Not that there's necessarily any reason to do so, but does anyone know if anyone ever studies Galois extensions of operads themselves?

Or, for instance, the "descent theory" for $E_n\to E_{n+1}$?

Which would require trying to look at $E_{n+1}\otimes_{E_n}E_{n+1}$?

Which, actually, I guess would just be $E_{n+1}\otimes E_1$

But so... still... formed as a bar construction that should maybe be some kind of cooperad, and we could ask about coalgebras over it??

And these would be descent data for that "extension"? And we could ask questions like "Given an $E_{n+1}$-algebra $A$, how many $E_n$-algebras are there whose induced $E_{n+1}$-algebra is equivalent to $A$?" And this should be controlled by some kind of "non-abelian Galois cohomology of operads."

is there a space X such that KU_p^0(X) consists of Z_p-valued measures on Z_p^*?

seems unlikely. if there is, it doesn't have a comparison map to K(Z, 2), which is a space consisting of Z_p-valued measures on Z_p. KU is the initial spectrum under K(Z, 2) with that cohomology, so if you had a comparison map then you'd be looking at a space whose suspension spectrum is KU-locally equivalent to KU

but stranger things have happened. after all, L_K(1) K(Z, 2) splits as a wedge of K-theories for the same reason, so it's not impossible to 'find K-theory' lying around

pardon the ignorance but how is K(Z,2) a space of measures and whence the interest in having some invariant arising as a space of measures?

nothing to pardon, it's pretty esoteric fact. i also misspoke: i meant that the p-adic K-cohomology of K(Z, 2) (not K(Z, 2) itself) is isomorphic to the collection of such measures on Z_p

the identification has three major benefits: (1) the p-adic K-homology is presented as just the function of continuous functions on Z_p; (2) the map K(Z, 2) --> KU presents the p-adic K-homology of KU as the collection of functions on Z_p^*, and the induced map is restriction of domain; and (3) the stable adams operations become recognizable as delta distributions, which is just aesthetically pleasing

and maybe (4) the kronecker pairing between K-homology and K-cohomology becomes integrating the function (coming from the element of homology) against the measure (coming from the element of cohomology)

@EricPeterson my question was motivated by the question of whether there's a space whose suspension spectrum is KU-locally equivalent to KU haha

it definitely seems unlikely but i don't know how to show that it doesn't exist

Oh, wait, does Bousfield's theta functor give a positive answer?

The Bousfield Kuhn Phi functor factors K(n)-localization through the space underlying the spectrum. It has a lesser-known cousin, the Theta functor, that iirc factors K(n)-localization through suspension spectra

So much for "unlikely"

does theta factor through suspension spectra?

Theta is a functor from spectra to pointed spaces and the composition \Sigma^\infty\Theta is K(n)-equivalent (or even T(n)-equivalent) to the identity

so T(n)-locally any spectrum is equivalent to a suspension spectrum, as Eric points out

(what i said didn't even make any sense)

thanks!

that's very interesting

@EricPeterson interesting, thanks!