Is MSO[1/2] a quotient of MU[1/2] (killing off the odd generators)?

@SaalHardali this seems pretty atypical to me, if X is pointed then this is asking when all modules over 𝕊[ΩX] are generated by actions of ΩX on 𝕊. in "the lingo" i think this is asking for 𝕊[ΩX] -> 𝕊 to be proper? that seems hard unless the multiplication on ΩX is genuinely nilpotent

Here's a thing which seems pretty basic, so someone might have written it up. Fix a simplicial set $S$. Consider a category $C$ whose objects are $(E, i_0\colon \Delta^0\to E, i_1\colon S\to E, \pi\colon E\to \Delta^1)$ so that $i_0$ and $i_1$ are base changes of $\pi$ along the vertex inclusions $\{0\}\subset \Delta^1$ and $\{1\}\subset \Delta^1$. That is, $C$ is a category of "correspondences $\Delta^0\Rightarrow S$".

There is an adjoint pair relating $C$ to $sSet_{/S}$. The left adjoint sends $(X\to S)$ to $E=(\Delta^0*X)\cup_X S$. The right adjoint sends $E$ to the basechange of $E_{i_0/}\to E$ along $i_1$.

Is there a model structure on $C$ which makes this a Quillen equivalence with the covariant model structure?

Basically, it's a baby form of straightening/unstraightening for left fibrations, without going all the way to enriched categories.

Might need $S$ to be a quasicategory for it to work this way.

@CharlesRezk This is Theorem 6.4 in my paper on Joyal's cylinder conjecture arxiv.org/abs/1911.02631

Following Joyal, I denote your category C by Cyl(Delta[0],S). This category admits a model structure created from Joyal's model structure for quasi-categories by the forgetful functor to sSet.

Theorem 6.4 shows that the adjunction you describe is a Quillen equivalence between this model structure on Cyl(Delta[0],S) and the covariant model structure on sSet/S, for any simplicial set S.

There's another description of the model structure on Cyl(Delta[0],S) (as the "ambivariant model structure") that makes this Quillen equivalence easy to prove. The hard thing to prove, which is equivalent to Joyal's cylinder conjecture, is that these two model structures coincide.

@AlexanderCampbell Oh cool, thanks. I dimly remember Joyal's cylinder conjecture, but didn't understand it when I heard it.

@PedroTamaroff what I would hope for is that $H_*(G)$ with the Pontryagin product should be Koszul dual to $H^*(BG)$ with the usual product, and the resulting equivalence of module categories should reflect "spaces with an action of $G$" and "spaces over $BG$, with the passage from the former to the latter even implemented by a bar construction. Can you check the case of $SU(2)$?

@pupshaw Ah, that makes a lot of sense. Thanks!

I'll try to look into that.