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added to orbit category a remark on what the name refers to (since I saw sonebody wondering about that)
Since Mackey functor is only a stub, I added a ’Guide’ in the references
Some small changes at orbit category (including making $\mathrm{Or}\, G$ into $\operatorname{Or}G$ throughout).
The difference between the results of \mathrm{}\backslash, and \operatorname{} is invisible to my eyes.
The space induced by the backslash-comma combination looked too big to me. But never mind.
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?
If you have time, please do!
added pointer to the original:
added pointer to:
maded more explicit that and where Dwyer-Kan gave the general definition:
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:
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…)
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?
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