@DenisNardin right, thanks -- but when is the ∞-topos of a topological space hypercomplete? is the best known criterion that it has finite homotopy dimension?

@JonathanBeardsley yes, totally. this is true if you work "model-independently" (see this note that i wrote: 1510.02402) and consider ∞-groupoids of such data, or alternatively if you work quasicategories and consider kan complexes of such data.

@DenisNardin, oh right, thanks. What if I take the target category to be connective S-algebra spectra? Does the left adjoint get more interesting (if it exists) ?

@AaronMazel-Gee thanks for the reference!

@DenisNardin just to clarify here: certainly an open cover gives a sieve. conversely, it seems that a sieve also defines a really big open cover (maybe modulo size issues, or maybe subobjects of the terminal object always form a small poset?). do you agree? this seems necessary in order to conclude that "descent for sieves = descent for covers".

@AaronMazel-Gee This is true for any object in any presentable $\infty$-category. See HTT 6.2.1.3.

@AaronMazel-Gee Yes, on topological spaces a covering sieve is a special kind of open cover (in general everything here works form small sites, if you want the discussion above to work for big sites you have to add a bunch of qualifiers etc)

Unstraightening has come to Kerodon twitter.com/kerodonmath/status/1352264988287766531

@DenisNardin are you sure about the claim for small sites? I think that for various finitary sites a cover might be only finite cover but then a sieve is anything that contains a finite cover in it.

I don't think you can recover the cover from the sieve. I thought one of the reasons for talking about sieves is that two families of sieves give rise to the same topology on a category iff the families are the same, but two families of coverings can given rise to the same topology without the families being the same.

@S.carmeli Hmm... I was talking only of topological spaces to be on the safe side. In general I am a bit loose with the term "cover" though...

And to be clear I hope I never claimed that you can reconstruct the generating cover from the sieve because as Tom notices you can't

@S.carmeli But to be clear, it depends on how you define the Grothendieck topology induced by a pretopology. IIRC a covering sieve is defined as one generated by a covering in the pretopology, so the existence of a generating covering is a non-issue

@S.carmeli @DenisNardin In a topology generated by a pretopology, there can indeed be more covering sieves than those generated by coverings, but this doesn't change the descent condition. In general descent wrt a collection of sieves stable under pullbacks is equivalent to descent wrt the topology it generates (for 1-categories this is SGA4 II 2.3, and the same proof works for ∞-categories).

I agree that all this discussion has no effect on the resulting topos at all. In my opinion, this is an indication that the best notion to consider is that of a topos, if one really wants it also makes sense to talk about the covering sieves version Grothendieck topology (which is like choosing a presentation by generators and relations to a ring) and the notion of topology in terms of covering families is basically there for historical reasons. Sieves are just better notion.

There's a definition of a Grothendieck pretopology on an infty-category in my paper on synthetic spectra, as well as the verification that the descent condition can be checked on covering families (A.1, A.9 in the appendix). Like Marc says, the proof is not at all different from the classical one.

@PiotrPstrągowski thanks good to know about this ref!

@AaronMazel-Gee I also just realized that this is 5.5.4.13 in Kerodon

I'm not entirely sure how to cite Kerodon right now I don't think...

ah i guess Kerodon has something for that...

@AaronMazel-Gee Uh sorry for some reason I did not see this question. The answer unfortunately is that yeah, the best criterion I know is when the space has finite homotopy dimension (e.g. because it has finite covering dimension or some such)

@Bbb The fiber sequence I wrote should be, for Y unpointed, S(Y) -> S_+(Y_+) -> S_+(*_+). (If Y is unpointed I don't even know how to apply S_+, e.g. to the empty set.)

I wrote this down -- there are 3 steps.

(1) The forgetful functor U: C_* -> C preserves pushouts and pullbacks. The definition of the first polynomial approximation is just in terms of pushouts in the source category and pullbacks+limits in the target category, and so if G is a functor C -> D where D is stable + differentiability assumptions, then P_1(GU) = P_1(G) U.

Hmm maybe I'm just using bad notation - in the above I had S_+ the stabilization of the unpointed category - this is a notation from higher algbera I borrowed, but maybe that's causing confusion

(2) The pushout of the diagram + <- X -> X_+ is always +_+, and so applying P_1 F we get that P_1 F(X) is the pullback of P_1 F(+) -> P_1 F(+_+) <- P_1 F(X_+).

(sorry, I have to switch to + uniformly b/c of markdown fighting me)

The definition of D_1 F(X) is the fiber of P_1 F(X) -> P_1 F(+), and so this tells is that there is a fiber sequence D_1 F(X) -> P_1 F(X_+) -> P_1 F(+_+).

(3) Then we apply the octahedral axiom to P_1 F(X_+) -> P_1 F(+_+) -> P_1 F(+), which is a diagram of pointed objects. That gives us a fiber sequence D_1 F(X) -> D_{1,+} F(X_+) -> D_{1,+} F(+_+).

(Sorry, I just misunderstood your notation.)

Me doing math: (*_+)

Just because I'm slow, can you clarify what you mean by S and S_+?

Well, maybe I shouldn't have said S because my assumptions were different (I wanted to linearize a functor with stable target).

But roughly, S should be the stabilization of the identity functor on unpointed spaces, and S_+ the stabilization of the identity functor on pointed ones.

We know S_+(X) would then be Σ^∞ X, and so this would assert a fiber sequence S(X) → Σ^∞(X_+) → S^0.

In particular, the stabilization of the empty set should be S^{-1}.

(consistent with its reduced homology)

ah, and by stabilization you mean 1st Goodwillie derivative?

Yeah, I mean the 1-layer in the Taylor tower.

Sorry for all the confusion. Does that make more sense?

Yeah this makes a lot more sense now - I was confused because I was playing with the "stabilizations" (in the sense of natural maps into the Sp(C)) of the pointed and unpointed categories, which is different

thanks for clarifying!

No problem. Missing basepoints in Calculus are something I've always found confusing

And with rings, you can't really pretend that the basepoints are going to work themselves out.