Thanks very much for this John. This paper of yours and Mark Hovey's is really fascinating. Do you know if more work has been done on this idea of Jack Morava's concerning a structure sheaf over $DL$?

Oh man, these both seem like great answers. I need to think about them both for a bit, to make sure I really do understand the situations. Thanks Charles for this also. I've never gotten so many concise answers to homotopy theory questions in my life since I started asking on MO.

Indeed this makes perfect sense, $H\mathbb{F}_p$ is a field. Thanks so much Tom. You have repeatedly come to my rescue on basic homotopy theory questions that have caused me to linger for hours in the middle of reading papers.

If I'm not mistaken, it's essentially a sort of set-theoretic condition on the class of morphisms we want to localize at? That is, we basically take (homotopy) colimits over acyclics w.r.t. to our class W of morphisms, and if that class is generated under homotopy colimits by a set, then our localization exists.

Oh no problem. Thanks to all parties involved. Very interesting though.

Thanks a lot. Forgot to look in MacClane, which is usually my first stop!

Thanks a lot. What is Ban?

Been thinking about this a little more. You seem to imply that what you say following "For example..." follows from the fact that $H$ preserves sums and filtered colimits. It seems that what we really need is the functor $M:A\to M(A,0)$ which takes $A$ to its Moore spectrum, to commute with filtered colimits, at least in this case, since we're replacing $\mathbb{Z}$ with $\mathbb{S}$, not $H\mathbb{Z}$. Is this correct?

Wow thanks so much. Ultimately my question really concerned the behavior of the functor H, and what you told me makes that really clear! Is there a good reference that proves that fact?

Like the thing above, I do "know" this I guess, but I can't see off the top of my head how to prove it.