(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
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 small -enriched category.
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 .
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.
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.
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.