Descent is decent.

It's a slippery slope from 'satisfies the sheaf condition' to 'satisfying certain coherence conditions' though

Hi everyone, I am trying to recall a rather basic result that I simply can't seem to find a reference for. I think it goes: if a map between two spaces induces isomorphisms on the first homotopy group and on all homology groups, it is in fact a weak homotopy equivalence. Googling this sentence and related ones has not helped me except that Corollary 4.33 on page 376 of Hatcher gives the case where the spaces are 1-connected. Am I missing out an assumption / have I dreamed this result?

@ArthurPanderMaat The statement you're looking for is not true. You're probably thinking of the most general version of Whitehead's theorem: if a map of topological spaces induces an isomorphism on the fundamental group for each basepoint and all homology groups with local coefficients for every local coefficient system, then it is a weak homotopy equivalence

(math.stackexchange.com/questions/750955/… for references)

@DenisNardin Thank you and I must apologize for my laziness: it turns out the failure of the version I was looking for is alluded to only a few pages after the Hatcher result I quoted. I was thinking of Whitehead's theorem for abelian spaces, although the result you refer to is actually much more interesting! Thanks again.

@DenisNardin did you get an answer to your Thom spectrum question?

@JonathanBeardsley I deleted shortly afterwards because I realized it was not the question I wanted to ask

@DenisNardin Ah okay. I'm curious about that particular spectrum.

(nothing specific, I just always wonder how much people know about it)

@S.carmeli I was just coming back to this, what is "C" here, when you say "for every C-module M"?

I suppose I should also say that I don't quite know what that identity is you're using there, or rather, I don't think I know how to obtain it off the top of my head.

Oh I think this must be 4.8.4.6 of HA

I have somehow reached a point on MO where I'm now getting alerts when people flag comments in chat as "spam/offensive."

@S.carmeli oh, by the way, does the same identity hold for algebras? i.e. Alg(M)≃Alg(X)⊗_X M, for X-modules M?

Wondering if there's some silly trick one can do with Δ^{op}... But anyway it doesn't matter I guess, since Piotr's proof still works in the algebra case.

Is there something general one can say about "algebras for a monad on a presentable ∞-category C" commuting with "tensoring with a mode" or something?

I seem to recall seeing something like this in the case of stabilization.

Ah I think this is Prop 5.4.2 in Lurie's Goodwillie I paper