@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.
@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
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
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
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.
