I know this is a very vague question, but I can't think of a better question to ask.
Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \mathscr{C}$ equipped with a monomorphism $\iota$ to a set $S : \mathbf{Set}$ are in one-to-one correspondence to filters $\mathcal{F}_{A}$ on $S$, where elements of $\mathcal{F}_A$ are supposed to be the subsets of $S$ containing $A$. Since $\mathscr{C}$ is a regular category, it has an exact completion $\mathscr{C}_{\mathrm{ex}}$ consisting of formal quotients by equivalence relations. If I've done things correctly, the full subcategory of $\mathscr{C}_{\mathrm{ex}}$ consisting of those with a regular epimorphism from a set is equivalent to the category of complete uniform spaces. Therefore, $\mathscr D = \mathrm{Ab}(\mathscr{C}_{\mathrm{ex}})$ should be an abelian category with a fully faithful embedding from the category of complete topological abelian groups.
Does $\mathscr D$ have a name? Has it been studied? Do the standard constructions involving abelian categories work correctly?
Note that $\mathscr D$ has a sort of similar spirit to condensed sets, but behaves differently with regards to kernels and quotients. The morphism $\mathbb{R}^\delta \to \mathbb{R}$ of maps from the underlying set of $\mathbb{R}$ to $\mathbb{R}$ as a topological space has no kernel but a complicated nontrivial quotient in the category of condensed sets. In $\mathscr D$, the quotient is trivial, and the kernel is the formal intersection of all neighborhoods of zero in $\mathbb R$, interpreted as subsets of the underlying set $\mathbb{R}^\delta$.
