Timeline for (∞, 1)-categorical description of equivariant homotopy theory
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 29, 2009 at 3:40 | comment | added | Reid Barton | To clarify for other non-experts like myself: the confusion between (1) and (2) probably arises from the fact that they are the homotopy theories of two model categories with the same underlying category, but different weak equivalences (the weak equivalences between Ω-spectra are in (1) levelwise equivalences of underlying spaces but in (2) levelwise equivariant equivalences). | |
| Oct 29, 2009 at 1:52 | comment | added | Reid Barton | Right, now I can clearly see that the guess in my first comment was wrong: the stabilization of G-spaces is (2) as Charles says. I do think there may some confusion about the term "naive G-spectrum" in the literature; for instance in Calculus III Goodwillie uses the phrase "what are sometimes called naive G-(pre)spectra" to describe his spectra with G-action, which are definitely (1). | |
| Oct 29, 2009 at 0:31 | comment | added | Reid Barton | Oh, hmm. I have been (naively?) using "naive G-spectrum" to refer to (1), but it seems that May uses it to mean (2). And now I would like a (∞,1)-categorical description of (3)... Maybe another question is in order... | |
| Oct 29, 2009 at 0:26 | comment | added | Charles Rezk | (1) is supposed to read "Ordinary spectra equipped ...". | |
| Oct 29, 2009 at 0:16 | comment | added | Charles Rezk | This always confuses me, since I know three kinds of G-spectra, I can never remember which one is supposed to be naive: (1) Ordinary equipped with a G-action. These are presheaves of spectra over BG. (2) G-spectra indexed on the universe built from the trivial representation: these are spectrum valued presheaves on O(G). (3) G-spectra indexed on a complete universe: these are something else ... | |
| Oct 29, 2009 at 0:02 | comment | added | Reid Barton | From the description of G-spaces as functors from O(G) to Spaces, it follows by generalities about presentable (∞,1)-categories that its stabilization is the category of functors from O(G) to Spectra. This seems unlikely to be the same as the category of naive G-spectra (which I believe is functors from BG to Spectra) so I guess it's probably the category of true G-spectra...? | |
| Oct 28, 2009 at 23:59 | vote | accept | Reid Barton | ||
| Oct 28, 2009 at 22:58 | history | answered | Mike Shulman | CC BY-SA 2.5 |