module over an enriched category


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Basic facts




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

ρ:BA opV \rho : \mathbf{B}A^{op} \to V

from the delooping one-object VV-enriched category BA\mathbf{B}A, corresponding to AA, to VV itself.

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

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


Let VV be a closed monoidal category. Let CC be a VV-enriched category.


A left module over VV is a VV-presheaf on CC, i.e. a functor of VV-enriched categories

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

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

Let CC and DD be VV-enriched categories.


A CC-DD-bimodule is a VV-enriched functor

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

i.e. a left CD opC \otimes D^{op}-module.

CC-DD-bimodules are also known as profunctors or distributors from CC to DD.


Modules over a ring

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

Then a left RR-module NN is equivalently an Ab-enriched functor

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

This makes manifest that the category RRMod is an Ab-enriched category, namely the Ab-enriched functor category

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

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

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

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


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

BGSet \mathbf{B}G \to Set

is nothing but a GG-set: a set equipped with a GG-action.

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

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


See the references at enriched category theory and profunctor.

Last revised on June 8, 2018 at 02:14:14. See the history of this page for a list of all contributions to it.