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Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof we consider the covering of a closed $S \subset X$ by open $U_i$. Why can't we just take a single $U_1$ covering $S$? A section $s$ over $S$ by definition is an element of a direct limit, so it should have a representative in some neighborhood of $S$ and we could just set $U_1$ to be that neighborhood. Or couldn't we? But the proof is more complicated than that and I'm confused...

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Direct limit presheaf is not a sheaf. Must sheafify it. (Direct sum is a special case of this.) – BCnrd May 12 '10 at 14:18
Sure, but I don't think we need that. – Kestutis Cesnavicius May 12 '10 at 15:38
@Kestutis: the issue is the same as for sheaf pullbacks: the initial construction is just a presheaf, not a sheaf, so need to sheafify. So a section over $S$ is not generally an element of a direct limit; a sheafification intervenes. That's the error. The example of direct limit sheaf is just a toy version of the same issue. – BCnrd May 12 '10 at 15:42
@Kestutis: It is proved in Godement's book on sheaf theory that on such nice spaces as you consider, a section over a closed set does actually arise from a section over an open around the closed set, but that is not the definition (for Godement) and its proof requires some real work. I don't know offhand what foundations Wells is using, but if he takes the theorem in Godement's book as a "definition" then maybe it shifts the burden of work to proving that the definition has good properties (e.g., behaves like a sheaf), so possibly Wells is running into that issue in the proof you ask about? – BCnrd May 12 '10 at 15:46
Oh, it looks like this is indeed a reason. Wells defines sections over a closed $S$ to be elements of the direct limit, and he doesn't really care if that is a sheaf or not (just an abelian group or whatever for each closed $S$). And in this later theorem he suddenly starts using this other viewpoint of a sheaf associated to a presheaf that you're pointing out. Thanks a lot! I'll have a look at Godement. – Kestutis Cesnavicius May 12 '10 at 18:15

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