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I was having an email conversation with Jacob Lurie yesterday, and he totally shocked me. He gave me an example of a topological space $Q$ and a basis $B$ of $Q,$ such that $Sh_\infty(Q)$ and $Sh_\infty(B)$ are NOT equivalent infinity topoi (where the latter is sheaves on $B$ with the induced Grothendieck topology). Note that by the Comparison Lemma (c.f. Elephant, Theorem 2.2.3) one always has that $Sh(Q)$ and $Sh(B)$ ARE equivalent 1-topoi, so this is quite surprising. In particular, this means that if we left Bousfield localize simplicial sheaves with respect to Cech covers, this does not always represent the infinity topos of sheaves. To see this, if simplicial sheaves always presented the infinity topos of sheaves, then any two Grothendieck sites whose associated topos of sheaves were equivalent would then have Quillen equivalent model categories of simplicial sheaves, and hence, one would have equivalent infinity topoi of sheaves, but, this cannot be the case.
For completeness, here is Jacob’s counterexample:
Let $Q$ be the Hilbert cube (a product of countably many copies of the interval $[0,1]$) let $X = Sh_\infty(Q)$, and let $D$ be the subcategory of $X$ given by those open subsets of $Q$ which are homeomorphic to $Q \times [0,1)$ (which you can view as subobjects of the unit object of $X$). Such open subsets form a basis for the topology of $Q$, so they can be used to cover any object of $X$. Consider the functor which assigns to each open set $U$ of $Q$ the complex of Borel-Moore chains on $U$. This defines a sheaf on $Q$ with values in the infinity-category of chain complexes of abelian groups. This sheaf vanishes on every object of $D$ but the sheaf does not vanish everywhere (the Borel-Moore homology of $Q$ is isomorphic to the usual homology of $Q$, since $Q$ is compact).
Any thoughts? This is very surprising to me. It also shows that the full and faithful embedding of 1-topoi into infinity topoi, whose essential image is 1-localic infinity topoi, does not always send the topos $Sh(C,J)$ to $Sh_\infty(C,J)$! (otherwise we’d again have a contradiction). But it does provided we chose $(C,J)$ to be a site with finite limits (Proof of Proposition 6.5.4.7, HTT).
if simplicial sheaves always presented the infinity topos of sheaves, then any two Grothendieck sites whose associated topos of sheaves were equivalent would then have Quillen equivalent model categories of simplicial sheaves, and hence, one would have equivalent infinity topoi of sheaves, but, this cannot be the case.
But the (Joyal) model structure on simplicial sheaves presents the hypercomplete $\infty$-topos. (It’s Quillen equivalent to the Jardine model structure.)
Isn’t that what makes the difference? The two plain $\infty$-sheaf $\infty$-toposes may differ, but their hypercompletion will coincide if their sites have the same sheaf 1-topos.
That’s, I think, also why the example that you quote invokes the Hilbert cube: the Hilbert cube is is an example of a site for which the $\infty$-sheaf $\infty$-topos differs from its hypercompletion.
@Urs: Indeed, I was aware that they had equivalent hypercompletions for the reason you stated. But, I was (apparently incorrectly) under the impression that if we localize simplicial sheaves with respect to Cech covers (instead of hypercovers) that this would model the infinity topos of infinity sheaves (not hypersheaves). However, this is apparently wrong. I have many questions now, but one is:
Given a full subcategory $D$ of a Grothendieck site $(C,J),$ under what conditions will $Sh_\infty(D,j_D)$ be equivalent to $Sh_\infty(C,J).$ Clearly, the usual comparison lemma is not sufficient. I would guess one would guess that this is true if and only if the Yoneda-embedded image of $D$ in $Sh_\infty(C,J)$ is dense, however, I would like something not directly involving the infinity topos.
Oh, I see.
You know, I have to think here: do we even have a global model structure on simplicial sheaves?
But of course, I understand, for your example you start with one (hyper)local model structure and then ask about further localization.
Okay, so I understand the question now. :-) I don’t know the answer. But clearly it’s a good question. Maybe one crisp way to put it would be:
what are the $\infty$-analogs of the notion of dense subsite?
