I looked in Mac Lane's CWM and found (on p. 155 of the 2nd ed.) that the "Crude Tripleability Theorem" (which gives sufficient but not necessary conditions for monadicity) is attributed to Barr-Beck. So I guess in some circles this pair of names got attached to the necessary and sufficient condition version of the monadicity theorem.

Ahh

$K(n)$ is small in the $K(n)$-local category. Is it small in the $T(n)$-local category? Or is that equivalent to the telescope conjecture?

Equivalently, $K(n)$ is a retract of the $K(n)$-localization of a finite spectrum. Is it also the retract of the $T(n)$-localization of a finite spectrum?

Er-- maybe that's not equivalent -- but it's a variant

What do you mean by small?

I don't think K(n) is a retract of a K(n)-localization of a finite, since K(n)_*K(n) is infinite dimensional.

@CharlesRezk If $X$ is an object of a cocomplete stable $\infty$-category $C$, then by "$X$ is small", I mean "$X$ is compact", which stably means that $Hom(X,-): C \to Spectra$ preserves coproducts (this is basically how the condition is usually formulated in a triangulated category). I'm going off of Theorem 8.5 of Hovey-Strickland, but I suppose the gap in my reasoning is that I didn't actually check that $E^\ast(K(n))$ is finite.

I gather from what you say that it's not

But I thought that $K(n)_\ast K(n)$ was a finite-dimensional algebra over $K(n)_\ast$?

E_*K(n) is basically the (completed) group ring of the Morava stabilizer group (more precisely, of a certain open subgroup of it).

And $E^\ast K(n)$ is even bigger?

I mean $E^*K(n)$.

They are all big.

Actually I mean $K(n)^*E$ lol.

Doesn't matter: it's far from being compact.

well that's embarrassing! Thanks

about time i upvote Dylan's comment

:)