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added a graphics (here) illustarting the map from a) unstable equivariant Cohomotopy of representation spheres to b) the Burnside ring to c) the representation ring
Now that I have done it I see that I should have made the labels “$G$” be explicitly “$\mathbb{Z}_4$”. Will fix later.
added this historical pointer (dug out by David C. in another thread) where the term “representation group” is used to refer to a group equipped with a permutation representation:
Sorry, wrong spot. Actually Burnside means by “permutation group” what we now call a “$G$-set”. Am moving the reference.
I wonder if anything interesting happens if we shift from $G$ acting on sets to it acting on the finite $\infty$-groupoids treated in
What plays the role of finite vector spaces there are the slices $\mathcal{F}/\alpha$, where $\mathcal{F}$ is the $(\infty, 1)$-category of finite $\infty$-groupoids. These slices are objects of $lin$, whose morphisms are derived from spans of finite $\infty$-groupoids, $\alpha \leftarrow \mu \to \beta$. There’s a ’cardinality’ map from $lin$ to $FinVect$.
Wild speculation: perhaps virtual permutation reps in this setting could close the gap with virtual linear reps.
Speaking of wild speculation, I keep wondering if it’s just a coincidence that the 2-site for global equivariant homotopy theory – namely the global orbit category – is that of connected finite 1-groupoids (connected finite homotopy 1-types). This suggests that the $\infty$-site for globally “higher equivariant” homotopy theory is that of all finite $\infty$-groupoids.
Moreover, the site for $G$-equivariant homotopy theory is the faithful slice of this site over the delooping groupoid of $G$, which suggests that the faithful slices $(\mathcal{F}/\alpha)^{faith}$ of finite $\infty$-groupoids over deloopings of finite $\infty$-groups $\alpha = B \mathcal{G}$ are the sites for higher $\mathcal{G}$-equivariant homotopy theory.
These ingredients are not-so-vaguely reminiscent of Goodwillie calculus, which works over the opposite and pointed site of finite $\infty$-groupoids (e.g. here).
Probably Goodwillie calculus is also a good thing to keep in mind when thinking of homotopy linear algebra.
Homotopy linear algebra certainly fits very neatly with polynomial and analytic functors in the non-Goodwillie sense, as in
Is there an $\mathbb{F}_1$ view on finite $\infty$-groupoids? I see in the notes by Barwick, $\mathsf{Perf}_{\mathbb{C}}$ is
the $(\infty,1)$-category of perfect $H \mathbb{C}$-modules – that is, of $H \mathbb{C}$-modules with finitely many homotopy groups, all of which are finite-dimensional.
$H \mathbb{F}_1$-modules would give us what? Chains of sets?