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Hartshorne requires (in his Algebraic Geometry) a presheaf (of abelian groups) to send the empty set to the zero group. But Wikipedia's definition doesn't have that condition (just a contravariant functor from the category of open subsets to the category of abelian groups). I don't think the condition automatically follows from definition. Is Hartshorn defining a presheaf specifically designed for varieties?

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The condition doesn't automatically follow from the definition; take any constant functor whose value isn't the zero group. –  Qiaochu Yuan May 25 '12 at 14:46
@QiaochuYuan I remember Vakil writing that $\mathscr F(\varnothing)$ is forced to be the final object in the appropriate category, but I've never understood why that's true. –  Dylan Moreland May 25 '12 at 14:49
@Dylan: that's true for sheaves but not (with Wikipedia's definition) for presheaves. –  Chris Eagle May 25 '12 at 14:50
@ChrisEagle Aha, I didn't read the question carefully. Probably it has to do with the empty product being the final object? –  Dylan Moreland May 25 '12 at 14:51
@Dylan: indeed. –  Chris Eagle May 25 '12 at 14:54

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up vote 11 down vote accepted

The definition in Hartshorne's book is wrong. Of course it's no harm to deal only with presheaves with this property, but in the general setting one should not force this as part of the definition. A presheaf in an abstract setting is just a contravariant functor $\tau^{op} \to ?$, where $?$ is some nice category and $\tau$ is some site; it may have no initial object at all, so that it doesn't make sense to talk about $\emptyset$. And even if it exists, the category of the presheaves sending $\emptyset \mapsto *$ won't be so well behaved as the Grothendieck topos of all presheaves.

However, sheaves automatically have the property that $F(\emptyset)=*$ (apply the sheaf condition to the empty covering of $\emptyset$), or in the abstract setting: $F$ must preserve the terminal object (that is, if it exists).

PS: Hartshorne's definitions of vector bundles (charts!), coherent sheaves (confused with finite type; only equivalent in the noetherian case) and projective morphisms ($\mathbb{P}^n$ instead of $\mathbb{P}(\mathcal{E})$, only equivalent over projective base) are also "wrong", although you can work with them quite well in the context of the book, of course. Probably this also motivated these ad hoc definitions.

In any case, for the "correct" definitions, see EGA, SGA, FGA ;) or the SP ;).

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Why F must preserve the terminal object?, F is just a contravariant functor. –  Gastón Burrull Jan 28 at 2:28
For me there are no contravariant functors, everything is covariant by definition and if necessary we use dual categories. A sheaf is a certain functor $\tau^{op} \to C$ and it preserves the terminal object, i.e. the initial object of $\tau$ (if it exists) is mapped to the terminal object of $C$. –  Martin Brandenburg Jan 28 at 9:40
A functor in the generic case does not preserve initial objects. I think one needs to define a sheaf as a contravariant functor which sends the empty set in a trivial group. –  Gastón Burrull Jan 29 at 2:45

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