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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $\mathrm{Or}\, G$ into $\operatorname{Or}G$ 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.
• CommentTimeNov 4th 2013
• (edited Nov 4th 2013)

There is also another construction which goes by the name of “orbit category”: when $T : \mathcal{A} \to \mathcal{A}$ is an additive automorphism, the orbit category $\mathcal{A}/T$ is the one with the same objects as $\mathcal{A}$, and in which morphisms are given by $\mathcal{A}/T(X, Y) = \bigoplus_{i \in \mathbf{Z}} \mathcal{A}(X, T^i(Y))$. 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)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMar 17th 2021

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMar 17th 2021

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMar 17th 2021

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

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 12th 2021

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

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

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

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

I have added a section (here) on tom Dieck’s “fundamental category” of a $G$-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:

• 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 $G$-space.

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

• 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 $G Orbt^{op} \xrightarrow{ \pi_0 \esh X^{(-)} } Set$.

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 $G Orbt$ which corresponds to the given $G$-space.

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