Alternative Construction of Sheaf from Sheaf on a Base

Question

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

$\bar{\mathscr F}(U)=\varprojlim_{B\subset U}\mathscr F(B)$, 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?

Answer 1

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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Edit (two years later!): A conceptual way to look at this construction is that the formula \begin{equation}\tag{1}\mathscr{F}(U)\cong\underset{\overset{\longleftarrow}{B\subset U}}{\lim}(\mathscr{F}(B))\end{equation} is precisely the usual formula computing right Kan extensions. Let $\mathsf{Open}(X)$ be the category whose

• Objects are the open subsets of $X$;
• Morphisms are inclusions.

Given a basis $\mathcal{B}$ of $X$, write $\mathsf{Open}(\mathcal{B})$ for the full subcategory of $\mathsf{Open}(X)$ spanned by the opens of $X$ in $\mathcal{B}$. Then we have a canonical inclusion $$ i\colon\mathsf{Open}(\mathcal{B})\hookrightarrow\mathsf{Open}(X), $$ and the presheaf associated to a presheaf $\mathscr{F}\colon\mathsf{Open}(\mathcal{B})^\mathsf{op}\to\mathcal{A}$ on the basis $\mathcal{B}$ taking values on a category $\mathcal{A}$ is precisely the right Kan extension of $\mathscr{F}$ along $i$:

You can go from this to Equation $(1)$ by computing $\mathsf{Ran}_i(\mathscr{F})$ as the limit in Eq. 6.2.3 of Riehl's book:

,

which in our context takes the form $$ \mathscr{F}(U)\cong\lim\left((\underline{U}\downarrow i)^\mathsf{op}\twoheadrightarrow\mathsf{Open}(\mathcal{B})^\mathsf{op}\overset{\mathscr{F}}{\longrightarrow}\mathcal{A}\right). $$ But $(\underline{U}\downarrow i)^\mathsf{op}$ is precisely the category whose objects are inclusions of the form $B\hookrightarrow U$ for $B\in\mathcal{B}$, so projecting from this to $\mathsf{Open}(\mathcal{B})^\mathsf{op}$ and then applying $\mathscr{F}$ recovers Eq. $(1)$.

For sheaves, you first take the right Kan extension of $\mathscr{F}$ on $\mathcal{B}$ as above and then sheafify. If $\mathscr{F}$ is already a sheaf on $\mathcal{B}$, then so is the resulting sheaf on $X$ (Görtz–Wedhorn, Prop. 2.20).

Answer 2

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...