@David So this is probably a ridiculous question, but I've seen it everywhere, and I'll probably feel stupid when I find out, but what does OP stand for?

I have heard that Rognes has a concept of field extensions in the homotopy category? If we wanted to be somewhat closed minded, I suppose couldn't we look at "extensions" of HQ as "number fields" Hk? And then somehow look at finite wedges of HZ (from the maximal order point of view)? This idea is probably rather silly and useless. I imagine we'd lose a lot of specificity in some sense, but I guess we might have some general framework within which HO_k might be an example. I guess what I'd really like to see is some use of homotopy theory to answer number theory questions in greater generality.

@Tyler Thanks very much for this exposition. I will probably spend some time thinking about it. I'm extremely interested in such issues, at least, in the questions, though I'm not sure my education has progressed far enough yet for me to really talk about answers. My advisor recommended asking you specifically to find out what sort of things made such generalizations difficult, so I was delighted to see that you had responded.

@Will I mean, okay yes there is the notion of modules over ring spectra in homotopy theory ($E$ is a module over $R$ if there is some map $E\wedge R\to E$ satisfying some diagrams, etc.). So for ideal I'm guessing you just use subspectra because we can always localize at that subspectrum? How is the definition of a principal ideal obvious?

I'm not sure I follow your use of manifolds?

...when referring to a "simplicial object" associated to a comonad, because there is not one. Perhaps I mean some kind of simplicial functor $F:\mathcal{C}\to \mathrm{SSet}(\mathcal{C})$. But then again, I haven't run into a lot of "simplicial functors," so perhaps this is not an interesting question...

...(I think above I mean co-unitality)... the $n$th level is $\bot^{n+1}A$ and the face operators at that level are given by the $n+1$ places I can delete a $\bot$ using $\epsilon$ and the degeneracy operators are given by the $n+1$ places we can insert a $\bot$ using $\delta$ (this perhaps non-standard as I refer to the face operators going down from level $n+1$ and the degeneracy operators going up from that same level, not TO that level). Anyway, it is tedious and unpleasant to check that these meet all the necessary conditions but they do. So you're right Tom, what the heck do I mean...

@Tom, you're right and perhaps the question was in fact imprecise since indeed an object is required to do anything at all. So I'm not sure, do we just require the category to have objects? Is that sufficient? The basic construction I'm referring to is as follows: We have a cotriple $(\bot,\epsilon, \delta)$, $\bot:\mathcal{C}\to\mathcal{C}$, $\epsilon:\bot\Rightarrow 1$, $\delta:\bot\Rightarrow\bot\bot$ which satisfy the necessary diagrams (associativity and unitality). Now, assume we have an object $A$ in $\mathcal{C}$. So we can construct a simplicial object where the $n$th level...

Thanks very much for the info. So, @Sean, I'm not sure if it's always acyclic, and the bar construction I'm familiar with I guess is very specific to either groups or $R$-modules, though I can see how it could generalized. At the moment I'm reading in Charles Weibel's book, and he refers to such simplicial objects as "aspherical" (which I think is essentially acyclic) if the object that we build the construction over is "$\bot$-projective". So I'm not entirely sure how it all works out but I think that this sort of thing is not always acyclic, necessarily.