nLab
Ab-enriched category

AbAb-enriched categories

Idea

An AbAb-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.

Definition

Explicitly, an AbAb-enriched category is a category CC such that for all objects a,ba,b the hom-set Hom C(a,b)Hom_C(a,b) is equipped with the structure of an abelian group; and such that for all triples a,b,ca,b,c of objects the composition operation a,b,c:Hom C(a,b)×Hom C(b,c)Hom C(a,c) \circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c) is bilinear. A ringoid is small AbAb-enriched category.

Remarks

  • AbAb-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 AbSet *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 AbAb-enriched category CC 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 CC.

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

Finite products are absolute

One of the remarkable facts about AbAb-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 AbAb-enriched functor.

Zero objects

In an AbAb-enriched category CC, any initial object is also a terminal object, hence a zero object, and dually. An object aCa\in C is a zero object just when its identity 1 a1_a is equal to the zero morphism 0:aa0:a\to a (that is, the identity element of the abelian group hom C(a,a)\hom_C(a,a)). Expressed in this way, it is easy to see that any AbAb-enriched functor preserves zero objects.

Biproducts

For c 1,c 2Cc_1, c_2 \in C two objects in an AbAb-enriched category CC, the product c 1×c 2c_1 \times c_2 coincides with the coproduct c 1c 2c_1 \sqcup c_2 when either exists. More precisely, when both exist, the canonical morphism

r:c 1c 2c 1×c 2 r : 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 2c_1\sqcup c_2 and c 1×c 2c_1\times 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 2c_1\oplus c_2 equipped with morphisms q i:c ic 1c 2q_i:c_i\to c_1\oplus c_2 and p i:c 1c 2c ip_i:c_1\oplus c_2 \to c_i such that p iq j=δ ijp_i q_j = \delta_{i j} and q 1p 1+q 2p 2=1 c 1c 2q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}. Expressed in this form, it is clear that any AbAb-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 RR, which can be defined as AbAb-enriched functors RAbR\to Ab and R opAbR^{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 AbProfAb Prof of AbAb-enriched profunctors. Modules over a ringoid also form an abelian category and thus have a derived category.

One interesting operation on ringoids is the (AbAb-enriched) Cauchy completion, which is the completion under finite direct sums and split idempotents. In particular, the Cauchy completion of a ring RR is the category of finitely generated projective RR-modules (aka split subobjects of finite-rank free modules). Every ringoid is equivalent to its Cauchy completion in the bicategory AbProfAb Prof, and two ringoids are equivalent in AbProfAb Prof if and only if their Cauchy completions are equivalent as AbAb-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 AbAb-enriched category with one object is precisely a ring.

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

References

Revised on March 7, 2014 09:00:05 by Zoran Škoda (161.53.130.104)