I need a clarification in the definition of presheaf. If $X$ is a topological space then we define presheaf on $X$ to be an assignment to every open subset $U \subset X$ and to every pair of open sets $U,V$ ($V \subset U$) a map (restriction map) $ res_{UV} : \jmath(U) \rightarrow \jmath(V) $ satisfying $$res_{VW} \circ res_{UV} \rightarrow res_{UW}$$ for every $W \subset V \subset U$ open sets.

Now, I have seen examples of presheaf where it is a set of differeniable functions on $U$. So here the assignment is a set of differentiable funcions. More generally , I am curious about the nature of this assignment. Can this assignment be say, assigning a measure on $U$ or even a topology on $U$ as long as the satisfy all the required conditions?

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    $\begingroup$ Yes, you can assign anything, as long as you define what restrictions are. $\endgroup$ – Crostul Jan 21 '17 at 6:20
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    $\begingroup$ And as long as that "anything" is a set. $\endgroup$ – Stahl Jan 21 '17 at 6:23
  • $\begingroup$ (One can define a notion of a $\mathcal D$-valued presheaf on $\mathcal C$ for any categories $\mathcal C$ and $\mathcal D$; this is simply a contravariant functor $F : \mathcal C\to\mathcal D$. In your case, $\mathcal C$ is the category of open subsets of $X$ with morphisms given by inclusions and $\mathcal D = \mathsf{Set}$.) $\endgroup$ – Stahl Jan 21 '17 at 6:56

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