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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?
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.
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.
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