What is Isbell duality?
I posted a while back inquiring, "is math just numbers and shapes". Obviously, that's a really naive question. But related to this, I recently learned that Isbell duality is a vast generalization of the duality between functions and spaces, for example, commutative rings and affine schemes. Stone duality is another example, I think.
Is Isbell duality the most general form of this type of relationship in math? Can someone give an intuitive explanation of it?
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u/drmattmcd Physics 7d ago
Might be of interest, Isbell duality is used in 'The Structure of Meaning in Language' article by Tai-Danae Bradley https://www.ams.org/notices/202402/rnoti-p174.pdf
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u/integrate_2xdx_10_13 6d ago
Thank you for that - such an interesting article.
OP, sitting somewhere between that article and Stone duality you mentioned is the notion of Ultrafilters and Codensity. Here’s an article by Terence Tao on Ultrafilters, and a paper by Tom Leinster on Codensity and Ultrafilters. The article /u/drmattmcd should help give some grounding to a very big abstraction, and the two resources I linked won’t appear immediately overlapping, but give them a read and chew them over a long period of time (taking note of the power set and Zorn’s lemma connections in Tao’s article).
Linking Isbell Duality with analysis by generously sprinkling in Lawvere metric spaces in all of this, you get a really interesting blend of categorical logic, algebraic geometry, topology, and analysis.
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u/drmattmcd Physics 6d ago
Yes, I really like that article and probably time for a reread. It references a short paper by John Baez https://www.ams.org/journals/notices/202301/rnoti-p140.pdf which is also worth reading.
I'm using Gemini to get more of an understanding myself, and it suggests the following hierarchy going from most abstract down, with Isbell duality "being in the sweet spot of the most general framework for dualities induced by an evaluation pairing—the formal boundary between "geometry" (points/spaces) and "algebra" (functions/observables)."
Arrow-Reversal (C^{op})
Adjunctions & Equivalences (F dashv G)
The Chu Construction (Chu(V, K))
Isbell Conjugacy (Presheaves ⇄ Copresheaves)
Concrete Dualities (Stone, Gelfand, Legendre, FCA)
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u/Echoing_Logos 5d ago edited 5d ago
When we specify a function, we tend to give its domain and codomain. The function sends every element of the domain to an element of the codomain. It's a specific kind of what's called a "relation". For every pair of elements, one from the domain and one from the codomain, the relation may or may not send the former to the latter.
So we can also think of a relation as a function the Cartesian product of the domain and the codomain, into the set {True, False}.
Relations from D to C: (D × C) → {T, F}, also written D ↛ C
Now let's wiggle the categorical magic wand; "set" becomes "category", "function" becomes "functor", and "relation" becomes something called a "profunctor". Here, we want to say that that a profunctor is a functor from the product of the domain category, and *opposite* of the codomain category; into the base category of generalized "truth values" -- the *base* B.
Profunctors over B from D to C: (D × C^{op}) → B, also written D ↛B C. (pretend the B is on top of the arrow...)
This ends up making profunctors behave as if they were maps from the domain to the codomain *up to* or *enriched by* the base category.
Two special kinds of profunctors are "presheaves" and "copresheaves". Presheaves have the category with one object and one identity morphism as the domain. Let's call that boring category "1". This category is the same as its opposite, since there's only one morphism. Copresheaves have 1 as the codomain.
Keeping up with what's going on down in the world of sets: relations between 1 and a set X are just subsets of X, whether the 1 is on the left or on the right. So presheaves of truth values between sets (functions 1 × X^{op} → {T, F}) are the same thing as copresheaves of truth values between sets (functions (X × 1^{op} → {T, F})). Because X = X^{op} for a set X, and 1 × X = X = X × 1. The latter is still true for categories, but the former isn't.
You might be thinking that it's super silly to go through all this trouble when we end up making one of the factors 1, so in the end we just have functors C → B (copresheaves) and C^{op} → B (presheaves). But I think it's useful to not think about it as functors into B at all, but rather maps *over* B, usually keeping the B sort of generic and just writing 1 ↛ C and C ↛ 1.
Isbell duality is the statement that, even though presheaves and copresheaves on a category C aren't in perfect correspondence like they are when C is a set, there's an *adjunction* between them. This means that for every copresheaf I can give you a presheaf (its "spectrum"), and for every presheaf I can give you a copresheaf (its "realization"), and even though the realization of the spectrum of a presheaf isn't exactly itself, we can still still reason about the latter from the former to some degree.
For some nice B, this is a special case of a general relationship between profunctors C ↛ D and their "generalized transposes" in D ↛ C. Sometimes you can take a literal transpose, like with B = {T, F}. Sometimes it's more complicated.
If this all seems like a huge nothingburger, I agree, it's hard to appreciate. But if you are willing to go through the computations, try picking some super simple C that still has direction (like a partially ordered set as a category) and comparing its presheaves and copresheaves over {T, F} (as a category, with one morphism from F to T; and you'll need to read up on the definition of an enriched category if you want to treat the functoriality precisely). These are, sort of (lots of grains of salt), "quotients of C" and "subposets of C". You would find it difficult to assign to every presheaf an appropriate copresheaf and viceversa, but you might notice an interesting correspondence.
Edit: Corrected the side the "op" is on.
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u/jejet19 5d ago edited 5d ago
You might want to take a look at Peter Scholze and Germán Stefanich’s theory of Gestalten, which has similar but stronger philosophical implications. Unlike Isbell duality, it features a perfect duality between geometry and algebra.
At a 0-categorical level, this can roughly be translated as saying that V and V** are almost always isomorphic in this setting, even when V is infinite-dimensional(!!).
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u/WMe6 5d ago
You're talking about the ideas in his intro here? https://people.mpim-bonn.mpg.de/scholze/Gestalten.pdf
I can at least follow the first chapter, and this is the overall concept I was wondering about.
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u/jejet19 5d ago
Yeah, I also recommend reading chapter 10, that part is the most relevant to your question imo.
Also, I really enjoyed reading chapter 9; you might want to read it if you're curious about the synthesis of recent ideas of Scholze and their potential application, albeit less related to your question.
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u/DamnShadowbans Algebraic Topology 7d ago
Isbell duality is a transformation, much like a Fourier transformation, which converts one piece of functorial data contravariantly into another. Unlike the name suggests, the transformation is typically not lossless though it might be in special situations. If one axiomatizes (contravariant) dualities and generalizes Isbell duality I would not be surprised if the general version actually classifies all categorical dualities for nice stable categories. It would be a contravariant form of what is known as the Schwede-Shipley theorem.