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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 3rd 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWatching a part of Lurie's Arizona State [lecture](https://vimeo.com/120708135), 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 $\mathbf{B} G$. 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?

   Watching a part of Lurie’s 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 $\mathbf{B} G$.

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 3rd 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItex> But that was just what we are being told in the slick homotopy type theoretic account of [[representation theory]], right? Work in the context $\mathbf{B} G$. Yes, but now at a further nested level: First there is the trivial $G$-action on the linear category $\mathcal{C}$ (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 $$ \array{ \mathcal{C} &\longrightarrow& (\mathbf{B}G) \times\mathcal{C} \\ && \downarrow \\ && \mathbf{B}G } \,. $$ The homotopy invariants of the action are equivalently the sections of this fibration, so here these are equivalently the maps $$ \rho \colon \mathbf{B}G \longrightarrow \mathcal{C} \,. $$ This is generally the expression for $G$-homotopy invariants of $G$ acting trivially on any $\mathcal{C}$. One then observes that this is of course itself a $G$ action, now on whatever $V \in \mathcal{C}$ the point of $\mathbf{B}G$ 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 haven't thought about.

   But that was just what we are being told in the slick homotopy type theoretic account of representation theory, right? Work in the context $\mathbf{B} G$.

   Yes, but now at a further nested level:

   First there is the trivial $G$-action on the linear category $\mathcal{C}$ (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

   $$\array{ \mathcal{C} &\longrightarrow& (\mathbf{B}G) \times\mathcal{C} \\ && \downarrow \\ && \mathbf{B}G } \,.$$

   The homotopy invariants of the action are equivalently the sections of this fibration, so here these are equivalently the maps

   $$\rho \colon \mathbf{B}G \longrightarrow \mathcal{C} \,.$$

   This is generally the expression for $G$-homotopy invariants of $G$ acting trivially on any $\mathcal{C}$. One then observes that this is of course itself a $G$ action, now on whatever $V \in \mathcal{C}$ the point of $\mathbf{B}G$ 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 haven’t thought about.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 4th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIt 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.

   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.