nLab
Ab-enriched category

Ab-enriched categories

Definition

An Ab-enriched category is a category enriched over the category Ab of abelian groups with its usual tensor product.

Sometimes Ab-enriched categories are called pre-additive categories, although sometimes that term also implies the existence of a zero object. They are also sometimes called ringoids, since the concept is a horizontal categorification (or ‘oidification’) of the concept of a ring.

Remarks

  • Explicitly, the definition means that 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 a,b,c:Hom C(a,b)×Hom C(b,c)Hom C(a,c) is bilinear.

  • There is a canonical forgetful functor AbSet * from abelian groups to pointed sets, which sends each group to its underlying set with point being the neutral element. Using this functor every pre-additive category C is in particular also a category that is enriched over pointed sets. This is sufficient for there to be a notion of zero morphism, kernel and cokernel in C.

  • In general abelian categories are the most important examples of Ab-enriched categories. See additive and abelian categories.

Finite products are absolute

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.

Zero objects

In an Ab-enriched category C, any initial object is also a terminal object, hence a zero object, and dually. An object aC is a zero object just when its identity 1 a is equal to the zero morphism 0:aa (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.

Biproducts

For c 1,c 2C two objects in an Ab-enriched category C, the product c 1×c 2 coincides with the coproduct c 1c 2 when either exists. More precisely, when both exist, the canonical morphism

r:c 1c 2c 1×c 2r : c_1 \sqcup c_2 \to c_1 \times c_2

defined by

(c ic 1c 2rc 1×c 2c j)={Id ci ifi=j 0 ifij,\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 1c 2 and c 1×c 2 do, is an isomorphism. This object is called a biproduct or (sometimes) a direct sum and is generally denoted

c 1c 2.c_1 \oplus c_2.

It can be characterized diagrammatically as an object c 1c 2 equipped with morphisms q i:c ic 1c 2 and p i:c 1c 2c i such that p iq j=δ ij and q 1p 1+q 2p 2=1 c 1c 2. Expressed in this form, it is clear that any Ab-enriched functor preserves biproducts.

Examples

  • 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.

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