You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
finegeometer (Jul 18 2023 at 17:05):I've come across the concept of "noncommutative geometry", but haven't been able to understand it.
But I know that the internal language of the classifying topos of commutative rings, or of commutative local rings, supports the important axioms of synthetic differential geometry. So does the internal language of the classifying topos of noncommutative local rings support some sort of "synthetic noncommutative geometry"?
If so, is there a paper explaining noncommutative geometry from this perspective?
John Baez (Jul 20 2023 at 08:45):I haven't seen such a paper, and I doubt noncommutative local rings would be very helpful in noncommutative geometry. (I could be wrong.)
John Baez (Jul 20 2023 at 08:55):Connes' work on noncommutative geometry is based on C*-algebras, and this is very popular, but if you want an approach that more closely resembles traditional algebraic geometry you might try Rosenberg's work on noncommutative algebraic geometry.
Xuanrui Qi (Jul 20 2023 at 14:28):Some non-commutative geometers once told me that homotopy theory is in some sense a type of NCG. While I don't really understand this statement, I suppose it's related to how spectra work. Anyways there definitely is hope!
Xuanrui Qi (Jul 20 2023 at 14:36):I don't think local rings are really used in NCG, as the basic machinery used for NCG is vastly different from that of regular AG. Namely, if you try to work with non-commutative rings the textbook way, construct its spectrum, equip it with the Zariski topology, etc., you soon run into unfixable problems, and that is futile. Instead you must work with things at a more abstract level: e.g. instead of constructing directly differential forms, etc., we define homological-algebraically the Hochschild algebra of an associative algebra, etc.
Xuanrui Qi (Jul 20 2023 at 14:36):These notes by Kaledin are very good, most of the little NCG I know I've learned here: http://imperium.lenin.ru/~kaledin/math/tokyo/final.pdf
Xuanrui Qi (Jul 20 2023 at 14:38):But then you see NCG is already "more synthetic" than regular AG! Instead of doing this analytically we're forced to define them by categorical constructions
John Baez (Jul 20 2023 at 14:47):Those notes by Kaledin looks nice! I've been attracted to Kontsevich and Soibelman's paper Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. Part I, and I wonder if Kaledin develops or explains any of those ideas. I don't know if Part 2 was ever written.