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Stream: learning: questions

Topic: (Isbell?) duality between algebra and geometry

Paolo Perrone (Jun 11 2025 at 08:07):

On the nLab, dualities of the type "Algebras = Spaces^op" are referred to as Isbell duality (see for example the link at the beginning of Gelfand duality).
I probably lack the necessary background, but I don't quite see why Isbell duality (presheaves, copresheaves, etc) has to do with, say, Gelfand duality, or the duality (by definition) between rings and affine schemes.
Could someone explain to me the idea?

Ivan Di Liberti (Jun 11 2025 at 08:12):

This is something that someone should have the energy to clarify on the nlab. I will write you something privately later hopefully. There is a relation.

David Corfield (Jun 11 2025 at 08:22):

There's some further material at [[space and quantity]] on Lawvere's outlook, but not directly addressed to your question.

fosco (Jun 11 2025 at 08:24):

Gel'fand duality and the duality between rings and affine schemes are basically the same thing, right?

Paolo Perrone (Jun 11 2025 at 08:25):

fosco said:

Gel'fand duality and the duality between rings and affine schemes are basically the same thing, right?

Yes, or at least they are "clearly" instances of the same abstract phenomenon. What I don't understand is whether that abstract phenomenon is Isbell duality, and in that case, why. (The nLab seems to say that, but it's unclear.)

Paolo Perrone (Jun 11 2025 at 08:27):

I also wonder, in case that the abstract phenomenon isn't quite Isbell duality, what it is then. Or does one just say, for example, "Gelfand-type duality", in analogy with Stone-type duality?

fosco (Jun 11 2025 at 08:27):

I'll give you two references but leave the final word to Ivan, who has thought about this more than me

John Baez (Jun 11 2025 at 08:28):

Paolo Perrone said:

On the nLab, dualities of the type "Algebras = Spaces^op" are referred to as Isbell duality (see for example the link at the beginning of Gelfand duality).
I probably lack the necessary background, but I don't quite see why Isbell duality (presheaves, copresheaves, etc) has to do with, say, Gelfand duality, or the duality (by definition) between rings and affine schemes.
Could someone explain to me the idea?

I think it's just not true. A bunch of good category theory students at U. Edinburgh and I tried for a couple of days to make it make sense somehow, and failed. I will delete that sentence unless someone clarifies it.

fosco (Jun 11 2025 at 08:28):

Porst, H.-E., Tholen, W.: Concrete dualities. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, pp. 111–136. Heldermann, Berlin (1991)

Kurz, A., Velebil, J. Enriched Logical Connections. Appl Categor Struct 21, 349–377 (2013). https://doi.org/10.1007/s10485-011-9267-y

Paolo Perrone (Jun 11 2025 at 08:29):

Thank you, let me take a look.

fosco (Jun 11 2025 at 08:30):

All the statements in Kurz paper are "formal" in the sense that they hold in a good Yoneda structure

fosco (Jun 11 2025 at 08:31):

(give or take; one needs assumptions here and there)
This is (was?) a work in progress of mine years ago that then I dropped because, speaking with Tholen, it emerged this problem is tied to the preliminary task to formalize topological 1-cells in a YS

fosco (Jun 11 2025 at 08:31):

which I will gladly leave to a hard-working student, wink wink

John Onstead (Jun 11 2025 at 09:16):

Interestingly, back in January, I brought up a similar point in my thread here. I don't know if I can help much, but I can try to summarize what I learned there. The consensus seemed to be that while it might be hard to make certain dualities (like Gelfand duality) a direct special case of Isbell duality, they both are special cases of a more underlying concept: contravariant dualities. This is basically an adjunction (or potentially an equivalence) between a category and the opposite of some other category generated by a "dualizing object", an object which can be thought to "live in both categories". In the case of Gelfand duality the complex numbers play this role, and in the case of Isbell duality the hom functor $\mathrm{Hom}(-,-)$ plays this role.

John Onstead (Jun 11 2025 at 09:16):

As for how this connects to spaces, if you think of the objects of $D$ as spaces and $C$ as algebras, and the dualizing object is $d$, you can think of the functor $F: D^{op} \to C$ as sending a space to the algebra of $d$-valued functions defined on that space. It's kind of like $\mathbb{R}$ in the category of manifolds, where there's a way to find the algebra of real-valued functions on any manifold. Anyways, I hope this helps somewhat!

Adrian Clough (Jun 11 2025 at 20:30):

Isbell duality and the algebra-geometry duality are not the same, but somewhat related. 

Everything I say will be for unenriched presheaves (i.e., valued in sets or homotopy types, whose $(\infty-)$ category is denoted by $\mathcal{S}$). Let $C$ be some small $(\infty-)$ category.  

