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nLab > Latest Changes: orbit category

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2013

    added to orbit category a remark on what the name refers to (since I saw sonebody wondering about that)

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2013

    Since Mackey functor is only a stub, I added a ’Guide’ in the references

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 1st 2013

    Some small changes at orbit category (including making OrG\mathrm{Or}\, G into OrG\operatorname{Or}G throughout).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 1st 2013
    • (edited Nov 1st 2013)

    The difference between the results of \mathrm{}\backslash, and \operatorname{} is invisible to my eyes.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 1st 2013
    • (edited Nov 4th 2013)

    The space induced by the backslash-comma combination looked too big to me. But never mind.

    • CommentRowNumber6.
    • CommentAuthoradeelkh
    • CommentTimeNov 4th 2013
    • (edited Nov 4th 2013)

    There is also another construction which goes by the name of “orbit category”: when T:𝒜→𝒜T : \mathcal{A} \to \mathcal{A} is an additive automorphism, the orbit category 𝒜/T\mathcal{A}/T is the one with the same objects as 𝒜\mathcal{A}, and in which morphisms are given by 𝒜/T(X,Y)=⨁ i∈Z𝒜(X,T i(Y))\mathcal{A}/T(X, Y) = \bigoplus_{i \in \mathbf{Z}} \mathcal{A}(X, T^i(Y)). This has appeared under the name “orbit category” in papers of Keller, Tabuada, and others. Perhaps we should mention this somewhere on the page?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2013

    If you have time, please do!

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2020
    • (edited Oct 24th 2020)

    added graphics showing the orbit categories of the first five cyclic groups (here)

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to the original:

    diff, v26, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to:

    diff, v27, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    maded more explicit that and where Dwyer-Kan gave the general definition:

    diff, v27, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2021

    added the remark (here) that, at least for discrete groups, the category of GG-sets is the free coproduct completion of the orbit category.

    diff, v32, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2021
    • (edited Dec 17th 2021)

    added some lines (here) on surjective group homomorphisms G^↠G\widehat {G} \twoheadrightarrow G inducing reflective subcategory inclusions of orbits GOrbt↪G^OrbtG Orbt \hookrightarrow \widehat{G} Orbt.

    diff, v33, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2021
    • (edited Dec 27th 2021)

    I have added a section (here) on tom Dieck’s “fundamental category” of a GG-space, for the moment mainly to record the observation that this is the Grothendieck construction on the system of fundamental groupoids of fixed loci.

    For a moment I was wondering if this note would become the first mentioning of this evident observation, but then I saw that it has been mentioned before, albeit just recently:

    diff, v35, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2021
    • (edited Dec 28th 2021)

    I have now also added (here) a section on tom Dieck’s “equivariant universal cover” together with the observation, analogous to the above, that – now under the ∞\infty-Grothendieck construction – this is equivalent to the 1-truncation unit of the equivariant homotopy type of a GG-space.

    (The note still needs some polishing, will first bring in some tikz-typesetting now…)

    diff, v36, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2021

    Further in this vein, tom Dieck’s “component category” (I.10.6 in Transformation Groups) is the Grothendieck construction on the connected component functor GOrbt op→π 0eshX (−)SetG Orbt^{op} \xrightarrow{ \pi_0 \esh X^{(-)} } Set.

    In summary, what section I.10 in tom Dieck 1987 is secretly describing is the image under the ∞\infty-Grothendieck construction of (the first few stages of) the Postnikov tower of the ∞\infty-presheaf on GOrbtG Orbt which corresponds to the given GG-space.

    Hm, is there a conveniently citable reference relating the ∞\infty-Grothendieck construction to Postnikov towers in presheaf ∞\infty-toposes?

1 to 16 of 16

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