Anyone know if there are conditions under which the colimit functor makes a functor category comonadic? I.e. such that colim:Fun(C,D)→D presents Fun(C,D) as comonadic over D? It admits a right adjoint (the diagonal functor). And it seems to probably be necessary to assume D is stable...

And I mean, I know we can just try to apply Barr-Beck, but I was just wondering if this was like, a common example or something.

@JonathanBeardsley Is it even conservative in general?

I thought he was talking about the functor in the other direction

Yeah, sorry. Apparently today I'm dumb

Is it known whether any Morava E-theories at heights >1 are E-infinity orientable?

@WilliamBalderrama I think we arrived at the point where "E_infty-orientation" is a slightly ambiguous term: arxiv.org/abs/1905.00072o

@SaalHardali As far as I can tell, it's only ambiguous if you say "E-infinity orientation by periodic complex cobordism".

@WilliamBalderrama I see

@WilliamBalderrama I see. I don't have anything intelligent to say anyway. Just wanted to mention this paper because it totally changed the way I think about complex bordism.

@SaalHardali no not in general

But I'm also thinking about... maybe the source also needs to be a groupoid or something... blergh I dunno.

Ah actually the target pretty clearly doesn't need to be stable, because the canonical example of this is $Fun(X,Top)\simeq Comod_X(Top)$.

@SaalHardali I am also interested in this question.... I feel like there's a very old archived algtop listserv conversation about this involving like, Tom Goodwillie...

It appears to be discussed in this paper, though there's a level of generality there that I'm confused by: arxiv.org/pdf/1612.08622.pdf

@WilliamBalderrama In forthcoming work with Jeremy Hahn, we show that the 2-completion of tmf_1 (3) admits an E_infty complex orientation, which implies that at least one form of height 2 E-theory admits an E_infty complex orientation at the prime 2.

For odd primes, you can show that the height 2 E-theory defined over the algebraic closure of F_p, which I'll call E_2, admits an E_infty complex orientation in the following way.

First use the fact that Dylan Wilson has shown that the Ochanine genus comes from a map of E_infty ring spectra to obtain an MSpin and thus MSU orientation of E_2.

This is the same data as a choice of nullhomotopy of J : bsu --> gl_1 (E_2). The obstruction to extending this nullhomotopy over bu is a map Sigma^2 HZ --> gl_1 (E_2).

But Rezk has computed that there are no such maps, so we win.

I don't know how to E_infty orient E-theories of heights 3 and above.