nLab
Ab-enriched category

Idea

An $\mathrm{Ab}$-enriched category (or 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.

Definition

Explicitly, a ringoid (or $\mathrm{Ab}$-enriched category) is a category $C$ such that for all objects $a,b$ the hom-set ${\mathrm{Hom}}_C(a,b)$ is equipped with the structure of an abelian group; and such that for all triples $a,b,c$ of objects the composition operation ${\circ }_{a,b,c}:{\mathrm{Hom}}_C(a,b)\times {\mathrm{Hom}}_C(b,c)\to {\mathrm{Hom}}_C(a,c)$ is bilinear.

Remarks

• $\mathrm{Ab}$-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 $\mathrm{Ab}\to {\mathrm{Set}}_{*}$ from abelian groups to pointed sets, which sends each group to its underlying set with point being the neutral element. Using this functor, every $\mathrm{Ab}$-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 $\mathrm{Ab}$-enriched categories. See additive and abelian categories.

Finite products are absolute

One of the remarkable facts about $\mathrm{Ab}$-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 $\mathrm{Ab}$-enriched functor.

Zero objects

In an $\mathrm{Ab}$-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 ${\mathrm{hom}}_C(a,a)$). Expressed in this way, it is easy to see that any $\mathrm{Ab}$-enriched functor preserves zero objects.

Biproducts

For $c_1,c_2\in C$ two objects in an $\mathrm{Ab}$-enriched category $C$, the product $c_1\times c_2$ coincides with the coproduct $c_1\bigsqcup c_2$ when either exists. More precisely, when both exist, the canonical morphism

$$r:c_1\bigsqcup c_2\to c_1\times c_2$$r : c_1 \sqcup c_2 \to c_1 \times c_2

defined by

$$(c_i\to c_1\bigsqcup c_2\stackrel{r}{\to }{c}_1\times c_2\to c_j)=\{\begin{array}{cc}{\mathrm{Id}}_c{}_i& \mathrm{if}i=j\\ 0& \mathrm{if}i\ne j\end{array},$$\left( c_i \to c_1 \sqcup c_2 \stackrel{r}{\to} c_1 \times c_2 \to c_j \right) = \left\{ \array{ Id_c_i & if i = j \\ 0 & if i \neq j } \right. \,,

which exists whenever $c_1\bigsqcup c_2$ and $c_1\times c_2$ do, is an isomorphism. This object is called a biproduct or (sometimes) a direct sum and is generally denoted

$$c_1\oplus c_2.$$c_1 \oplus c_2.

It can be characterized diagrammatically as an object $c_1\oplus c_2$ equipped with morphisms $q_i:c_i\to c_1\oplus c_2$ and $p_i:c_1\oplus c_2\to c_i$ such that $p_i{q}_j={\delta }_{ij}$ and $q_1{p}_1+q_2{p}_2=1_{c_1\oplus c_2}$. Expressed in this form, it is clear that any $\mathrm{Ab}$-enriched functor preserves biproducts.

As a generalisation of rings

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 $\mathrm{Ab}$-enriched functors $R\to \mathrm{Ab}$ and $R^{\mathrm{op}}\to \mathrm{Ab}$. Bimodules over ringoids have a tensor product (the enriched tensor product of functors) under which they form a bicategory, also known as the bicategory $\mathrm{Ab}\mathrm{Prof}$ of $\mathrm{Ab}$-enriched profunctors. Modules over a ringoid also form an abelian category and thus have a derived category.

One interesting operation on ringoids is the ($\mathrm{Ab}$-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 $\mathrm{Ab}\mathrm{Prof}$, and two ringoids are equivalent in $\mathrm{Ab}\mathrm{Prof}$ if and only if their Cauchy completions are equivalent as $\mathrm{Ab}$-enriched categories. This sort of equivalence is naturally called Morita equivalence.

See also dg-category.

Examples

• The category Ab is closed monoidal and hence canonically enriched over itself.

• An $\mathrm{Ab}$-enriched category with one object is precisely a ring.

• For any small $\mathrm{Ab}$-enriched category $R$, the enriched presheaf category $[R^{\mathrm{op}},\mathrm{Ab}]$ is, of course, $\mathrm{Ab}$-enriched. If $R$ is a ring, as above, then $[R^{\mathrm{op}},\mathrm{Ab}]$ is the category of $R$-modules.

Blog resources

• John Baez, Ringoids