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Watching a part of Lurieβs Arizona State lecture, I realise I had never thought before that a representation of a group G, can be thought as a G-invariant of a linear category under the trivial action of G.
But that was just what we are being told in the slick homotopy type theoretic account of representation theory, right? Work in the context BG.
So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of G and that of the representations of the dual (for some kinds of group), is what in the HoTT picture? Is it better to be in a linear HoTT setting?
But that was just what we are being told in the slick homotopy type theoretic account of representation theory, right? Work in the context BG.
Yes, but now at a further nested level:
First there is the trivial G-action on the linear category π (take, without restriction of generality, its core maximal groupoid if we want to stay within homotopy theory proper) and that is exhibited by the split homotopy fiber sequence
πβΆ(BG)ΓπβBG.The homotopy invariants of the action are equivalently the sections of this fibration, so here these are equivalently the maps
Ο:BGβΆπ.This is generally the expression for G-homotopy invariants of G acting trivially on any π. One then observes that this is of course itself a G action, now on whatever Vβπ the point of BG goes to.
So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of G and that of the representations of the dual (for some kinds of group), is what in the HoTT picture?
This I havenβt thought about.
It sounds like something we ought to want to do. Right at the end, Lurie suggests that the self-duality of Laurent series as shown by the Weil symbol gives rise to some part of geometric Langlands.
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