3:00 AM
In algebra, the category of modules over a Hopf algebra $H$ has this funny property that it's monoidal, because of the coalgebra structure of $H$. Does the same thing happen if I look at modules in spectra over $\Sigma^\infty_+\Omega X$, for some pointed space $X$? That is, of course, only an $E_1$-algebra, so we would not expect its module category to be monoidal, but maybe we can do the same thing we do in the discrete case?
Where, here, module means "left module," rather than "bimodule," to be clear.
I believe, if I recall correctly, the trick in the discrete case is to look at $H\overset{\Delta}\to H\otimes H\to Aut(M)\otimes Aut(N)\to Aut(M\otimes N)$ or something like this?

3:22 AM
@JonathanBeardsley inverse.com/article/…

1 hour later…
4:33 AM
Hm, well it seems to me like $4,000 could actually be a lot of money to some people, especially if it was easy to steal. I don't think thieves generally feel at a loss if the monetary value is small relative to the sentimental value. 7 hours later… 11:24 AM Does anyone have a reference for Künneth formulas in equivariant (borel type) homology/cohomology? 12:14 PM "Sheaf Theory" by Bredon has a section on Künneth formula for Borel-Moore homology probably also something in "Cohomology of Sheaves" by Iversen, but don't have that one in front of me 12:38 PM Thanks! I'll go to the library tomorrow 12:51 PM @AaronMazel-Gee odd, i didn't see one mention of this on twitter 1:25 PM @Greebo I don't think Borel-Moore homology is the same thing as Borel equivariant homology. Compare ncatlab.org/nlab/show/Borel-Moore+homology and ncatlab.org/nlab/show/Borel+equivariant+cohomology Or do those books also address the equivariant case? 1:56 PM I think you're right, I just saw borel homology and have been reading about BM-homology a lot I see Bredon has written a book on equivariant cohomology as well though "Equivariant cohomology theories" Geoghegan's "Topological Methods in Group Theory" deals with BM-homology and at least mentions equivariant homology, but not in the same chapter and does not reference between them, so probably no connection yeah @Greebo Bredon cohomology is not quite the same as Borel cohomology, unfortunately (there are links, but I don't think I believe in a Künneth formula for Bredon cohomology) man why must there be so many homologies/cohomologies, such confuse @greebo: Because there are many spaces anyway I reserved also the equivariant cohomology book let's see 7 hours later… 8:47 PM @ThomasRot You will have tough luck getting a product formula in Borel cohomology. Let$X$and$Y$be two spaces with an action by a compact Lie group$G$. Then you are trying to calculate$\text{Tor}_{C_* G}(C_* X, C_* Y)$, using the notation/definitions from Gugenheim-May "differential torsion products", though you can just define this using a bar construction. They explain that things are best behaved when you equip one of your terms with a "Kunneth resolution", essentially meaning 1) a resolution built out of copies of$C_* G$, such that 2) the homology is flat over the ground ring. In this case, you have a spectral sequence$\text{Tor}_{HG}(HX, HY)$to your desired output groups. To obtain something like this, you probably want to put some assumptions on your group (torsion-free homology) and the homology of your orbits$BH$, and then replace$X$with$X \times EG$. Another good reference that contains many of the results in GM is "6 model structures on dg-algebras", by Barthel-May-Riehl. A great test case for all of this is to let$G = SU(2)$and make sure all of your orbits are of type$*, S^2$, or$SU(2)$. Then all cohomology rings involved are polynomial, and things are calculable. I would write$C_* X$in some Morse-Bott type way, built out of 'critical orbits' (or a G-CW decomposition), the same for$Y$, and try to understand the resulting way to build up$X \times Y$out of products of orbits. You get a spectral sequence calculating the equivariant homology coming from the equivariant homologies of products of orbits, which you can determine more easily using the tor-spectral sequence above, especially under the nice assumptions we make on the homologies. These are basically the same story. I am not sure if there is anything useful in there, but I suggest tom Dieck's book transformation groups. There might be something around the section on localization theorems, in the case that your groups are sufficiently nice, the intuition being that we mainly need to figure out how to glue together the reducible subspaces. Addendum to "nice test case" above: Try$S^3$as the one-point compactification of$\mathfrak{su}(2)$, so a spherical suspension of the$S^3$action on$S^2\$, for one of your spaces. I think it is a fun one.

2 hours later…
11:21 PM
Here are three things which all sound right but can't all be true:
1. a spin structure on a vector bundle V -> X is equivalent data to a trivialization of V over the 2-skeleton of X
2. S^2 has a spin structure
3. the hairy ball theorem
actually, never mind. #1 is of course the wrong one, but I was reading some hypotheses incorrectly. sorry about that