additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
An $Ab$-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 $Ab$-enriched category is a category $C$ such that for all objects $a,b$ the hom-set $Hom_C(a,b)$ is equipped with the structure of an abelian group; and such that for all triples $a,b,c$ of objects the composition operation $\circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c)$ is bilinear. A ringoid is a small $Ab$-enriched category.
$Ab$-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 $Ab \to Set_*$ from abelian groups to pointed sets, which sends each group to its underlying set with point being the neutral element. Using this functor, every $Ab$-enriched category $C$ 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 $C$.
In general, abelian categories are the most important examples of $Ab$-enriched categories. See additive and abelian categories.
One of the remarkable facts about $Ab$-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 $Ab$-enriched functor.
In an $Ab$-enriched category $C$, any initial object is also a terminal object, hence a zero object, and dually. An object $a\in C$ is a zero object just when its identity $1_a$ is equal to the zero morphism $0:a\to a$ (that is, the identity element of the abelian group $\hom_C(a,a)$). Expressed in this way, it is easy to see that any $Ab$-enriched functor preserves zero objects.
For $c_1, c_2 \in C$ two objects in an $Ab$-enriched category $C$, the product $c_1 \times c_2$ coincides with the coproduct $c_1 \sqcup c_2$ when either exists.
For example, if $c_1 \times c_2$ exists, with projection maps $p_i\colon c_1 \times c_2 \to c_i$, then according to the universal property of products, there are unique maps
such that $p_i q_i = 1_{c_i}$ and $p_j q_i = 0$ for $j \neq i$, and these maps $q_i$ are the coproduct coprojections, i.e., they realize $c_1 \times c_2$ as the coproduct of $c_1$ and $c_2$. Indeed, for any maps $r_1\colon c_1 \to e$ and $r_2\colon c_2 \to e$, it is easily checked that
satisfies $r q_1 = r_1$ and $r q_2 = r_2$, and is the unique map satisfying these equations. The full argument is spelled out at additive category.
By a dual argument, if the coproduct $c_1 \sqcup c_2$ exists, then it may also be realized as the product of $c_1$ and $c_2$. 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 $c_1\oplus c_2$ equipped with morphisms $q_i \colon c_i\to c_1\oplus c_2$ and $p_i \colon c_1\oplus c_2 \to c_i$ such that $p_i q_j = \delta_{i j}$ and $q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}$. Expressed in this form, it is clear that any $Ab$-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 $R$, which can be defined as $Ab$-enriched functors $R\to Ab$ and $R^{op}\to Ab$. Bimodules over ringoids have a tensor product (the enriched tensor product of functors) under which they form a bicategory, also known as the bicategory $Ab Prof$ of $Ab$-enriched profunctors. Modules over a ringoid also form an abelian category and thus have a derived category.
One interesting operation on ringoids is the ($Ab$-enriched) Cauchy completion, which is the completion under finite direct sums and split idempotents. In particular, the Cauchy completion of a ring $R$ is the category of finitely generated projective $R$-modules (aka split subobjects of finite-rank free modules). Every ringoid is equivalent to its Cauchy completion in the bicategory $Ab Prof$, and two ringoids are equivalent in $Ab Prof$ if and only if their Cauchy completions are equivalent as $Ab$-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 $Ab$-enriched category with one object is precisely a ring.
For any small $Ab$-enriched category $R$, the enriched presheaf category $[R^{op},Ab]$ is, of course, $Ab$-enriched. If $R$ is a ring, as above, then $[R^{op},Ab]$ is the category of $R$-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 $Ab$-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.