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1. • CommentRowNumber101.
   • CommentAuthorUrs
   • CommentTimeOct 21st 2019
   • (edited Oct 21st 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a graphics ([here](https://ncatlab.org/nlab/show/permutation+representation#ComparisonMapFromBurnsideRingToRepresentationRing)) 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. <a href="https://ncatlab.org/nlab/revision/diff/permutation+representation/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/permutation+representation/41">v41</a>, <a href="https://ncatlab.org/nlab/show/permutation+representation">current</a>

   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.

   diff, v41, current

2. • CommentRowNumber102.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2020
   • (edited Jun 3rd 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this historical pointer (dug out by David C. [in](https://nforum.ncatlab.org/discussion/7282/burnside-ring/?Focus=85074#Comment_85074) 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](https://doi.org/10.1112/plms/s1-34.1.159)) <a href="https://ncatlab.org/nlab/revision/diff/permutation+representation/43">diff</a>, <a href="https://ncatlab.org/nlab/revision/permutation+representation/43">v43</a>, <a href="https://ncatlab.org/nlab/show/permutation+representation">current</a>

   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

3. • CommentRowNumber103.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry, wrong spot. Actually Burnside means by "permutation group" what we now call a "$G$-set". Am moving the reference.

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

4. • CommentRowNumber104.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 15th 2021
   • (edited Jun 15th 2021)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI wonder if anything interesting happens if we shift from $G$ acting on sets to it acting on the finite $\infty$-groupoids treated in * [[Imma Gálvez-Carrillo]], [[Joachim Kock]], and [[Andrew Tonks]]. _Homotopy linear algebra_. Proc. Roy. Soc. Edinburgh Sect. A, 148(2):293–325, 2018 ([arXiv:1602.05082](https://arxiv.org/abs/1602.05082)) 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.

   I wonder if anything interesting happens if we shift from $G$ acting on sets to it acting on the finite $\infty$-groupoids treated in

   • Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks. Homotopy linear algebra. Proc. Roy. Soc. Edinburgh Sect. A, 148(2):293–325, 2018 (arXiv:1602.05082)

   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.

5. • CommentRowNumber105.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexSpeaking 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](https://ncatlab.org/nlab/show/model+structure+for+excisive+functors#TheUnderlyingCategories)). Probably Goodwillie calculus is also a good thing to keep in mind when thinking of homotopy linear algebra.

   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.

6. • CommentRowNumber106.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 15th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHomotopy linear algebra certainly fits very neatly with polynomial and analytic functors in the non-Goodwillie sense, as in * [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469)). Is there an $\mathbb{F}_1$ view on finite $\infty$-groupoids? I see in the [notes](https://www.msri.org/workshops/689/schedules/18234/documents/2047/assets/20469) 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?

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

   • David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, (arXiv:1712.06469).

   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?