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    • CommentRowNumber101.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2019
    • (edited Oct 21st 2019)

    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 “GG” be explicitly “ 4\mathbb{Z}_4”. Will fix later.

    diff, v41, current

    • CommentRowNumber102.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020
    • (edited Jun 3rd 2020)

    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:

    • William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

    diff, v43, current

    • CommentRowNumber103.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020

    Sorry, wrong spot. Actually Burnside means by “permutation group” what we now call a “GG-set”. Am moving the reference.

    • CommentRowNumber104.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 15th 2021
    • (edited Jun 15th 2021)

    I wonder if anything interesting happens if we shift from GG 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 /α\mathcal{F}/\alpha, where \mathcal{F} is the (,1)(\infty, 1)-category of finite \infty-groupoids. These slices are objects of linlin, whose morphisms are derived from spans of finite \infty-groupoids, αμβ\alpha \leftarrow \mu \to \beta. There’s a ’cardinality’ map from linlin to FinVectFinVect.

    Wild speculation: perhaps virtual permutation reps in this setting could close the gap with virtual linear reps.

    • CommentRowNumber105.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2021

    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 GG-equivariant homotopy theory is the faithful slice of this site over the delooping groupoid of GG, which suggests that the faithful slices (/α) faith(\mathcal{F}/\alpha)^{faith} of finite \infty-groupoids over deloopings of finite \infty-groups α=B𝒢\alpha = B \mathcal{G} are the sites for higher 𝒢\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.

    • CommentRowNumber106.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 15th 2021

    Homotopy linear algebra certainly fits very neatly with polynomial and analytic functors in the non-Goodwillie sense, as in

    Is there an 𝔽 1\mathbb{F}_1 view on finite \infty-groupoids? I see in the notes by Barwick, Perf \mathsf{Perf}_{\mathbb{C}} is

    the (,1)(\infty,1)-category of perfect HH \mathbb{C}-modules – that is, of HH \mathbb{C}-modules with finitely many homotopy groups, all of which are finite-dimensional.

    H𝔽 1H \mathbb{F}_1-modules would give us what? Chains of sets?

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