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I created a new entry on Laurent Lafforgue so as to point out his views on the importance of Olivia Caramello’s work on topos theory. I also updated her site.
The Caramello page has a problem with the links: it’s two times a link to the second paper.
it seems rather misleading to have this thread here titled “Langlands program”, could you change it to something that would lend itself more to satisfactory search results.
The link has been fixed and I changed the title here.
Thanks!
By the way, it may happen that a Fields medalist has a wrong opinion. Not that I think at all it is the case here, but sometimes I wish the mathematical community would rely less on proof-of-relevance-by-authority and more on technical expertise. For instance if I were Olivia I’d rejoice if somebody wrote on the Lab: “O’s work is great, because it achieves this and that, which is useful because of such and such.” Contrast that with “O’s work is great because the community said that X is great and X said that O’s work is great.” While in this case it’s nice for O (and well-deserved) in general it adds to a vicious circle of mathematical judgement by hearsay.
Fields medallists often have wrong opinions. (I have person experience of this with two of them.) The point is rather that someone who is in the know about Langlands is thinking it worth learning some topos theory in order to try to generalise those results hoping to prove something of interest to Langlandsists (or whatever they are called).;-)
I was also hoping to stimulate someone to explain how her work fitted in with other nLab topics.
Olivia Caramello is working on a book ‘Lattices of theories’. I added a link to the first chapter. It works up to her toposes as bridges technique which uses Morita equivalence of toposes to transport theorems from one field of mathematics to another. Strangely enough the notion of Morita equivalence of toposes does not exists in the nlab yet.
That’s because they’re more commonly known as simply “equivalences of toposes”.
But there doesn’t seem to be a page for that either, nor a link from Morita equivalence to that concept.
An equivalence of toposes is (essentially) the same as an equivalence of the underlying categories, though it might be defined differently.
One doesn’t really speak of Morita-equivalent toposes so much as Morita-equivalent theories – two theories (of the same type, e.g. cartesian, regular, coherent, geometric…) are defined to be Morita-equivalent if their classifying toposes are equivalent. This is a slightly weaker notion than requiring the syntactic categories to be equivalent: algebraic theories can have the same classifying topos without having equivalent syntactic categories (= Lawvere theories) – the most famous example of this being theories of (left) modules for rings.
Yes, but we still need the references to connect this to Morita equivalence of, say, topological groups. All this is conveniently collected in the chapter by Olivia. It just needs to be put there. I could put in some links if noone else does.
I’d say go right ahead.
I see Lafforgue here is wondering whether Olivia’s Morita topos techniques could be extended to the Langlands Program.
Isn’t it likely that geometric Langlands will be easier to crack than ordinary Langlands? I see they’re looking to investigate the Morita nature of the Fourier-Mukai transform, at the Fourier-Mukai, 34 years on symposium.
If it could be made to work, we might expect S-duality to be a Morita-equivalence phenomenon too.
Isn’t Mukai transform a kind of derived Morita by the very definition ? I mean, in the cases when it is a (derived) equivalence at all (some people nowdays look at somewhat more general derived push-pull transforms as Fourier-Mukai transforms, than the case when it is a derived equivalence). Correspondences/bimodules in geometric cases are more or less sheaves over … On the other hand, any equivalence of bounded derived categories of coherent sheaves on smooth projective varieties is given via push-pull transform. Theorem of Orlov in
This is now considered one of the starting points in that subject. So it is puzzling to me, what secret zou are pointing in “If it could be made to work”…I think the conference states “relation to Morita theory” just in the sense that some other derived equivalences of non-geometric origin and in abstract homotopy theoretic context might benefit from the geometric insight in the case of Fourier-Mukai transform. The fact that the geometric case fits into Morita-kind phenomena is more or less triviality.
I agree, Zoran. The precise relation with derived Morita theory was explained by Toen in his legendary paper referenced at derived Morita equivalence.
As for the relation with the “conjectural interpretation of the Geometric Langlands correspondence” mentioned in the workshop description, I assume they are referring to Gaitsgory’s interpretation of one side of the correspondence as ind-coherent sheaves (there is a minimal page about this here).
The ’it’ was ambiguous, pointing back further.
I’m really wondering how deep Morita equivalence goes. Do we expect geometric Langlands to be Morita-like?
