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After digesting the Presheaf definition by the very first time, one feels (at least I felt) a strange sensation noticing the existence and uniqueness conditions to graduate that Presheaf as a sheaf, but although some "natural" examples are given to show that the existence condition is not garanted (bounded functions is the canonical one), all examples that I occur are bizarre and absolutely unnatural, in the text books I've seen I found nothing.

So the question is: Is there some "interesting" and/or "natural" Presheaf (I mean a Presheaf useful for something at least pedagogically) which supports existence and fails only the uniqueness condition?

Thanks

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    $\begingroup$ The answer, of course, depends on which presheaves you count as natural or non-pathological. If you're only willing to consider presheaves F on X of the type "F(U) = {functions on X satisfying some condition}" then you're always going to have uniqueness. $\endgroup$
    – Tom Leinster
    Commented Nov 16, 2012 at 3:52
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    $\begingroup$ Well, all "natural" presheafs are presheafs of functions, for which uniqueness is automatic. However, the presheaf quotient of a sheaf by a subpresheaf need not satisfy uniqueness. For example, consider at the presheaf quotient of the sheaf of locally constant functions on a space by the subpresheaf of constant functions. $\endgroup$
    – anon
    Commented Nov 16, 2012 at 3:54

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