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Stream: theory: category theory

Topic: Linear analogues of topos

Fernando Yamauti (Oct 29 2022 at 23:57):

I'm not sure if I'm following the social protocol here regarding the creation of new topics (first time writing anything here). Has anyone seen any work on enriched versions of topos? I'm interested in the geometric point of view (so topos as classifying spaces of theories and geometric morphisms) and not in the elementary topos (or even logos) point of view.

In non-commutative geometry, one often sees Grothendieck abelian categories and locally presentable stable categories (probably with an additional monoidal structure) as avatars for spaces up to Morita equivalence. Still, I have never seen a translation of the usual topological concepts in topos theory to that setting. For instance, in that linear setting, what's the analogous of a localic morphism? Or, say, an open morphism? I'm aware about récollement, torsion theories and some of the several spectrum constructions, though. But that's pretty much as far as I know the analogy goes.

I wouldn't be surprised, however, if maybe someone working with cosmos and linear logic has already done implicitly something like that in a very general setting.

Thanks in advance.

fosco (Oct 30 2022 at 08:47):

I'm aware of the existence of "linear" analogues for some constructions in CLTT, akin to the hyperdoctrine approach, https://arxiv.org/pdf/1405.0033.pdf (and by no means an expert of even a knowledgeable reader). The approach, however, is very logical in nature. I am interested in this question, indeed motivated by the same idea -more naive than yours- that Grothendieck categories behave like toposes in some respect.

Sam Speight (Oct 30 2022 at 13:28):

I have thought about this too. I think (but need to check the details) that the usual realizability topos construction makes sense using a linear partial combinatory algebra (in the sense of Abramsky) and you get something elementary-topos-like.

John Baez (Oct 30 2022 at 17:09):

Given a topos of sheaves it's natural to consider the category of sheaves of abelian groups, or vector spaces, or more generally modules for some sheaf of rings as a kind of "linear version" of that topos. And we can define all these things for any elementary topos, e.g. the category of modules of a ring object in our elementary topos. So this would be one source of examples for any attempted formalization of the idea of a "linear analogue of a topos" - and these examples are incredibly important in algebraic geometry.

Morgan Rogers (he/him) (Oct 30 2022 at 21:20):

There were some foundations laid by the Australian school for enriched topos theory, where one substitutes the category of presheaves on a locally small category for the V-enriched category of V-functors to V from a V-category (for V a suitable enriching category; I forget what requirements were imposed). As far as "translating topological concepts" goes, this direction doesn't provide a useful parallel to the classic theory, since we build the topos of sheaves as a subcategory of the category of presheaves on the poset of opens, which isn't enriched in anything interestingly linear. What @John Baez suggests may be more interesting: if I have a ringed space $(X,\mathcal{O}_X)$ and a map $f:Y \to X$, the induced geometric morphisms will lift to an adjunction between the categories of modules for $\mathcal{O}_X$ and $f^*(\mathcal{O}_X)$. It makes sense to topological properties with the induced properties of this adjunction (although the properties are likely to be more coarse-grained along this translation than the original properties).

Peter Arndt (Oct 30 2022 at 22:03):

Hi Fernando, good question!

I'll just throw out a few thoughts. In the following "tensor category" is intended to mean abelian tensor category, or stable tensor category, possibly closed, depending on your context of choice.

Peter Arndt (Oct 30 2022 at 22:03):

Martin Brandenburg in his thesis analyzes some properties of the functors $QCoh(X)\to QCoh(Y)$ between quasicoherent module categories of schemes and stacks that come from open/closed/general immersions and affine morphisms -- these are probably good to know if you look for a definition of open/closed/general immersion of tensor categories (note Brandenburg's Remarks 5.7.15 and 5.7.16, to see that something more general is going on than just talking about $QCoh(X)$ for a scheme $X$). See also Section 2 of his earlier article with Alexandru Chirvasitu.
Thinking in terms of the 2-algebraic geometry of Chirvasitu and Johnson-Freyd is probably helpful, and Brandenburg does this in a way.

