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Recently on glancing through Hartshorne's description of Cartier divisors I started pondering the definition of sheafification which led me to a question I can't answer. Neither can I find a discussion in the standard texts.

First let's set up some notation. Let $\mathcal{F}$ be a presheaf (let's say of sets) on the topological space $X$. It has a sheafification $\mathcal{F}^+$. We want to describe the global sections of $\mathcal{F}^+$. A global section of $\mathcal{F}^+$ consists of a map $f:x\mapsto f_x$ where $f_x\in\mathcal{F}_x$, the stalk of $\mathcal{F}$ at the point $x\in X$. This map $f$ also obeys a local compatibility condition: there is an open covering $(U_i)_{i\in I}$ of $X$, and sections $g_i\in\mathcal{F}(U_i)$ with the property that whenever $x\in U_i$ then $f_x$ equals the germ of $g_i$ at the point $x$. Let's call such a covering $(U_i)$ and sections $(g_i)$ a representing system of sections for the global section. (Is there a standard term for this concept?)

My question is this:

For each global section of $\mathcal{F}^+$ is there always a representing system of sections for it having the stronger compatibility property that $$g_i|_{U_i\cap U_j}=g_j|_{U_i\cap U_j}\in\mathcal{F}(U_i\cap U_j)$$ for all $i$, $j\in I$? If not, is there some reasonable condition on the presheaf $\mathcal{F}$ that will guarantee this?

Another motivation is to find a good "pointless" description of the sheafification functor (in the sense of "pointless topology" or locale theory). The definitions of presheaf and sheaf only use the complete lattice structure on the collection of open sets of $X$ and so are thoroughly "pointless", but the usual description of the sheafification functor uses the definitely "pointy" notion of stalk.

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+1. The associated sheaf may be constructed also with respect to other Grothendieck topologies. If $F'$ denotes the presheaf $H^0(X,F)$ (Cech cohomology), then $F''$ is the associated sheaf of $F$. However, your question about Cartier divisors was not answered yet, because in this context it is often assumed that global sections of $F^+$ look like above ... –  Martin Brandenburg Jul 11 '10 at 13:50
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2 Answers

up vote 4 down vote accepted

The following is a counterexample for $\mathcal{F}$ a presheaf of abelian groups (or sets, if you like).

Let $X=\lbrace a,b,c,d\rbrace$ with nontrival opens given by $\lbrace a \rbrace,\lbrace b \rbrace,U=\lbrace a,b,c \rbrace,V=\lbrace a,b,d \rbrace, U\cap V$.

Define the presheaf $\mathcal{F}$ by

$\mathcal{F}(\lbrace a \rbrace)=\mathcal{F}(\lbrace b \rbrace)=\mathbb{Z}/2\mathbb{Z}$,

$\mathcal{F}(U)=\mathcal{F}(V)=\mathcal{F}(U\cap V)=\mathcal{F}(X)=\mathbb{Z}$,

with the obvious restriction maps.

Then $\mathcal{F}^+(X)=\lbrace (x,y)\in\mathbb{Z}\oplus\mathbb{Z}| x\equiv y\text{ (mod 2)}\rbrace$, since the germs at $a$ and $b$ are determined by those at $c$ and $d$, and the only restrictions on $c$ and $d$ are that they give the same germs at $a$ and $b$.

Consider $(0,2)\in\mathcal{F}^+(X)$. The germs at $c$ and $d$ cannot come from a common section of $\mathcal{F}(X)$. Any system of sections which does not include $X$ in the cover must include both $U$ and $V$, having sections 0 and 2, respectively. Of course these do not agree when restricted to $U\cap V$. QED

This construction relies crucially on the fact that the presheaf is not separated (i.e. gluing is not unique). If the presheaf $\it{is}$ separated, the condition described in the question is clearly satisfied.

This construction was shown to me by Paul Balmer.

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Thanks Kevin, that's very neat! –  Robin Chapman Jul 11 '10 at 10:07
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As Kevin points out, one can't ask for a representing family of sections that are equal on every intersection. However, the description can certainly be restated pointlessly.

For $f,g \in \mathcal{F}(U)$, define $f \approx g$ if there's a cover $U = \bigcup_i U_i$ such that for each $i$, $f|_{U_i} = g|_{U_i}$. Read this as “$f$ and $g$ are equal on a cover”.

In pointy terms, $f \approx g$ iff all their germs are equal: for every $x \in U$, $f_x = g_x$.

Now, every section $f \in \mathcal{F}^+(U)$ of the sheafification can be represented by a weakly matching family in $\mathcal{F}$: that is, a cover $U = \bigcup_i U_i$, and sections $f_i \in \mathcal{F}(U_i)$, such that for each $i,j$, we have $f_i|_{U_i \cap U_j} \approx f_j|_{U_i \cap U_j}$. (This is immediate from the representing families you exhibit in the question.) Similarly, two weakly matching families are $(f_i), (g_j)$ are equal as sections of the sheafification iff for each $i, j$, we have $f_i|_{U_i \cap V_j} \approx g_j|_{U_i \cap V_j}$.

So the sections of $\mathcal{F}^+$ can be described exactly as equivalence classes of weak matching families from $\mathcal{F}$. This is a standard way of constructing sheafification on general sites, known as the “double plus-” or “$(-)^{++}$-construction”; see e.g. the Mac Lane and Moerdijk book Sheaves in Geometry and Logic for more context.

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It's notable that the single-plus construction turns presheaves into separated presheaves and separated presheaves into honest sheaves (whence comes the "double-plus" construction). –  Harry Gindi Jul 11 '10 at 14:34
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Indeed! — and this sheds useful light on how non-separated presheaves can (and must) be used to find counterexamples like Kevin's. (Background: the “single plus” construction is defined as “(strictly) matching families, identified up to $\approx$”; it successfully sheafifies separated presheaves, and “separatifies” all presheaves, so applying it twice always suffices for sheafification.) –  Peter LeFanu Lumsdaine Jul 11 '10 at 14:54
    
A further comment: All of this generalizes straightforwardly to fibered categories/separated fibered categories/stacks. –  Harry Gindi Jul 11 '10 at 16:36
    
Thanks Peter for the reference. I'd looked at M&M's chapter II but had no joy --- I should have persevered and contniued with chapter III. :-) –  Robin Chapman Jul 11 '10 at 17:16
    
@Robin: Yep, III.5 is where this comes up :-) @Harry: yes! but do you know a good reference/exposition for that? What I know of that direction is all folklore, nlab, etc. --- I'd be grateful if you can recommend a good proper source for reading up on it. –  Peter LeFanu Lumsdaine Jul 12 '10 at 9:11
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