Name for abelian category in which every short exact sequence splits

Question

What is the name of the class of abelian categories defined by the following property: every short exact sequence splits?

Answer 1

The abelian categories in which all short exact sequences split I would call "split abelian categories", reserving the term "semisimple abelian category" for a more restrictive condition. Roughly, perhaps an abelian category should be called "semisimple" if all objects in it are coproducts of simple objects (a nonzero object is called simple if it has no nonzero proper subobjects).

A related question: is every semisimple abelian category split? (Answer: yes, it is; see Jeremy Rickard's comment.) The converse implication is certainly not true.

Grothendieck abelian categories in which all short exact sequences split are known as "spectral categories". The word "spectral" here refers roughly to the spectral theory of operators in Hilbert spaces etc. (in functional analysis), where there is a fundamental opposition between the "discrete" and "continuous" spectrum.

A spectral category in which every object is a coproduct of simple objects is called discrete. A spectral category having no simple objects is called continuous. Every spectral category has a natural, unique decomposition into the Cartesian product of a discrete spectral category and a continuous spectral category.

It is a remarkable and unexpected fact that nondiscrete spectral categories (and in particular, nonzero continuous spectral categories) exist.

Spectral categories $\mathcal A$ with a chosen generator $G$ are in bijective correspondence with right self-injective von Neumann regular associative rings $R$. To a spectral category $\mathcal A$ with a generator $G$ one simply assigns the ring $R=\operatorname{Hom}_{\mathcal A}(G,G)$.

To a right self-injective von Neumann regular ring $R$, one assigns the full subcategory $\mathcal A$ in the category of right $R$-modules $Mod{-}R$ consisting of all the direct summands of products of copies of the (injective) right $R$-module $R$. It may be better to think of $\mathcal A$ as a quotient category of $Mod{-}R$, as in the Gabriel-Popescu theorem; so there is an exact, colimit-preserving localization functor $Mod{-}R\longrightarrow\mathcal A$ which has a fully faithful right adjoint $\mathcal A\longrightarrow Mod{-}R$. The image of this fully faithful functor is described above.

To produce an example of a nondiscrete spectral category, one has to come up with a "nontrivial enough" example of right self-injective von Neumann regular ring. E.g., complete Boolean algebras with no atoms correspond to some continuous spectral categories.

References:

1. P. Gabriel, U. Oberst. Spektralkategorien und reguläre Ringe im von-Neumannschen Sinn. Math. Zeitschrift 92, #5, p.389-395, 1966.

2. B. Stenström. Rings of quotients. An introduction to methods of ring theory. Springer, 1975. Sections V.6-7 and XII.1-3.

Answer 2

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every left $A$-module is a direct sum of simple objects, or equivalently that every object of $A$ is a finite direct sum of simple objects) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective.)

Answer 3

In section III.2.3 in the book "Methods of Homological Algebra" by Gelfand and Manin such an abelian category is indeed called semisimple as Donu Arapura suggested in the comment. One might also call them "abelian categories of global dimension 0", since every object is projective.

Answer 4

In On the representations of 2-groups in Baez-Crans 2-vector spaces, such categories are called “split abelian” (§2.4, p. 910), as suggested in Leonid Positselski’s answer.

You could also call them abelian categories which satisfy the external axiom of choice (all epimorphisms split) or that satisfy the von Neumann axiom (for all arrows $f\colon A\to B$, there is a $g\colon B\to A$ such that $f = fgf$). In an abelian category, those axioms are equivalents to all short exact sequences splitting. See also Abelian categories in dimension 2, Theorem 306, p. 247, for a list of equivalent conditions.

N.B. In the On von Neumann varieties paper cited above, von Neumann varieties are those for which the full subcategory of either finitely presentable objects or finitely presentable projective objects satisfies the von Neumann axiom, not the whole category, so it wouldn’t be an subextension of their (underdetermined) terminology to talk of a von Neumann abelian category in your case.