First, let’s understand Isbell duality in more conceptual terms. In the adjunction $[C,\mathcal{S}]^{\mathrm{op}} \leftrightarrows [C^{\mathrm{op}},\mathcal{S}]$ the top arrow is is the coYoneda extension of the inclusion $C \hookrightarrow [C^{\mathrm{op}},\mathcal{S}]$, and the bottom arrow is similarly the Yoneda extension of $[C,\mathcal{S}]^{\mathrm{op}} \hookleftarrow C$. 

Now, specialise to $C = \mathbf{CRing}^{\mathrm{fp}}$ being the $(\infty-)$ category of finitely generated, commutative ordinary / derived / spectral rings (the finite generatedness condition should become clearer below, but one a priori reason for it is that the category of rings is not small), then the full subcategory of $C^{\mathrm{op}}$ spanned by $\mathbb{A}^n$ $(n \geq 0)$, denoted by $A$, is the Lawvere theory of commutative rings. The obvious embedding $A \hookrightarrow [C^{\mathrm{op}},\mathcal{S}]$ preserves products so that we again by a variant of the coYoneda extension obtain an adjunction $\mathbf{CRing}^{\mathrm{op}} \leftrightarrows [C^{\mathrm{op}},\mathcal{S}]$. The left adjoint is not fully faithful, but this can be addressed by replacing $[C^{\mathrm{op}},\mathcal{S}]$ with the category of locally ringed topological spaces / locales / toposes / $\infty$-toposes, which is typically regarded as the “usual” algebra-geometry duality. 

I think the above relationship reflects a deeper interplay between limits and colimits in geometry, which I will attempt to make more precise in the following two paragraphs (largely following DAG V):

A $(\infty-)$ topos is often presented as the category of sheaves on an $(\infty-)$ category $G$ of “geometric” local objects. (Modulo issues of hyperdescent in the homotopical setting) the $(\infty-)$ category of sheaves on $G$ is obtained by taking the presheaf topos on $G$ and then forcing some colimits we really liked (those reflecting reasonable ways to glue together objects in $G$) to remain colimits. This is reminiscent of presenting a $(\infty-)$ topos in terms on generators and relations. But this is not quite right. When constructing objects using generators and relations one should construct the free object of ones generators and then take relations. To construct the free $\infty$-topos one first takes the finite completion of $G$, and only then the cocompletion of this finite completion. Thus, one is led to the question of whether there were any limits in $G$ that we really liked and want to keep…

Now, one can show that product preserving functors out of $A$ are the same as functors out of the $(\infty-)$ category of smooth affine / derived / spectral schemes which preserve pullbacks along smooth morphisms, and these are definitely pullbacks which are already good and we want to keep. Moreover, we see that the universal finitely complete $(\infty-)$ category receiving a functor out of the $(\infty-)$ category of smooth affine ordinary / derived / spectral schemes which preserves pullbacks along smooth morphisms is thus the opposite of the category of finitely generated affine ordinary / derived / spectral schemes. Moreover the $(\infty-)$ topos $\mathcal{E}$ of Zariski sheaves on $(\mathbf{CRing}^{\mathrm{fg}})$ is then precisely the classifying $(\infty-)$ topos for locally ringed $(\infty-)$ toposes. From now on I will work with $\infty$-categories in order not to have to keep track of decreasing truncatedness. For any finitely generated affine scheme $X$ the $\infty$-category of Zariski local open embeddings $Y \to X$ is an $\infty$-topos, and clearly admits geometric morphism to $\mathcal{E}$ exhibiting its local ring structure. The $\infty$-category of locally ringed $\infty$-toposes is complete and the spectrum functor is simply given by extending the functor from $\mathbf{Aff}^{\mathrm{fp}} = (\mathbf{CRing}^{\mathrm{fp}})^{\mathrm{op}}  \to \mathbf{LocRingTop}$ to $\mathrm{Pro}(\mathbf{Aff}^{\mathrm{fp}}) = \mathbf{CRing}^{\mathrm{op}}  \to \mathbf{LocRingTop}$, where $\mathbf{LocRingTop}$ denotes the $\infty$-category of locally ringed $\infty$-toposes.

I'm not entirely sure how to go from this back to Isbell duality, but it feels to me like somehow it is simply reflecting that both limits and colimits are useful for building new geometric objects out of simpler ones, and the attendant universal properties of the categories constructed this way is naturally related via the Isbell duality. In Convenient Categories of Smooth Spaces (Baez, Hoffnung) a bunch of notions of generalised smooth manifolds are discussed, some using presheaves, other coprosheaves, and some both.