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David Michael Roberts (Oct 22 2025 at 01:06):I really like theorems like this one of Anna Giulia Montaruli:
http://tac.mta.ca/tac/volumes/41/40/41-40abs.html namely
...how to embed in a constructive way a small abelian category into the category of sheaves of modules over a ringed space.
This might be the paper I was thinking of the whole time, but there's a niggling feeling that I was actually recalling a slightly stronger result that was not just an embedding theorem, but an actual reconstruction, analogous to an abelian category version of Giraud's theorem (recreating the category as equivalent to some kind of abelian sheaves on a "linear site", perhaps?). Can anyone point to such a paper, or else disabuse me of thinking there is such a result?
David Michael Roberts (Oct 22 2025 at 01:09):Hmm, maybe I was thinking of https://doi.org/10.1007/s10485-017-9511-1 / https://arxiv.org/abs/1707.07453 (by @Julia Ramos ?) which remarks
The fact that the categories of sheaves over linear sites (or linear Grothendieck topoi)
are precisely the Grothendieck categories can be deduced from Gabriel–Popescu theorem
[9] together with the main result in [5]. Indeed, Gabriel–Popescu theorem characterizes
Grothendieck categories as the localizations of presheaf categories of linear sites, that is
subcategories of presheaf categories whose embedding functor has a left exact left adjoint.
On the other hand, from [5, Thm 1.5] one deduces that categories of sheaves are precisely
the localizations of presheaf categories of linear categories. Thus, the combination of the
two results provides a linear counterpart of the classical Giraud Theorem that characterizes
Grothendieck topoi in the classical setting.
David Michael Roberts (Oct 22 2025 at 01:12):Prior work here is by Wendy Lowen: https://win.uantwerpen.be/~wlowen/pdf%20papers/JPAA-GabrielPopescu.pdf
Hmm. I might have solved my first problem.
David Michael Roberts (Oct 22 2025 at 01:13):The second problem is this: is there work on the internal logic of a (Grothendieck) abelian category from this point of view, keeping in mind the usual internal logic of a (Grothendieck) topos?
Morgan Rogers (he/him) (Oct 22 2025 at 08:03):I'll raise that question at my local category theory working group next week, since we have an expert on abelian categories and several logicians around :blush:
David Michael Roberts (Oct 22 2025 at 08:27):Thanks!
Fernando Yamauti (Oct 26 2025 at 15:35):Have you already seen this paper of Melliès?
I just skimmed through, but it seems some initial attempt at extracting some linear logic out of modules over a precosheaf on comm rings. Presumably, it shouldn't be hard to extend everything to modules over a ringed topos, which is not that far from an arbitrary Grothendieck abelian cat with a symmetric monoidal structuture inside loc pres cats.
The the last section, though, says that a linear logic for alg geom still doesn't exist.
Fernando Yamauti (Oct 26 2025 at 15:41):Also I just remembered there exists an analogous of a Lawvere-Tierney topology for abelian cats, but I don’t know to what extent that fully internalises (what's the analogous of a subobject classifier?)