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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 20th 2023
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   Author: David_Corfield
   Format: MarkdownItexNow we have linear HoTT, we might extend to a 'homotopy type [[linear representation]] theory'. Is there anything novel here? Would something like character theory be conveniently expressed by linear HoTT?

   Now we have linear HoTT, we might extend to a ’homotopy type linear representation theory’. Is there anything novel here? Would something like character theory be conveniently expressed by linear HoTT?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 25th 2023
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   Author: Urs
   Format: MarkdownItexFirst of all we can record that $R$-linear $\infty$-representations are subsumed as those $\infty$-actions in $R$-linear tangent $\infty$-toposes, in the sense of the table, whose incarnation as a fibration over $\mathbf{B}G$ is a $\natural$-counit -- as in Exp. 2.22 of *[[schreiber:Entanglement of Sections|Entanglement of Sections]]*.

   First of all we can record that $R$-linear $\infty$-representations are subsumed as those $\infty$-actions in $R$-linear tangent $\infty$-toposes, in the sense of the table, whose incarnation as a fibration over $\mathbf{B}G$ is a $\natural$-counit – as in Exp. 2.22 of Entanglement of Sections.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 25th 2023
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   Author: David_Corfield
   Format: MarkdownItexI guess a good path to character theory in this way is through the homotopic reformulation in, say, pp. 3-5 of Raksit's [Characters in global equivariant homotopy theory](https://ncatlab.org/nlab/show/transchromatic+character#Raksit15).

   I guess a good path to character theory in this way is through the homotopic reformulation in, say, pp. 3-5 of Raksit’s Characters in global equivariant homotopy theory.