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Leopold Schlicht (Jul 29 2021 at 08:51):As described in Part 0 of Lambek & Scott, many duality theorems between algebra and geometry, such as Stone duality, Pontrjagin duality and Gelfand duality, arise as follows: there's a special mathematical entity E that can be regarded both as an object of an algebraic category Alg and as an object of a geometric category Geo (so it's both a space and an algebraic structure). (In the context of Stone duality E is Z/(2), which is both a ring and a topological space; in the context of Pontrjagin duality E is R/Z, which is both an abelian group and a topological abelian group.) Then one considers two functors: to each space X in Geo one assigns the object Hom(X,E) of Alg (its algebraic structure is inherited by that of E) and, similarly, to each "algebra" A one assigns the object Hom(A,E) of Geo. Hopefully, these two contravariant functors Alg -> Geo and Geo -> Alg constitute a pair of adjoint functors and restrict to an equivalence between two interesting subcategories of Alg and Geo^op.
It's interesting that this pattern shows up again and again, the nLab has a whole page about it: https://ncatlab.org/nlab/show/duality+between+algebra+and+geometry. My question is: can we make the analogy between all these dualities precise? Is there a hidden categorical principle behind all these dualities that generalizes the phenomenon described above?
I picked up the book Stone spaces by Johnstone and saw that chapter VI contains an approach to generalizing these kinds of dualities via ind- and pro-completions. But this doesn't look like what I'm searching for (I want to see a generalization of the process of picking the special object E and then considering Hom(-,E)).
Also, what I find a bit mysterious: how to pick the right E?
Amar Hadzihasanovic (Jul 29 2021 at 10:44):You may be interested in Yoshihiro Maruyama's PhD thesis.
Amar Hadzihasanovic (Jul 29 2021 at 10:45):It's mainly about this question.
Simon Burton (Jul 29 2021 at 13:30):At least in some cases, E is the tensor unit object.
Leopold Schlicht (Jul 29 2021 at 15:19):@Amar Hadzihasanovic Thanks! So it's a current research topic and there's no established approach for a general duality "framework"?
Nathanael Arkor (Jul 29 2021 at 15:21):Duality theorems have been well-studied in category theory. Take a look at some of the references at the bottom of the nLab page for dualising object, for instance.
Jon Awbrey (Jul 29 2021 at 15:45):Amar Hadzihasanovic said:
You may be interested in Yoshihiro Maruyama's PhD thesis.
This is probably far afield from what's afoot here, but since you linked to a very broad context, I'll go ahead and link to a point where one brand of duality has returned to my thoughts. This time it was a post on Gödel's Lost Letter, so now we both know who to blame.
Cf: Animated Logical Graphs • 45
Regards,
Jon
John van de Wetering (Jul 30 2021 at 12:47):In effectus theory you have a class of such duality-like statements that Bart Jacobs refers to as "state-state triangles". https://arxiv.org/abs/1703.09034
Here the dualizing object is the underlying set of scalars of your theory (for instance {0,1} or [0,1]).
John van de Wetering (Jul 30 2021 at 12:47):Relatedly, you also get a Stone duality between directed-complete effect monoids and Stonean spaces with a designated clopen subspace. Here the dualizing object is [0,1]. See https://arxiv.org/abs/1912.10040
Leopold Schlicht (Aug 06 2021 at 16:19):In the context of Makkai duality, E is Set, which is both an ultracategory and a pretopos. Just to add one more example. :grinning_face_with_smiling_eyes:
Leopold Schlicht (Aug 26 2021 at 00:25):Update: I just found out that in his 1987 paper "Stone Duality for First Order Logic", Makkai proposes the exact same idea of making the analogy between all dualities precise: this is discussed in chapter 8, especially in the paragraph that begins with: "Although the idea behind the general concept of a Stone adjunction is very simple, we do not have a satisfactory general formulation of it."
Leopold Schlicht (Mar 19 2022 at 18:23):The paper Concrete Dualities by Porst and Tholen looks very related to my original question.
Leopold Schlicht (Mar 19 2022 at 19:44):I just realized that Nathanael Arkor already mentioned it implicitly.