Let $\mathcal{C}= {\bf{Set}}$, and let $X= \mathbb{C}$, and the category of open sets of $X$ equal $X$. We can define a presheaf on this space by the contravariant functor $\mathcal{F}: \mathbb{C} \to {\bf{Set}}$. By Axioms (a) and (b) in the first page of http://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/lecture-notes/MIT18_726s09_lec03_sheaves.pdf we can define this to become the sheaf of open complex sets. One can see that this presheaf is also a sheaf. Does this hold in general, i.e. when is a presheaf a sheaf?
When is a presheaf a sheaf?
0
$\begingroup$
algebraic-geometry
sheaf-theory
-
3First, a presheaf is a *contravariant* functor, not a covariant one. Second, there are tons of notes on sheaf theory, why don't you google for them? – 2012-12-01
-
1A sheaf is by definition a presheaf that satisfies certain additional properties... – 2012-12-01
-
2I'm not sure what you are asking. Are you asking if every presheaf is a sheaf? The answer is no. Are you asking what the conditions are for a presheaf to be a sheaf? It's written in the PDF you linked. – 2012-12-01