@MarcHoyois I see, thanks!

@AlexanderCampbell So I found that Theorem 6.1 of Rezk's arxiv.org/abs/math/9811038v2 is precisely what I needed to avoid the use of quasi-fibrations in that proof of Dwyer and Kan on group completion I brought up the other day.

@AlexanderCampbell fwiw I think 'sharp' might be the same as 'quasi-fibration after every base change' or even 'quasi-fibration after base change along every map from a disk', but I agree it's better behaved than quasi-fibrations more generally

What I'm trying to understand really is if F is an étale sheaf on some spdm stack such that F is the limit of its own postnikov tower, is the same true for the slice spdm stack. I think I reduced it to this (possibly untrue) statement.

@DenisNardin gotcha, thanks. i haven't thought almost at all about PL manifolds, but that's a great point. that actually makes me start to think that the kan complexes are more fundamental than the topological spaces.

@SaalHardali on a first pass, i think the fact that all basic algebraic invariants -- homology, cohomology, and homotopy -- carry weak homotopy equivalences to isomorphisms is sufficient justification for studying weak equivalences. i agree that on a deeper level this is a subtle question.

@AdrianClough thanks for these comments. i agree with your first message; i used the word "algebraic topology" because that's the nominal title of the course. i also definitely agree that manifolds should be considered as a/the primary example of topological spaces in this context! (of course, this is ignorant of the sorts of topological spaces that show up in algebraic geometry.)

@AaronMazel-Gee I wouldn't go as far as that. I do think that Kan complexes are more fundamental as a representation of homotopy types, but topological spaces do a lot more work. They have a bit of a "jack of all trades, master of none" vibe: you can use topologies to do all sorts of stuff, although often they are suboptimal at the job...

haha

And for example the underlying topological space of Emb(M,N) for M,N smooth manifolds is important, because you want to endow it with an infinite dimensional manifold structure (although diffeological spaces are better at this job, so we go back again to my previous comment...)

i lost some faith in both topological spaces and kan complexes when i learned that posets can do it too

Well, but poset do it in a much more indirect way. For example definining the homotopy groups of a poset is tricky

yeah, i suppose there's not a model structure or anything for them? actually, come to think of it, is it indeed the case that $PoSets [ whe^{-1}] \to Spaces$ is an equivalence? all i really know is that it's surjective.

regarding Emb(M,N), i am unaware of most of the literature, but the little i've read seems to suggest that people effectively treat it as some sort of sheaf/stack (whether or not they use that language). do you know of any interesting differential topology style invariants of it, like de rham cohomology?

@AaronMazel-Gee No idea, I'm sorry. Its homotopy type is important in stuff like surgery theory (e.g. there's that famous open problem to determine when $Emb(M,\mathbb{R}^n)\neq\varnothing$), but the manifold is also important. Unfortunately I'm not very familiar with it so I cannot give you more pointers

@HarryGindi No, a counterexample is T = the classifying topos of the profinite group lim_n (Z/2)^n and X = the constant sheaf on the product of all Eilenberg-Mac Lane spaces K(Z/2, n), n ≥ 1.

Then X is obviously connected but its Postnikov completion is the infinite product of Eilenberg-Mac Lane sheaves K(Z/2, n), which is not connected.

(because BZ/2 has infinitely many nonzero Z/2-cohomology groups)

Ah, crud. Thanks, @MarcHoyois . So I guess being a 'protruncated étale sheaf' just won't be invariant under all basechanges, I guess. That's a shame, but it makes sense.

Actually this topos T is very close to being the etale topos of a field (might be even, I'm not sure if lim_n (Z/2)^n is an absolute Galois group), and this X would be a higher DM-stack over it, so you can probably get a counterexample to your other statement.

Yeah, that's more or less what I had in mind. I know exactly how to build that stack as an étale stack.

@MarcHoyois Well, anyway, the thing I was trying to do was understand when the underlying étale topos of a spectral DM stack is bounded. Is there any answer more satisfying than 'its underlying étale topos is bounded, and it is stable under slicing over truncated objects'?