https://calendar.math.cas.cz/content/nonabelian-%C4%8Dech-cocycles-noncommutative-geometry
Prague-Hradec Kralove Seminar “Cohomology in algebra, geometry, physics and statistics” at Czech Academy of Sciences
Wednesday, 19. April 2023 - 13:30 to 14:30 online
Nonabelian Čech cocycles in noncommutative geometry
Gluing of sheaves, bundles and similar objects is the subject of descent theory. In this talk we shall focus on a kind of noncommutative geometry where spaces are represented by Abelian categories which locally look like (= are glued from) full categories of one sided modules over noncommutative “coordinate” rings. Locality may be in the sense of localizations, but also more generally in the sense of faithfully flat covers presented by corings. Principal bundles will have such global spaces with categorified action analogous to a principal action of a structure group and the cover of the total space will have to be in an appropriate category respecting the action. A very general (nonaffine) principality or Galois kind of condition will be formulated in terms of categorical adjunctions. Then the cocycles will be introduced in several levels of generality; in the main special case related to structure groups coming from Hopf algebras, and locally cleft extensions, we use comonads in a pair of new auxiliary bicategories found in 2019. Comparing different cocycles is first done for the case involving the same cover of the base. Comparison for different base covers, studied in a project with M. Stojić, uses refinements which involve additional data in general. Passing to the colimit defining cohomology classes requires some set-theoretical care regarding that a priori such refinements form a proper class.
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