# Realization/embedding for (weakly) finite linear categories

I am trying to determine the status of the following claim. I know how to prove this (unless I made a stupid mistake), so the question is mostly

• Is it in the literature?
• If not, is there something similar that could be useful?

Let C be a linear category over a field k (that is, an abelian category enriched over k-spaces). Suppose that it satisfies the following finiteness relations:

1. C is essentially small
2. Every object of C has finite length
3. All hom spaces are finite-dimensional.

Claim. C is equivalent to the category of finite-dimensional modules over a pro-finite-dimensional k-algebra.

Here a pro-finite-dimensional k-algebra is a filtered projective limit of finite-dimensional k-algebras, or in more classical terms, a topological k-algebra that is complete and whose topology has a basis given by two-sided ideals of finite codimension. And of course, the modules must agree with this structure (i.e., they are topological).

Background. If we assume in addition that C has finitely many simple objects (up to isomorphism) and enough projectives, this becomes a known statement. Such categories are called finite (abelian) categories, and the claim is that C is equivalent to the category of finite-dimensional representations of a finite-dimensional algebra. (By the way, the statement is attributed to Deligne on nLab; do you know the source of this attribution? The reference on nLab is to a paper by Etingof and Ostrik, which as far as I see simply gives the statement.)

• This can fail due to size considerations; a necessary condition for this to be true is that $C$ must have a set of simple objects. As a counterexample, take $C$ to be the category of finite-dimensional ordinal-graded vector spaces. Jul 15, 2016 at 17:06
• Thanks, of course I assume that C is essentially small (forgot to put it in the question, now corrected). Jul 15, 2016 at 17:25
• First, apparently arxiv.org/abs/1406.4204 contains a proof of the background statement you cite. Second, can you sketch the proof you have in mind? I have a strategy in mind but I don't know how to make it work with the given hypotheses (although I'm not sure what more hypotheses are needed). Jul 17, 2016 at 20:34
• Thanks for the reference! (Still wondering why the claim was attributed to Deligne...) I will be happy to discuss the statement/proof, but I'd rather do it over email if you don't mind --- it's easier to just send the file. Jul 18, 2016 at 14:51