Peter Arndt (Oct 30 2022 at 22:03):

The localic morphism question is tricky. Here is a thought, but maybe that's not really what you want: A topos can be seen as a generalized topological space, and if that space carries extra geometric structure then that is encoded in objects of the topos (e.g. a ring object, thought of as function algebra on the space). In contrast, in the linear setting, the "function algebra" is already built into the category (e.g. the base ring is the endomorphism ring of the tensor unit), and it is somehow more natural to ask about geometric notions instead of topological notions (e.g. Brandenburg sketches how to define smooth, unramified and étale - see his Def. 5.12.18).

Peter Arndt (Oct 30 2022 at 22:03):

For localic morphisms, maybe one should start with the non-relative version? The localic topoi - if we ignore the question of there being enough points or not - are the "generalized spaces" that are actually usual, non-genralized spaces. They are the topoi for which the morphism to the terminal topos is localic.
Now about "localic morphisms" for tensor categories, you might first ask, what the non-relative version should be, i.e. what is a "localic" tensor category. Given that tensor categories generalize scheme-like things with function algebras, rather than topological spaces, maybe it should be a tensor category corresponding to the motivating example, i.e. one of the form $QCoh(X)$. Daniel Schäppi gives an answer, if you allow $X$ to be an Adams stack.
Now you could try and formulate that a "localic morphism of tensor categories" is fiberwise such a thing as Schäppi describes. That requires talking about pullbacks, which is somewhat tricky - see Brandenburg's Remark 5.1.21.

Peter Arndt (Oct 30 2022 at 22:03):

Or you try to imitate the fact that a localic morphism of topoi corresponds to an internal locale in the codomain topos. In the tensor category setting, if you were more restrictive, you could correspondingly say say that a localic morphism to $\mathcal{C}$ is an algebra object $A$ in $\mathcal{C}$. This should correspond to a functor $Mod(A) \to C$ as in Brandenburg's Corollary 5.3.3.
Now this is really what an affine morphism should be, and you might want to globalize it to a notion "2-affine morphism" (borrowing the terminology of Chirvasitu and Johnson-Freyd), but I don't know how...

Spencer Breiner (Oct 30 2022 at 23:14):

This work by Heunen & Barbosa might also be relevant:
Sheaf representation for monoidal categories
https://arxiv.org/abs/2106.08896

Fernando Yamauti (Oct 31 2022 at 00:39):

Thanks for all the replies! I will need more time to read everything carefully. But, for now, let me just comment on some things.

fosco said:

I'm aware of the existence of "linear" analogues for some constructions in CLTT, akin to the hyperdoctrine approach, https://arxiv.org/pdf/1405.0033.pdf (and by no means an expert of even a knowledgeable reader). The approach, however, is very logical in nature. I am interested in this question, indeed motivated by the same idea -more naive than yours- that Grothendieck categories behave like toposes in some respect.

The problem with that approach is that a geometric morphism doesn't usually preserve first-order logic (unless it's open), so the hyperdoctrine approach seems to go the wrong direction unless I'm overseeing something.

Regarding your naive interest, Grothendieck abelian categories are exactly the the category of sheaves over a small $Ab$-enriched site (where the definition of a topology should be the obvious one; it's explicitly written down here https://win.uantwerpen.be/~wlowen/pdf%20papers/JPAA-GabrielPopescu.pdf ).

Fernando Yamauti (Oct 31 2022 at 00:58):

Morgan Rogers (he/him) said:

There were some foundations laid by the Australian school for enriched topos theory, where one substitutes the category of presheaves on a locally small category for the V-enriched category of V-functors to V from a V-category (for V a suitable enriching category; I forget what requirements were imposed). As far as "translating topological concepts" goes, this direction doesn't provide a useful parallel to the classic theory, since we build the topos of sheaves as a subcategory of the category of presheaves on the poset of opens, which isn't enriched in anything interestingly linear. What John Baez suggests may be more interesting: if I have a ringed space $(X,\mathcal{O}_X)$ and a map $f:Y \to X$, the induced geometric morphisms will lift to an adjunction between the categories of modules for $\mathcal{O}_X$ and $f^*(\mathcal{O}_X)$. It makes sense to topological properties with the induced properties of this adjunction (although the properties are likely to be more coarse-grained along this translation than the original properties).

