8:33 AM
dunno. but I should add that I think that also my second formulation of the thing is wrong, and it should be $S[CP^\infty] \to bu$ (not ku)

9:04 AM
@TomBachmann @DenisNardin as far as I've ever thought about it, the derived cotensor is just the (homotopy) limit of the obvious cosimplicial object. I don't really think this type of stuff is written down carefully anywhere though. There's a paper by someone whose name I can't remember that talks about $\infty$-categorical Hopf-algebras and stuff, but only in the Cartesian monoidal case.
But, yeah, I think "Hopf-algebroid" sort of stops meaning anything in the derived setting, in a sense, because it's just a cosimplicial object, or rather, it probably should be.
Well, maybe not "stops meaning anything" because there's still the Segal-type condition. But anyway, nobody should listen to me.

1 hour later…
10:26 AM
@TomBachmann yes, I believe that's correct.

10:49 AM
@JonathanBeardsley Perhaps Torii's paper? arxiv.org/abs/1612.03265

3 hours later…
1:31 PM
@JonathanBeardsley doesn't your thom objects are cotorsors paper basically answer my question affirmatively? :D
@TylerLawson Thanks!

1:48 PM
@TomBachmann haha possibly.
i'm decently sleep deprived, and that paper is kind of a mess in retrospect

2 hours later…
3:37 PM
@SaalHardali i believe this map to be an n-equivalence when n! is inverted
re: S[CP^infty], i'm not aware of a description beyond "the thing freely generated by line bundles under formal sums", where "freely generated" (= Q = Loops^infty S^infty) hides quite a lot of homotopy theory

3:52 PM
to alleviate some of my own confusion: there are Susp^infty CP^infty, Q CP^infty = Loops^infty Susp^infty CP^infty, and P(Susp^infty CP^infty) = Sym^* Susp^infty CP^infty = Susp^infty Q CP^infty. the middle one is the one "freely generated" under formal sums. one version of the splitting principle says that it splits off a BU, and one also finds that its complement is a space with finite homotopy groups
the stable map P(CP^infty) --> Susp^infty BU is indeed a (2n)-equivalence after inverting n!, but that's not saying very much: it's hard to attach any collection of even cells in [0, 2n] if n! is inverted
i hope that's more sound

4 hours later…
8:00 PM
one further moment of lucidity: the rational homology of CP^∞ shows that there is a family of torsion-free classes in the even degrees of the stable homotopy of CP^∞, and the nontrivial statement about n! that i was struggling to remember is: the usual homology generator in degree 2j is a j! division of the hurewicz image of the torsion-free stable homotopy class in the same degree. i guess that doesn't sound so tightly related to what you're asking about, but i also guess it's good to know

8:20 PM
For a geometric interpretation, do Pontryagin-Thom constructions give anything for $Map(M, \Sigma^n CP^\infty)$? Naively, it seems like we could try to lift to $CP^\infty \times S^n$ to get a relationship between the mapping space and the space of line bundles on $M$ together with a framing of a codimension $n$ submanifold.