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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 ∞-groupoids treated in
What plays the role of finite vector spaces there are the slices ℱ/α, where ℱ is the (∞,1)-category of finite ∞-groupoids. These slices are objects of lin, whose morphisms are derived from spans of finite ∞-groupoids, α←μ→β. 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 ∞-site for globally “higher equivariant” homotopy theory is that of all finite ∞-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 (ℱ/α)faith of finite ∞-groupoids over deloopings of finite ∞-groups α=B𝒢 are the sites for higher 𝒢-equivariant homotopy theory.
These ingredients are not-so-vaguely reminiscent of Goodwillie calculus, which works over the opposite and pointed site of finite ∞-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 𝔽1 view on finite ∞-groupoids? I see in the notes by Barwick, Perfℂ is
the (∞,1)-category of perfect Hℂ-modules – that is, of Hℂ-modules with finitely many homotopy groups, all of which are finite-dimensional.
H𝔽1-modules would give us what? Chains of sets?
Added the second sentence here:
The image of the comparison morphism β=K(k[−]) (Def. \ref{ComparisonMapBurnsideRingRepresentationRing}) may be called the virtual linear permutation representations. Any virtual linear permutation representation is a formal difference, in the representation ring of G, of elements coming from permutation representations.
Added proof of an exercise in Serre’s book Linear Representations of Finite Groups:
(nonsurjectivity of the map from virtual permutation representations to virtual linear representations)
Suppose G is a finite group with a linear representation ρ such that:
Then the map β:A(G)→R(G) is not surjective.