In the following recent talk Laurent Lafforgue gives a beautiful overview of the role of categories in Grothendieck's work and a presentation of some recent developments involving toposes:
Thanks, it was a great talk.
i was wondering about the categorification of Galois theory, namely the equivalence between the covers of an object S and the category of G-set. In fact, i know it's possible to obtain a von Neumann algebra from an action of group. That's why i'm asking for a categorification of some operator algebras, if this makes sense.
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