This arose in the context of wondering about the Morita-like nature of ’dualities’ in physics. I wrote a contribution to a philosophy journal edition on such dualities. I chose to write about mathematical duality, but am coming now to think that Morita equivalence would have more appropriate.
I’ve been adding to the section Duality in physics: Relation to Morita equivalence various sources, including the claim by Albert Schwartz:
I am convinced that the mathematical notion of Morita equivalence of associative algebras and its generalization for differential associative algebras should be regarded as the mathematical foundation of dualities in string/M-theory.
15 I am glad Adeel has joined the discussion, he is certainly more competent about derived equivalences than I am. David, thanks for digging out the citation from Schwarz (beware the English spelling of his name, however! https://www.math.ucdavis.edu/~schwarz ).
We already have an entry Albert Schwarz. I have fixed the links.
Regarding “topological open string T-duality”, I suppose this was meant to refer to topological T-duality. This is (as discussed there) not T-duality for topological strings, but is something like: that part of T-duality of type II strings which may be captured mathematically without reference to the metric. I have edited this accordingly at duality in physics.
David,
while I agree that Morita equivalence is a good analogy for duality in physics, and that it is an omission on our part, and maybe on my part specifically, not to have mentioned this earlier, I feel like the idea that duality in physics should generally be identified with Morita equivalence is, depending on how widely one intereprets “Morita equivalence”, either false or else true by definition.
For instance we had discussed those articles, whose pointer you may have more easily at hand than I have right now, which discuss two QFTs as being in duality explicitly if they have Lagrangians both of which are obtained from one single Lagrangian by integrating out either this or that Lagrange multiplier.
This is precisely an example of the concept of duality in physics as we have discussed it as recorded in the Lab entry. But this will hardly come down to an example of a Morita equivalence of algebras (of any sort) in all cases.
I still think the formalization as we have it in the entry (equivalence between QFTs presented by two different Lagrangians/action functionals) is absolutely accurate to the phenomenon at hand. If one regards the Lagrangian/action functional here as presenting the QFT that they give under quantization, then this is an instance of two presentations being regarded as equivalent if the structures they present are equivalent. As such this may be thought of as generalized Morita equivalence, I suppose.
I notice that even so, one says “Morita equivalence” instead of “Morita duality”, so the key point that “duality in physics” is not “duality in mathematics” remains true, to my mind. Even if, of course, there may be many instances in which an actual duality is implemented by a Morita equivalences. For the generality that is necessary for the concept “two Lagrangians giving equivalent QFTs” as discussed in the entry, such special cases seem to not be of general enough impact, it seems to me.
depending on how widely one intereprets “Morita equivalence”, either false or else true by definition.
So if one just takes “Morita equivalence” to be something like a homotopified equivalence relation, then it’s true by definition, but if one requires the equivalence to be mediated by equivalence of models/modules/representations, then there will be cases of equivalence in physics not covered?
I guess Mike is indicating something along the lines of the former here
well, it probably depends on how wide a notion of “presentation” one is willing to admit…
On the other hand, is this so generous an interpretation? If we have a case of a generalized equivalence relation, and let’s say it’s of the form of duality in physics, i.e., a 1-epimorphism such as
won’t I be able to classify it by a map to the object classifier
In which case, couldn’t I liken this map, which sends a Langrangian QFT to the collection of Lagrangian data which give rise to it, to a map sending a ring to its modules, or a theory to its models?
That’s probably not right, is it?
It’s the downwards arrow that’s supposed to resemble the formation of category of modules
Anyway, I’d like to know what distinguishes Morita situations.
Yes, as I said, the epi is exactly analogous to and as such both their Cech nerves may be thought of as equivalences between presentations induced by equivalences of what they present, and hence of generalized Morita equivalences. But with Morita equivalence interpreted in this generality, then it is true essentially by definition that duality in physics is Morita equivalence, with Morita equivalence interpreted in this generality, then there would be no need for Schwarz to say “I am convinced that..”, because of this nobody needs special convincing, it’s true by fiat.
I am not saying that this is bad, actually I think it’s a good point. But it doesn’t imply anything on the relation of duality in physics to Morita equivalence understood in more specific terms. So I am thinking one needs to be a bit careful here.
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