@Urs: Yes, this is what lead me to the discussion I was having with Jacob. In particular, his counter-example shows the following straight-forward generalization of a very classical piece of $1$-category theory breaks for infinity categories:
Let $D$ be a full subcategory of $C,$ and suppose that $C$ has pullbacks and arbitrary coproducts. If for every object $c$ of $C,$ the canonical morphism $\coprod\limits_{f:d \to c} d \to c$ is an effective epimorphism, where the coproduct is indexed over morphisms with domain in $D,$ then $D$ is dense in $C.$
Counter-example: Let $C=\Sh_\infty\left(Q\right)$ be sheaves on the Hilbert cube, and $D$ the subcategory described in 1. $D$ is not dense.
However, Lurie’s notion of a hypercovering in an infinity topos makes sense in a wider context, and one can easily show that if every hypercover of an object in $C$ is effective, then $D$ is still dense. Moreover, every hypercovering of each object of an infinity topos is effective, if and only if the infinity topos is hypercomplete.
I think this effect exist already at $n=2$. In the times of Giraud’s book there were known examples of different sites with the equivalent topoi, but such that the categories of stacks are not equivalent. Of course, if one works internally, in the topos and takes the regular topology then one has trivially the same category of stacks. I think that Mike once convinced me that something was stupid about this kind of examples, but I do not remember the upshot now.
@Zoran: I’m tempted not to believe this. For stacks of $n$-groupoids for any finite $n,$ one may use hypercovers instead of covers and get the same answer. So, one could construct the associated model categories on simplicial sheaves for hypersheaves. These are Quillen equivalent, hence they have equivalent infinity categories of hypersheaves. But now, by passing to $1$-truncated objects, one has an equivalence between their associated $(2,1)$-categories of stacks. (Each stack of groupoid is a hypersheaf).
(comment removed, since I was being silly)
Like Urs, I kind of thought we already knew this, at least the bit about simplicial sheaves. But on the other hand, I’m pretty sure people have claimed to me things that contradict your last paragraph, and I haven’t been sure enough of myself to refute them — so this is clearly not as well-known as it should be. There should be a big warning label somewhere: you cannot reconstruct the $\infty$-sheaf topos on a site from its 1-sheaf topos! If only we knew the right place to put it….
(Just coming back online after Hurricane Sandy here in Princeton…)
Glad to hear you are safe, Mike.
And the same from me. How has Princeton fared?
Well, here at IAS we lost power for a good long time, about 4 days. Other parts of town were better off and got their power back after a day or two, or maybe even never lost it. Princeton University apparently has its own generator. Lots of downed trees were blocking the roads, though, in addition to knocking down power lines. But I’ve only seen a couple of houses that got hit.
Well, here at IAS we lost power for a good long time, about 4 days.
That’s actually not long, compared to other parts of New Jersey, and even many parts of Connecticut. (My house got back utility power on Saturday night, and with it cable and internet.) My brother, who has a house in the Atlantic Highlands, has been living in a hotel in Pennsylvania, and will probably have to wait several weeks for power to return. A terrible mess.
And a Nor’easter is due to hit in a few days, with yet more high winds!
Yeah, so I’ve heard! I didn’t realize at first how lucky we were.
@Urs #4
Yes, in fact we have a “Čech” model structure structure on simplicial sheaves: define the weak equivalences and trivial fibrations as in presheaves, and define the cofibrations by the left lifting property. The fact that the adjunction unit is a natural weak equivalence allows us to apply Kan’s recognition theorem for cofibrantly generated model structures. This works for both the projective and injective versions of the “Čech” model structure of simplicial presheaves. The same arguments work for the local Jardine (“injective”) and Blander (“projective”) model structures, recovering the Joyal (“injective”) and Blander (“projective”) model structures.
Thanks for picking up this old thread. In the spirit of the $n$Lab the idea would be that you write this out in a paragraph of a relevant $n$Lab entry.
I created Čech model structure on simplicial sheaves and wrote a very brief sketch of the proof.
Thanks!!
I have briefly edited the formatting a little (made the floating TOC come out right, made Definition numbers come out, added some hyperlinks). Also cross-linked at model structure on simplicial sheaves and topological localization.
@Zhen:
I don’t think the proof can possibly be correct. See the first entry in this discussion. I give an example of two Gorthendieck 1-sites which have equivalent categories of sheaves, but nonequivalent infinity categories of infinity sheaves. If there were a model category on simplicial sheaves which were Quillen equivalent to the local model structure on simplicial presheaves that this would cause a contradiction, as the latter presents the infinity category of (Cech) infinity sheaves.
It’s not defined intrinsically in terms of the topos, so I don’t see the problem.
@Zhen: Ah, I see- the underlying categories of the model categories are equivalent, but this equivalence need not be a Quillen equivalence, got it :)
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