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As far as I know if $C$ be a category, then a presheaf on $C$ is simply a functor $F:C^{op}\longrightarrow Set$. Now, let $X$ be a topological space and let $O(X)$ be the set of opens of $X$. Now, if we look at $O(X)$ as a category, then, as far as I know a presheaf on X, is simply a presheaf on $O(X)$ which is a functor $F:O(X)^{op}\longrightarrow Set$. For some people, for a presheaf on a topological space, it is also needed that $F(\emptyset)=\{ * \}$. Can we obtain it from the definition that I know? If not, why and how it is true that we need also this condition? Actually, I am looking to see why in the definition of a presheaf on a topological space with values in an abelian category, it is needed that $F(\emptyset)=0$?

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No, that does not follow from the definition as a contravariant functor, and it is incorrect to add it to the definition of a presheaf. I've never seen a source that does this, and it would be interesting if you would include a citation. The condition does follow from the definition of a sheaf, though.

EDIT: There is no more reason to make this requirement if the values are in an abelian category than if the values are in sets. A presheaf is simply a contravariant functor, full stop.

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    $\begingroup$ If I remember right, Hartshorne himself makes the mistake of including this condition in the definition of presheaf! $\endgroup$ – Alex Kruckman Aug 11 at 20:19
  • $\begingroup$ Thank you for your answer. I have changed my question and you are right there is no source which has this definition of a presheaf, there are just not good articles with this definition which I guess their authors are not familiar with category theory. $\endgroup$ – Pouya Layeghi Aug 11 at 20:20
  • $\begingroup$ @AlexKruckman As I see, almost everywhere when the second category is an abelian category, that condition is needed. $\endgroup$ – Pouya Layeghi Aug 11 at 20:23
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    $\begingroup$ @AlexKruckman Wow, looks like I’ve revealed how well-used my copy of Hartshorne is, whoops! $\endgroup$ – Kevin Arlin Aug 11 at 20:52

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