symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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The action on a module over a monoid in a closed monoidal category may be equivalently encoded in terms of a -enriched functor
from the delooping one-object -enriched category , corresponding to , to itself.
More generally it makes sense to replace by any -enriched category – regarded as the horizontal categorification of a monoid, a “monoid-oid” – and think of a -enriched presheaf as a module over the category .
From this perspective a --bimodule is a -enriched functor , which is in this context known as a profunctor from to . The notion of the bicategory of -enriched categories, -profunctors between these and transformations between those is then a generalization of the category of monoids in and bimodules between them.
Let be a closed monoidal category. Let be a -enriched category.
A left module over is a -presheaf on , i.e. a functor of -enriched categories
Dually a right module is a -enriched functor .
Let and be -enriched categories.
--bimodules are also known as profunctors or distributors from to .
For a ring, write for the Ab-enriched category with a single object and hom-object .
Then a left -module is equivalently an Ab-enriched functor
This makes manifest that the category Mod is an Ab-enriched category, namely the Ab-enriched functor category
The right -modules can be considered as -functors . Then the usual tensor product of abelian groups of left and right -modules can be considered as a functor
The coend computes then to .
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 -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 (with no extra structure, i.e. just a set with group structure) is a monoid in Set. A module over in the sense of Set-enriched functor (just an ordinary functor)
is nothing but a -set: a set equipped with a -action.
is the small category that is the delooping groupoid of , which has a single object and . The functor takes the single object to some set and takes each morphism to an automorphism of that set, such that composition is respected. This is just a representation of on the set .
Of course for this story to work, 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 06:14:14. See the history of this page for a list of all contributions to it.