Necessity of hypercovers for sheaf condition for simplicial sheaves

Question

I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.

For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that $F$ is a 'local object' with respect to the maps of presheaves $colim(\coprod U_{ij} \underrightarrow{\rightarrow} \coprod U_{i}) \rightarrow X$'for every open cover $\{ U_{i} \rightarrow X \}$. That is to say, HOMing $colim(\coprod U_{ij} \underrightarrow{\rightarrow} \coprod U_{i}) \rightarrow X$' into $F$ gives an isomorphism, and that is easily seen to be the usual sheaf condition. (This seems to be a fairly well-known way to say the sheaf condition, and I learned it from the nlab.)

Now for $F$ a presheaf of simplicial sets, people require more for the sheaf condition, namely that $F$ is local with respect to $hocolim(U_{*}) \rightarrow X$', where $U_{*}$ is a so-called hypercover, and these are some sort of refinement of simplicial diagrams that can be built from multiple intersections of elements of an open cover $U_{i} \rightarrow X$. (I don't really know how to draw a simplicial diagram here. Sorry.)

My question is why in the definition of simplicial sheaf it is necessary to consider not just double intersections of the $U_{i}$, but all intersections (organized into a simplicial diagram). I've read people say, 'The condition is satisfied for spaces, so should be satisfied for general sheaves', but I don't find this a completely convincing reason, since we don't need (?) the whole simplicial diagram to reconstruct a space (at least if we take naive colimits instead of homotopy colimits).

I imagine good answers would be something like 'it is necessary for homotopy invariance of constructions' or 'here is a presheaf F that we'd like to be a sheaf, but HOMing only double intersections into it doesn't see enough information'.

Answer 1

Actually, for simplicial sheaves, and to be more accurate infinity sheaves of infinity groupoids, you do not need hypercovers. Your "naive" idea about multiple intersections (actually fibered products) is correct. If you instead use hypercovers, you get the notion of a *hyper*sheaf. Both infinity sheaves and hypersheaves form an infinity topos, and the infinty topos hypersheaves is a left exact localization of the infinity topos of infinity sheaves. They COULD be the same, but this depends on your Grothendieck site. They are the same if and only if the infinity topos of infinity sheaves satisfies Whitehead's theorem internally with respect to internal homotopy sheaves; for any infinity sheaf $X$ over $\mathcal{C}$, one has sheaves $\pi_n(X) \in Sh_{\infty}\left(C\right)/X$ (it actually is a discrete object thereof) analogous to homotopy groups of spaces. The infinity topos of infinity sheaves is hypercomplete if and only if any morphism $g:X \to Y$ which induces an isomorphism on $\pi_n$ for all $n$ must be an equivalence. If any arbitrary infinity topos is not hypercomplete, we may hypercomplete it by localizing with respect to those morphisms which induces isomorphisms on all homotopy sheaves, and this is again an infinity topos, called its hypercompletion. So the more precise statement is that for an infinity topos of infinity sheaves over a site $\mathcal{C}$, its hypercompletion can be identified with the infinity topos of hypersheaves.

One should note that an infinity topos is defined to be a left exact (accessible) localization of an infinity category of infinity presheaves on some infinity category, and that this DOESN'T imply that you are the infinity topos of infinity sheaves on some site, in the infinite case; the infinity topos of hypersheaves over some site need not be equivalent to ordinary infinity sheaves over any site.

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Now to answer the question in the comments below:

Why do we need the full simplicial Cech diagram for infinity sheaves?

Well, let go back to ordinary sheaves of sets first. A Grothendieck topology should actually be defined in terms of covering sieves not in terms of covering families. What most people call a Grothendieck topology is really a basis for a Grothendieck topology. A sieve on an object $C$ of $\mathcal{C}$ is a subobject $S$ of the representable presheaf $y\left(C\right)$ in the category $Set^{\mathcal{C}^{op}}$. If we are given a basis for a Grothendieck topology, and a cover $\mathcal{U}:=\left(f:U_\alpha \to C\right)$ of $C,$ then we can consider the sieve $S_\mathcal{U}$ to be the subobject which assigns an object $D$ the morphisms $D \to C$ which factor through some $f_\alpha$. A sieve $R$ is a called a covering sieve if there is some cover $V$ such that $S_V \subset R$. Now a presheaf $F$ is said to be a sheaf if for any covering sieve $R \subset y(C),$ the induced map $$F(C) \cong Hom(y(C),F) \to Hom(R,F)$$ is an isomorphism. One can check that this is equivalent to requiring this condition on sieve of the form $S_\mathcal{U}$ for some cover. But, $S_\mathcal{U}$ can be described as the colimit of the diagram $$\coprod\limits_{\alpha,\beta}{y\left(U_\alpha \times_C U_\beta\right)}\rightrightarrows \coprod\limits_{\alpha}{y\left(U_\alpha\right)}$$ in presheaves (you can check this easily by computing this "object-wise"). But this means that $$Hom(S_\mathcal{U},F) \cong \varprojlim \left(\prod \limits_{\alpha}{Hom(y\left(U_\alpha\right),F)} \rightrightarrows \prod\limits_{\alpha,\beta}{Hom(y\left(U_\alpha \times_C U_\beta\right),F)}\right).$$ After Yoneda, this recovers the ordinary definition of sheaf with covers.

