@AaronMazel-Gee It turns out I the answer to my question wasn't that helpful for what I wanted to do... There are many nonequivalent so-called q-oscillator algebras that only look alike in a very specific representation and of which only one seems to carry a Hopf algebra structure. It is unclear if I can induce the Hopf structure on q-deformed Lie algebra by constructing them via a Schwinger-like construction.
Random question: if X is a pointed space, ΣX has a natural coassociative comonoid structure. Is it known what is the totalization of its cobar construction? I would assume it is equivalent to X, at least in nice cases, but I'm not sure how to prove it.
@DenisNardin could this maybe follow from the geometric construction of the Eilenberg-Moore spectral sequence?
Anyone know how well (or poorly) a tensor triangulated functor translates to a functor of Balmer spectra? I feel like this is one of Balmer's papers, but I'm not sure which one.
@DenisNardin oh actually i think my comment about the EMSS is wrong
I guess I feel like the totalization of that cobar construction (with coefficients on both side in a contractible space) should be equivalent to loops on \Sigma X
Presumably you're using the $\Sigma X$-comodule structure on $\ast$ given by the point $\ast\hookrightarrow \Sigma X$, so the cobar construction should be computing the pullback of the cospan $\ast\to \Sigma X\leftarrow \ast$, I think...
@JonathanBeardsley This doesn't seem right: if you take the geometric realization of ΩX you get the connected component of the basepoint of X, not ΣΩX. I think I can prove that the totalization is equal to X in the case X=S^0. I don't understand why you expect a pullback?
i thought i recalled someone saying that the Loops/B thing is very far from transferable to Susp/Tot, but i'm struggling to find where i read this discussion
I don't know if I even understand that statement (specifically the word "transferrable") but I sort of understood these things to be basically the same in the stable setting, but not elsewhere?
Are you saying that there is some functor Tot which is somehow... adjoint to suspension?
I thought that was the topic of discussion: Loops as targeting E_1 spaces has an adjoint B, given by the realization of the bar construction, that factors through connected spaces & on those the pair for an equivalence. Susp targets co-E_1-spaces, and you can form a functor back by taking the totalization of the cobar complex (which I just abbreviated to Tot, since I thought the cosimplicial object in play was clear), and my recollection was that this doesn't work very well at all
I'm not considering anything not already on the table, just saying I have this memory
maybe the trouble is that spaces are already coassociative coalgebras in a way commuting with whatever extra structure you get from suspension, and that somehow dampens the new information you thought you were picking up via eckmann-hilton. Single-cell spaces are probably not a good test case; something like the cone on twice unstable eta would do better
Maybe the confusion is with respect to which monoidal structure on spaces you have in mind. I think Denis means the wedge and Jon means the Cartesian product. Maybe I am wrong though.
Are we talking about the coalgebra structure coming from the pinch map on S^1 here? That's definitely not what I'm talking about. I'm talking about the coalgebra structure $\Sigma X\to \Sigma X\times \Sigma X$
