Abelian categories satisfying AB5*

Question

I could name on the spot a bunch of abelian categories satisfying AB5 but I cannot think of any that satisfies AB5*. That is, it should have all limits and the cofiltered limits are exact. Is there any example that I should have in mind?

Answer 1

The snarky response would be "the opposite category of any of the categories you could name on the spot". The less-snarky response is to observe that some of these are quite natural. For example, $\mathrm{Vect}^{\mathrm{op}}_{\mathbb{F}_p}$ agrees with the category of profinite $\mathbb{F}_p$ vector spaces, since $\mathrm{Vect}_{\mathbb{F}_p}$ is the $\mathrm{Ind}$-category of finite $\mathbb{F}_p$ vector spaces, finite vector spaces are self-dual, and profinite $\mathbb{F}_p$ vector spaces are the $\mathrm{Pro}$-category of finite vector spaces.

Answer 2

The classic situation for this is given in U. Oberst, Duality theory for Grothendieck categories and linearly compact rings, Journal of Algebra, 15, (1970), 473 – 542, whilst for examples look at A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Alg, 4, (1966), 442–470.

The discussion starts with Gabriel's thesis: P. Gabriel, Des catégories abèliennes, Bull. Soc. Math. France, 90, (1962), 323–448. and then with J.-E. Roos, 1969, Locally Noetherian categories and generalized strictly linearly compact rings. Applications, in Category Theory, Homology Theory and Their Applications II , 197 – 277, Springer.