Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections

Question

I would be very grateful for any references I might be led to, from a categorical point of view for the functors:

• $\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which leads to $\mathcal{O}(\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-))$,
• $\textsf{Spec}_{\mathscr{G}\textrm{elfand}}(-)$, related to $\mathcal{C}(-,\mathbb{C})$, which leads to $\mathcal{C}(\textsf{Spec}_{\mathscr{G}\textrm{elfand}}(-),\mathbb{C})$

These references would deal with "good" equivalences of categories, at the levels of algebraic geometry and operator algebras, which would be raised in a purely unifying categorical construction.

Many thanks in advance for any knowledge of books and articles written with this in mind.

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In particular, dear @DmitriPavlov, thank you very much for your articles, your high categorical point of view and your answer, 01/13/2022, 11:51 p.m., to the internal question which prompted my question from personal research related to the relationship between Gelfand spectrum and Zariski spectrum, to which your answer to question : https://mathoverflow.net/a/413776/502369 begins to reply.

You wrote this answer very brilliantly in terms of category theory and localization for this question of the relationship between the Gelfand spectrum and the Zariski spectrum, and the corresponding locales, topologies and sheaves, which therefore coincide in a more general setting .

I am also very sensitive to your arguments, Dear @DmitriPavlov, which appear in your article on Gelfand duality, stating the equivalences of categories with Hyperstonean spaces, Von Neumann algebras and three other categories, as well as in your text on category theory.

My first general question about possible references, therefore, continues, Dear @DmitriPavlov, with the following two questions, if you allow me to address you directly, but also to anyone who would be kind enough to answer them:

First :

I have not yet found the statement of your answer by localization and construction of the sheaf, in your own publications and available texts, or elsewhere.

I found links with localization and construction of the sheaf, in particular in the Remark 12.4.19 by Henning Krause: Homological Theory of Representations, Cambridge University Press, and the Ziegler spectrum and the Zarisky spectrum by Henning Krause: The spectrum of a module category, Memoirs of the AMS, volume 149 , Number 707, which both send to Melvin Hochster: Prime ideal structure in commutative rings, and to Mike Perst: Remarks on elementary duality, 1993. But the link with C*-algebras, spatial locales and the categorification of the subject of your answer in very few words which are very enlightening, leads me, if you allow me, to ask you for references for this construction, which would possibly be your own publications themselves, where I would not have found yet this statement, or in some other papers and references?

Moreover, would this luminous construction come from bibliographical references that could even be cited?

It would indeed be very valuable for me to go deeper into the study of this question which arises for me in categorical terms. Thank you all in advance for your enlightning.

Secondly :

It seems that it was asked to you earlier on the same MathOverFlow page: https://mathoverflow.net/a/413776/502369, but I'll rephrase it for all, because you cite, Dear @DmitriPavlov, many application cases, including Von Neumann algebras and Hyperstonean spaces in a categorification of Measure Theory, to which you devote an equally brilliant article.

Thus: What could be the minimum assumptions about the category for this construction of localization and structural sheaf to remain rigorously perfect? But maybe this question is contained in the references that could be indicated, about the previous question.

Thank you in advance for your attention to my question and your interest in answering it.

Best regards

Answer 1

I have not yet found the statement of your answer by localization and construction of the sheaf, in your own publications and available texts, or elsewhere.

To the best of my knowledge, there is nothing published on this topic yet. I once wrote some very rough unfinished notes on how to treat the algebraic, differential, and holomorphic cases simultaneously: http://dmitripavlov.org/notes/cart.pdf, which do discuss the construction of the spectrum and the structure sheaf, but do not go much further. I am not sure how general this approach is, for example, I have not tried to use it with continuous and/or measurable maps, or connect it to C-algebras or von Neumann algebras, although such a connection might exist. (I expect the construction of the structure sheaf to go through just fine for commutative C-algebras and von Neumann algebras.)

What could be the minimum assumptions about the category for this construction of localization and structural sheaf to remain rigorously perfect?

That's a good question, and I am pretty sure the answer has not been written up (yet?). In principle, it should not be too difficult to look at the construction of the Zariski spectrum and sheaf and abstract away the relevant properties that make the construction work. I would not be surprised if Malcev/protomodular/homological/semiabelian categories make an appearance at this point (see the book of Borceux and Bourn), since these are designed as abstractions of the usual categories of groups/modules/rings from algebra. But the precise conditions have not been pinned down yet, I think.