is an Ab-enriched category, namely the Ab-enriched functor category

$$R Mod \simeq [\mathbf{B}R,Ab] \,.$$

The right $R$-modules can be considered as $Ab$-functors $\mathbf{B}R\to Ab$. Then the usual tensor product of abelian groups $M\otimes N$ of left and right $R$-modules can be considered as a functor

$$\mathbf{B}R^{op}\otimes \mathbf{B}R\to Ab.$$

The coend $\int^R M\otimes N$ computes then to $M\otimes_R N$.

$G$-Sets

Classically the notion of module is always regarded internal to Ab, so that a module is always an abelian group with extra structure. But noticing that such abelian ring modules are just enriched presheaves in Ab-enriched category theory, it makes sense to consider enriched presheaves in general $V$-enriched category theory as a natural generalization of the notion of module.

For that generalization the case of Set-enriched category theory plays a special basic role:

a group $G$ (with no extra structure, i.e. just a set with group structure) is a monoid in Set. A module over $G$ in the sense of Set-enriched functor (just an ordinary functor)

$$\mathbf{B}G \to Set$$

is nothing but a $G$-set: a set equipped with a $G$-action.

$\mathbf{B}G$ is the small category that is the delooping groupoid of $G$, which has a single object and $Hom_{\mathbf{B}G}(\bullet,\bullet) = G$. The functor $\mathbf{B}G \to Set$ takes the single object to some set $S$ and takes each morphism $(\bullet \stackrel{g}{\to} \bullet)$ to an automorphism $\rho(g) : S \to S$ of that set, such that composition is respected. This is just a representation of $G$ on the set $S$.

Of course for this story to work, $G$ need not be a group, but could be any monoid.

Related concepts

• enriched presheaf category

References

See the references at enriched category theory and profunctor.