But now one can simply say that an infinity sheaf is an infinity presheaf $F$ of infinity groupoids such that for any covering sieve $R \subset y(C),$ the induced map $$F(C) \simeq Hom(y(C),F) \to Hom(R,F)$$ is an equivalence of infinity groupoids (i.e. a weak homotopy equivalence of simplicial sets). Again, it suffices to check this condition for sieves of the form $S_\mathcal{U}$. Now consider the infinity colimit (i.e. homotopy colimit) of the Cech diagram coming from $\mathcal{U}$. Call it $S'$. We can compute its 0-truncation $S'_0$, and since this functor is left-adjoint to the inclusion of ordinary presheaves, the unit of the adjunction furnishes us with a map $l:S' \to S'_0$. But since it is a left-adjoint, it preserves colimits, so it is equal to the colimit of the same diagram in ordinary presheaves. But, the "2-stage" diagram is cofinal in this case, so we recover $S_\mathcal{U}$. But, the homotopy fibers of $l:S' \to S'_0=S_\mathcal{U}$ are discrete, so this means $l$ is both 0-connected and 0-truncated, hence an equivalence. But this means that $S_\mathcal{U}$ is the homotopy colimit of the whole simplicial diagram, when we consider it as an object in infinity presheaves. This in terms tells us that $Hom(S_\mathcal{U},F)$ is a homotopy limit of a cosimplicial diagram, and envoking the infinity Yoneda lemma, we get that this diagram is equivalent to the one involving applying $F$ to iterative fibered products.

Answer 2

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} F(C_\bullet)$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.

If you just use the "first two steps of the diagram", you would be saying that the map $F(X)$ to the "homotopy equalizer" of $F(C_0) \rightrightarrows F(C_1)$ is a weak equivalence. This is a different condition, and is the wrong way.

Here is one way to see why it is wrong (I'll be a bit imprecise here): start with the constant presheaf whose value is an Eilenberg-MacLane space $K(G,n)$, and "localize" it to get a simplicial presheaf $F$ which has descent for covers. I want to be able to say that $F(X)$ has something to do with the cohomology of $X$. With the correct definition (full simplicial diagram), you can see that you are seeing something that looks similar to the classical definition of Cech cohomology. In fact, if $\{U_i\}$ is a "good cover", so that all finite intersections are either empty or contractible, this is what you get. If you just use the "two-step" definition, it's not clear that $F(X)$ will know anything about 3-fold intersections of open sets in the cover, so presumably can't compute Cech cohomology.

Answer 3

This might become clearer when you know that you can obtain the category of sheaves from the category of presheaves by localizing: you invert maps that are isomorphisms locally for the topology. The localization functor can then be identified with sheafification. Note that this description of the category of sheaves makes no reference to the sheaf condition at all.

Now for simplicial sheaves, there is an obvious way to define the correct notion of a "local equivalence of simplicial presheaves": it should just be a map which, locally in the topology, is a weak equivalence of simplicial sets. So the category of simplicial sheaves, whatever it is, should be the localization (preferably in the $\infty$-categorical sense) of the category of simplicial presheaves at these maps. Hypercovers are special cases of local equivalences, and it turns out that there are enough of them: localizing at hypercovers gives you the same category of sheaves. If you localize only at Cech covers, however, you get something different. This is why for example Cech cohomology does not always coincide with sheaf cohomology (and any site in which they disagree gives you an example where the Cech localization is different), but sheaf cohomology is always the same as hypercover cohomology.

This paper by Dugger, Hollander, and Isaksen, is a good reference for hypercovers. Theorem 6.2 is the fact that I mention above.

EDIT: I was wrong when I mentioned Cech cohomology above. Cech cohomology has nothing to do with either localization (see this relevant nForum post).