I recently came across several stack exchange questions concerning the research program of Caramello, its impact, and its relation to other research programs. The quality of the answers to these questions is mixed, but the questions themselves seem like good springboards for discussion, and for that a forum such as this (which Prof Caramello has easy access to!) seems like a good platform for discussion. I'll summarise and number the questions for reference in the discussion.
- This question quotes Laurent Lafforgue (without naming him for some reason) comparing Caramello's program to the Langland's program. Their specific question is: "I wonder if there were any papers confirming the overall sentiment of [Laurent Lafforgue], i.e. papers where Caramello's theory is applied to Langlands. Were there at least some long-standing problems that historically were not believed to have any serious connection with logic or topoi, and were solved using Caramello's techniques?"
Ignoring their dubious claim that,
"The entire mathematical world has produced few, if any, correspondences which unify lots of non-trivial mathematics as elegantly as the Langlands program does," it would be nice to know what the current state of progress of linking these two programs is. - This question, by far the highest profile, is asking for insight into the quote "one can generate a huge number of new results in any mathematical field without any creative effort," and whether it is an exaggeration. The answers are divided based on their interpretation of the quote, but Caramello's book (published since the question was asked back in 2010) certainly contains numerous illustrations of the kinds of result that can be extracted from the topos-theoretic bridge framework. As someone who works with Grothendieck toposes myself under Caramello's supervision, I know that "without any creative effort" doesn't mean "without any effort": extracting the results in a form that is comprehensible takes significant work, but the reason that effort is not 'creative' is because the toposes act as a guide, avoiding the need for one to rely on great leaps of inspiration to progress. I hope Prof Caramello will have more to say on the subject.
- This question asks how Olivia's program relates to Jacob Lurie's work on higher topos theory and higher categories, specifically asking, "Has there been any work connecting these schools of thought?"
Most of the results in Lurie's Higher Topos Theory are extensions to oo-toposes of results for Grothendieck (1-)toposes, so there is some immediate relation between the background on which the programs rest, but it would be interesting to get some insight into how Prof Caramello views the relation between their research directions.
[I made a common topic to discuss these questions in parallel, so to make sure the discussion is coherent, please follow the numbering system!]
I may eventually report any interesting answers and discussion provided here as answers to the original questions.
@morgan-rogers Thanks for posing these questions! Let me start by addressing the second and third; for the first, Laurent Lafforgue (@l-lafforgue) might want to reply himself.
Concerning 2, in saying that results could be obtained “without any creative effort” through the ‘bridge’ technique, I wanted to emphasize that, once an equivalence between two different presentations of the same topos is established, the calculation of how invariants express in terms of the two presentations is essentially canonical and can be automatized in many cases (descriptions of classes of invariants for which such calculations can be performed in a semi-automatic way are provided for instance in this paper and this other one). Of course, the complexity of such calculations can greatly vary depending on the given invariant (many topos-theoretic invariants can be readily calculated, while cohomological invariants tend to be very difficult to compute even in special cases); still, conceptually, such 'unravelings' are just computations, which, at least in principle - as well as in practice for many invariants! - do not require a creative effort to be performed. On the other hand, identifying the ‘right’ toposes, and the most appropriate invariants on them, which incarnate the “essence” of a certain mathematical situation or capture some specific feature of a given problem, is in general quite subtle and may require an imaginative effort. Here are a couple of relevant excerpts from my habilitation thesis, where these issues are thoroughly discussed:
We have already hinted above to the fact that there is an strong element of automatism implicit in the ‘bridge’ technique. In fact, in order to obtain insights on the Morita-equivalence under consideration, in many cases one can just readily apply to it general characterizations connecting properties of sites and topos-theoretic invariants. Still, the results generated in this way are in general non-trivial; in some cases they can be rather ‘weird’ according to the usual mathematical standards (although they might still be quite deep) but, with a careful choice of Morita equivalences and invariants, one can easily get interesting and natural mathematical results.
…
Starting from a topos or a Morita-equivalence, the calculation or expression in terms of sites or theories of presentation of topos-theoretic invariants is often technically non-trivial but feasible. On the contrary, trying to go in the other direction from a very sophisticated concrete mathematical result to a Morita-equivalence which could generate it is in general very difficult, if not impossible. In other, more metaphorical, words, this methodology generates a ‘rain’ of results falling in a territory surrounding a given problem whose essential aspects have been encoded by means of suitable topos-theoretic invariants. It is difficult to predict exactly where the single drops will fall, but, as the rain will eventually cover more and more of the wet space, so the application of this methodology is liable to bring a lot of concrete insights on aspects related to the original problem which could eventually lead to its solution.
As to question n. 3, the technique of toposes as ‘bridges’ is based on the possibility of presenting a topos in multiple ways (for instance, by using sites, geometric theories, topological or localic groupoids, etc.) and of investigating topos-theoretic invariants from the points of view of such different presentations. Going from ordinary toposes to higher toposes, many classical topos-theoretic tools and techniques can be naturally extended (as shown in Lurie’s book “Higher Topos Theory”), but the theory of ‘presentations’ for higher toposes is more complex than its classical counterpart: for example, higher sites are no longer sufficient to capture all left exact localizations of a higher topos. Also, from a logical point of view, the correct analogue of geometric logic has not yet been identified (most research efforts have concentrated so far on identifying the higher analogue of the internal logic of a topos, rather than on generalizing geometric logic to the higher setting). I certainly expect the technique of toposes as ‘bridges’ to be fully applicable in the higher setting, once an effective presentation theory for higher toposes (using ‘computationally effective’ generalizations of higher sites, or higher geometric theories, or any other suitable mathematical objects susceptible of presenting higher toposes) will be developed.
