Timeline for Is a representation sphere dualizable inside naive G-spectra?
Current License: CC BY-SA 3.0
8 events
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| Jul 15, 2012 at 4:47 | comment | added | John Klein | Just a remark: in the naive category there's another kind of duality which uses $S[G]=$ suspension spectrum of $G_+$ instead of the sphere to dualize with. In this variant $S^0[G]$ (which represents $G_+$ as a stable object) is always self-dual. This is since $F(S^0[G],S^0[G])^G = S^0[G]$, where I've taken equivariant function spectra on the left of the equals sign. | |
| Jul 14, 2012 at 16:15 | comment | added | user25092 | @TL: Is that the same as spectra over BG? In this category the analog of $G_+$ has a dual? | |
| Jul 14, 2012 at 15:48 | vote | accept | user25092 | ||
| Jul 13, 2012 at 19:07 | comment | added | Tyler Lawson | @Neil: My apologies. My comment was based on misinterpreting the question: I was considering a different weak equivalence structure on naive G-spectra (using underlying weak equivalences, rather than equivariant ones). | |
| Jul 13, 2012 at 17:10 | comment | added | Peter May | Naive Spanier-Whitehead works fine equivariantly and gives the full subcategory of finite G-CW spectra (up to equivalence) in any other good construction. Neil's answer is fine, with R the regular representation, but Gaunce's result I cited is valid for orbits of compact Lie groups, not just finite ones. The pre-EKMM Lewis-May foundations work for compact Lie groups and any universe, the suspension G-spectra of spheres with trivial $G$-action are cofibrant there, and space level maps can be used via adjunction. Passing to EKMM doesn't clarify things here. | |
| Jul 13, 2012 at 16:53 | comment | added | Neil Strickland | @Tyler: what foundations do you have in mind? For finite groups I think we can do something very similar to Boardman's original setup, where a naive $G$-spectrum consists of $G$-CW complexes with equivariant structure maps $\Sigma X_n\to X_{n+1}$. A genuine $G$-spectrum has $X_n$ defined only when $|G|$ divides $n$, and structure maps $S^{\mathbb{R}[G]}\wedge X_n\to X_{n+|G|}$. My argument is valid in that context. With EKMM foundations you would need cofibrant replacement wrt a model structure based on the trivial universe, which would not make a difference at the end of the day. | |
| Jul 13, 2012 at 13:19 | comment | added | Tyler Lawson | Neil, in the naive category don't we have to take a cofibrant replacement of $S^n$ to calculate homotopy classes of maps? | |
| Jul 13, 2012 at 10:30 | history | answered | Neil Strickland | CC BY-SA 3.0 |