Thanks Fernando. Yeah, in general the notion of what is needed for an internal hom in higher categorical structures seems non-obvious to me. Cool!

So, to me, an interesting question then might be, for what category of compactly generated spaces (or quasicategories, or whatever) can we guarantee that every mapping space is already an object of the category. Or maybe it's not an interesting question...haha.

*compactly generated...

@Dylan, okay, I'm sort of just thinking very naively here. But I'm pretty sure you've answered my question, or at least sort of led me to the obvious answer. If we just think in terms of topologically enriched categories, then all homs are topological spaces, even homs between higher morphisms. So in that case, unless our category is compact generated spaces or something, we're don't really even have internal homs.

Thanks for all the help @Greg and @Fernando. Checking out that Neeman paper now!

Yeah. I'm especially wondering if this is the case for the derived category of a Noetherian ring. I'll have to do some more reading on it.

Since it's my current object of obsession, I might as well mention the large body of work on Bousfield lattices and lattices of (co)localising subcategories. Don't know if acyclic categories are really applicable, but there are certainly a lot of posets around (which we can realize trivially as sites in the way you mentioned).

@Tyler Sort of. Hovey's paper on the chromatic splitting conjecture was where I found (i) above. I have not yet found more information regarding this question specifically, but I have not entirely read these papers, though they are two of my favorites! But thanks for mentioning them again, I will check thru them more carefully I think

@Craig Does a localization of the spectrum always induce an actual localization of the cohomology rings?

@Sean, localization always preserves ring structure, so does that necessarily mean the localization map is a map of ring spectra? And I guess I'm asking if there are any interesting formal group laws that come from precisely such maps, though it appears not.

@Eric I'm going to write up the (now obvious) proof to the proposition, unless you want to write an answer or something. Also, I've been looking at the notes you wrote up on the geometry of formal varieties, and they've been super helpful!