additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
An -enriched category (or, if small, ringoid) is a category enriched over the monoidal category Ab of abelian groups with its usual tensor product.
Sometimes they are called pre-additive categories, but sometimes that term also implies the existence of a zero object.
Explicitly, an -enriched category is a category such that for all objects the hom-set is equipped with the structure of an abelian group; and such that for all triples of objects the composition operation is bilinear. A ringoid is a small -enriched category.
-enriched categories are called ringoids since the concept is a horizontal categorification (or ‘oidification’) of the concept of a ring.
There is a canonical forgetful functor from abelian groups to pointed sets, which sends each group to its underlying set with point being the neutral element. Using this functor, every -enriched category is in particular also a category that is enriched over pointed sets (that is, a category with zero morphisms). This is sufficient for there to be a notion of kernel and cokernel in .
In general, abelian categories are the most important examples of -enriched categories. See additive and abelian categories.
One of the remarkable facts about -enriched categories is that finite products (and coproducts) are absolute limits. This implies that finite products coincide with finite coproducts, and are preserved by any -enriched functor.
In an -enriched category , any initial object is also a terminal object, hence a zero object, and dually. An object is a zero object just when its identity is equal to the zero morphism (that is, the identity element of the abelian group ). Expressed in this way, it is easy to see that any -enriched functor preserves zero objects.
For two objects in an -enriched category , the product coincides with the coproduct when either exists.
For example, if exists, with projection maps , then according to the universal property of products, there are unique maps
such that and for , and these maps are the coproduct coprojections, i.e., they realize as the coproduct of and . Indeed, for any maps and , it is easily checked that
satisfies and , and is the unique map satisfying these equations. The full argument is spelled out at additive category.
By a dual argument, if the coproduct exists, then it may also be realized as the product of and . Either way, the product or coproduct is called a biproduct or (sometimes) a direct sum and is generally denoted
It can be characterized diagrammatically as an object equipped with morphisms and such that and . Expressed in this form, it is clear that any -enriched functor preserves biproducts.
When using the term ‘ringoid’, one often assumes a ringoid to be small.
Ringoids share many of the properties of (noncommutative) rings. For instance, we can talk about (left and right) modules over a ringoid , which can be defined as -enriched functors and . Bimodules over ringoids have a tensor product (the enriched tensor product of functors) under which they form a bicategory, also known as the bicategory of -enriched profunctors. Modules over a ringoid also form an abelian category and thus have a derived category.
One interesting operation on ringoids is the (-enriched) Cauchy completion, which is the completion under finite direct sums and split idempotents. In particular, the Cauchy completion of a ring is the category of finitely generated projective -modules (aka split subobjects of finite-rank free modules). Every ringoid is equivalent to its Cauchy completion in the bicategory , and two ringoids are equivalent in if and only if their Cauchy completions are equivalent as -enriched categories. This sort of equivalence is naturally called Morita equivalence.
See also dg-category.
The category Ab is closed monoidal and hence canonically enriched over itself.
An -enriched category with one object is precisely a ring.
For any small -enriched category , the enriched presheaf category is, of course, -enriched. If is a ring, as above, then is the category of -modules.
The original articles on ringoids:
Barry Mitchell: Rings with several objects, Advances in Mathematics
8 1 (1972) 1-161 [doi:10.1016/0001-8708(72)90002-3]
Po-Hsiang Chu, Theory of Ringoids, PhD thesis, McGill University (1984) [escholar:q811kn193, pdf]
Daniel Murfet, Localisation of ringoids, notes (2006) [pdf]
Exposition:
Textbook accounts on -enriched categories (mostly: abelian categories) and their application in (homological) algebra:
Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press (1973) [MR0340375]
Charles Weibel: An Introduction to Homological Algebra Cambridge Univ. Press (1994) [doi:10.1017/CBO9781139644136]
Last revised on July 19, 2024 at 07:27:10. See the history of this page for a list of all contributions to it.