Well. The "linear category" in the non-linear case is the category of sup-lattices. Then sheaves are a kind of $Set$-valued modules over locales. I'm not sure, however, if modules valued in sup-lattices would recover the respective topoi (if modules valued in sup-lattices are exactly the internal sup-lattices that will be true, though).

Regarding the suggestion of @John Baez , that's exactly what's done in SGA4. The problem, however, is to work the other way around. Certainly, things like $QCoh (X, \mathscr{O}_X)$, $Ab (X)$ and $Perf (X, \mathscr{O}_X)$ should be instances of what I'm after.

Fernando Yamauti (Oct 31 2022 at 01:30):

Sam Speight said:

I have thought about this too. I think (but need to check the details) that the usual realizability topos construction makes sense using a linear partial combinatory algebra (in the sense of Abramsky) and you get something elementary-topos-like.

If that theory of linear partial combinatory algebras generalises further to triposes, then maybe the quantalic version would give an answer to the linear localic case in my question (although, sadly, in the wrong category).

Peter Arndt (Oct 31 2022 at 11:18):

Fernando Yamauti said:

Well. The "linear category" in the non-linear case is the category of sup-lattices. Then sheaves are a kind of $Set$-valued modules over locales. I'm not sure, however, if modules valued in sup-lattices would recover the respective topoi (if modules valued in sup-lattices are exactly the internal sup-lattices that will be true, though).

Pitts has worked out the sup-lattice enriched picture of the category of relations of a topos (and from that you can get the actual topos).

Fernando Yamauti (Nov 01 2022 at 03:01):

Peter Arndt said:

Hi Fernando, good question!

I'll just throw out a few thoughts. In the following "tensor category" is intended to mean abelian tensor category, or stable tensor category, possibly closed, depending on your context of choice.

Hi, Peter. Nice to see you around here!

I was only able to check the references more carefully now. Thanks for all the suggestions!

I was not aware of Schäppi's Tannakian reconstruction. I don't really know, however, how his reconstruction relates to all the other ones, though (as you probably now there exists a bunch: Lurie's, Iwanari's, Bondal-Orlov's, Balmer's, Rosenberg's, Gabriel's). The general pattern seems to be that abelian categories. recover 1-categorial stuff (schemes for instance), while, for $\infty$-categorial stuff (derived Artin stacks for instance), one needs the tensor structure. I will need more time to connect everything.

I've seen Brandenburg's thesis in the past, but I haven't really given much attention to it. Skimming though it now, I could see that he was trying to generalise things to arbitrary monads and, in that paper with Chirvasitu, they actually extended some things to Durov's generalised rings.

Regarding your suggestion to the localic case, I agree with you. If I was taking (locally) ringed topoi as the standard model of whatever a theory of linear topos should be, a spectrum constructions would likely be the linear analogous of a localic topoi. Still, I'm tempted towards a $-1$-truncated version ($0$-ring?) of $2$-rings in the sense of $2$-algebraic geometry, which seems to be a quantale, as a possible definition of a linear locale. I might be hallucinating, though.

Regarding the paper of Pitts, if I remember correctly he proves that the quantaloid of relations recovers the topoi. Simon Henry's thesis seems to generalise that further. He proved that the quantale of relations recovers any bounded topoi and the category of modules over the respective quantale is exactly the category of internal sup-lattices in the topoi. In that case, a topoi could in some sense be considered linear localic. Maybe I'm again being delusional.

Fernando Yamauti (Nov 01 2022 at 03:27):

Spencer Breiner said:

This work by Heunen & Barbosa might also be relevant:
Sheaf representation for monoidal categories
https://arxiv.org/abs/2106.08896

Thanks for the reference. Let me see if I've got it correctly. It seems that they are generalising by using the fact that $1$-topos can be characterised as a locally presentable categories where colimits of monos are Van-Kampen. Their case dealing with finite joins seems to generalise the case of coherent topos (or, equivalently, pretopos). The unique thing lacking would be an extension to the case where the lattice of central idempotents need not to be spatial. Is that spaciality really a big deal? Has anyone attempted to generalise that?