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In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.

Suppose $\mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $U\subset X$ where $U$ is open, define

$\bar{\mathscr F}(U)=lim_{B\subset U}\mathscr F(B)$, where the $B$s are basic open sets.

This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?

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    $\begingroup$ What kind of limit is that? $\endgroup$ – Lord Shark the Unknown Jul 28 '18 at 9:02
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    $\begingroup$ An inverse limit. $\endgroup$ – Jehu314 Jul 28 '18 at 9:17
  • $\begingroup$ An inverse limit? Over what ordered set/category? $\endgroup$ – Lord Shark the Unknown Jul 28 '18 at 9:18
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    $\begingroup$ Thus if you have been able to show that $\overline{\mathscr{F}}$ is a sheaf, you're done! They're isomorphic on the level of basic opens so they're isomorphic as sheaves. $\endgroup$ – Samir Canning Jul 30 '18 at 15:52
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    $\begingroup$ For reference, this is Proposition I-12 in Eisenbud-Harris, "Geometry of Schemes". $\endgroup$ – DKS Nov 3 '18 at 17:53
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Such a construction works: here's a reference (title: "Construction of a sheaf from the data on a basis of open sets"). Here (pp. 26) is the one pointed out by @DKS in the comments.

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Yes, your construction works and indeed this is how Grothendieck defines the sheaf induced by a sheaf on a base in EGA I.

Let $\mathcal{K}$ be a category. In paragraph (3.2.1) (p. 26), given a sheaf $\mathscr F$ on a base $\mathscr B$ of open sets, he defines a sheaf $\mathscr F'$ by $$ \mathscr{F}'(U) = \varprojlim_V \mathscr{F}(V) $$ where $V$ ranges over the ordered set of sets $V \in \mathscr{B}$ with $V \subseteq U$.

He remarks in paragraph (3.2.6):

Assume still that $\mathcal{K}$ admits projective limits. Then the category of sheaves on $X$ with values in $\mathcal{K}$ also admits projective limits;...the verification of axiom (F) [the sheaf axiom] results from the transitivity of projective limits...

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