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I am a newcomer to non-commutative geometry and really enjoying the literature. I have read several of the classic papers, assisted by the excellent lecture notes Homological Methods in Non-commutative Geometry by Kaledin. I have also read the papers by Reyes, Heunen, and van den Berg on the obstructions to extending the Zariski site to $\mathsf{Ring}$ in various ways. Now I see on the nLab page for noncommutative algebraic geometry there is the added statement:
This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).
Are there a few names people can drop for me, or even specific papers, which are most significant for following up on this hint? Before coming across this page I worked out some things for sheaf theory on bicategories (e.g. of noncommutative algebras), but feel I must be on well-trodden ground.
This passage means to be pointing to 2-algebraic geometry (pointer added now).
But I am unsure how non-commutative such 2-algebraic geometry really is.
I think the royal road to non-commutative geometry via higher categories is the result by Schwede-Shipley (here) that stable $\infty$-categories that have a small set of generators are equivalently the categories of modules over $A_\infty$-ring(oid)s, hence of “quasicoherent sheaves over ’spaces’ including but being more general than affine non-commutative spaces”.
I think this provides also the transparent picture behind Kontsevich-style NCG, where non-commutative spaces are modeled as formal duals of $A_\infty$-categories: These $A_\infty$-categories in turn are but models for stable $\infty$-categories as in Schwede-Shipley’s theorem.
Thank you Urs, I appreciate the link. Actually, I find my thinking is getting split between Connes and Kontsevich for NCG, since I am essentially tackling problems in non-commutative analysis, but in the style of Kashiwara, Sato, etc. for algebraic analysis. In particular, I am developing a theory of sheaf cohomology in Clifford analysis which will support a kind of Riemann-Hilbert correspondence (absence of holomorphy nowithstanding). So my goal is to build a de Rham functor for certain (higher) categories of non-commutative algebras. Since Clifford algebras form a 2-cat, this is why I was drawn to sheaves on bicategories; in fact your Cafe post on categorified Clifford algebras and Shulman/Baez’s later commentary really encouraged me to think along these lines.
The split in my thinking happens because while Clifford algebras form a von Neumann algebra (essential ingredient of spectral triples), it is not clear what geometric object should be associated with it, as a spin manifold is in the classical case; on the other hand, differential envelopes of Clifford algebras are nice to work with and one can define a Grothendieck topology on the category of such superalgebras. But again, it is not clear where the underlying geometry should come from! So that’s one thing I’m struggling with…in a Clifford-analytic R-H correspondence, what should play the role of variety/manifold?
It seems like that Shwede-Shipley paper is working a full level of abstraction above what I’m saying: given a dg algebra $A$, the derived category $\mathcal{D}(A)$ of (left) dg $A$-modules is a triangulated category with a homotopy theory, although I am not sure it will always be stable in their sense. Of course I mean this only in a positive light, this seems like a powerful paper.
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