Semisimple Abelian categories with infinite sums

A semisimple category is an abelian category in which every object is a finite direct sum of simple objects.

A) Why does one impose the finiteness condition here?

B) If one condsiders infinite direct sums does something go wrong?

C) If B) works with no problems, then is this equivalent to an abelian category where exact sequences split?

• Commented Jun 17, 2020 at 15:56

A) It depends on what you are interested in. If you do not impose the finiteness condition, then it means that you are describing a different class of abelian categories. Which class is that, depends on additional conditions which you may want to impose instead of finiteness of the direct sums.

B) Nothing goes wrong, but you have to make some decisions. Firstly, if you want every object to be a (possibly infinite) direct sum of simple objects, then it is natural to impose the condition that all (set-indexed infinite) direct sums exist in your category. Further, you may want the condition that isomorphism classes of simple objects form a set. Under the previous assumptions, this is equivalent to the condition that your category has a generator, or a set of generators.

Imposing these conditions allows you to describe precisely what the objects of your category are. To describe the morphisms in a natural way, you may want to impose a further condition that, for any simple object $$S$$ in your category $$\mathcal A$$, the functor $$\operatorname{Hom}_{\mathcal A}(S,{-})\colon\mathcal A\to \mathcal Ab$$ preserves infinite direct sums. Under the previous assumptions, this is equivalent to the condition that $$\mathcal A$$ satisfies the axiom Ab5, or in other words, that $$\mathcal A$$ is a Grothendieck abelian category (as we've already assumed that $$\mathcal A$$ has a generator).

Then your category $$\mathcal A$$ is equivalent to the Cartesian product, taken over some set $$X$$, of the categories $$D_x{-}Mod$$ of (possibly infinite-dimensional) modules/vector spaces over some division rings (skew-fields) $$D_x$$, $$\,x\in X$$.

It seems to be an open question whether the condition that $$\mathcal A$$ is Ab5 can be dropped (i.e., whether it follows from the conditions that $$\mathcal A$$ has infinite direct sums, every object is a direct sum of simple objects, and there is only a set of isomorphism classes of simple objects).

C) No, it is not equivalent. In the classical terminology going back to 1960's, a Grothendieck abelian category in which every short exact sequence splits is called "spectral". The term comes from functional analysis and suggests an analogy with the distinction between the discrete and continuous spectrum in the spectral theory of operators in a functional space.

A spectral category in which all objects are direct sums of simple objects is called discrete. A spectral category having no simple objects is called continuous. It is known that there are many nonzero continuous spectral categories.

On the other hand, I am not aware of any example of a category with a generator, with infinite direct sums, in which all short exact sequences split, but which is not Grothendieck.

References:

1. A related question was discussed on MO in Name for abelian category in which every short exact sequence splits

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

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

4. L. Positselski, J. Šťovíček. Topologically semisimple and topologically perfect topological rings. Electronic preprint https://arxiv.org/abs/1909.12203, Section 2.

• "On the other hand, I am not aware of any example of a category with a generator, with infinite direct sums, in which all short exact sequences split, but which is not Grothendieck.": taking the opposite category of any non-zero spectral Grothendieck category gives an example. Commented Jun 18, 2020 at 6:24
• @AurélienDjament Yes, you are right. Thank you. Commented Jun 18, 2020 at 11:13