Does presheaf send the empty set to zero? 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?
 A: 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 ;).
