A sheaf $F$ of sets on (the category of open subsets of) a topological space $X$ is flabby (flasque) if for any open subset $U\subset X$, the restriction morphism $F(X)\to F(U)$ is onto. Equivalently, for any open $U\subset V\subset X$ the restriction $F(V)\to F(U)$ is surjective. In mathematical literature in English, the original French word flasque is still often used instead of flabby here.
The concept generalizes in a straightforward manner to flabby sheaves on locales.
An archetypal example is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle $E\to X$; regarding that every sheaf over a topological space is the sheaf of sections of an etale space, every sheaf can be embedded into a flabby sheaf $C^0(X,F)$ defined by
where $F_x$ denotes the stalk of $F$ at point $x$. This construction assumes that all stalks $F_x$ are inhabited, and also that the law of excluded middle is available. In the absence of either, the refined construction
works, where $P_{\leq 1}(F_x)$ is the set of subsingletons of $F_x$.
Let $F$ be a sheaf on a topological space (or locale) $X$. Then the following statements are equivalent.
$F$ is flabby.
For any open subset $U \subseteq X$ and any section $s \in F(U)$ there is an open covering $X = \bigcup_i V_i$ such that, for each $i$, there is an extension of $s$ to $U \cup V_i$ (that is, a section $s' \in F(U \cup V_i)$ such that $s'|_U = s$). (If $X$ is a space instead of a locale, this can be equivalently formulated as follows: For any open subset $U \subseteq X$, any section $s \in F(U)$, and any point $x \in X$, there is an open neighbourhood $V$ of $x$ and an extension of $s$ to $U \cup V$ (that is, a section $s' \in F(U \cup V)$ such that $s'|_U = s$).)
From the point of view of the internal language of the topos of sheaves over $X$, for any subsingleton $K \subseteq F$ there exists an element $s : F$ such that $s \in K$ if $K$ is inhabited. More precisely,
The canonical map $F \to \mathcal{P}_{\leq 1}(F), s \mapsto \{s\}$ is final from the internal point of view, that is
Here $\mathcal{P}_{\leq 1}(F)$ is the object of subsingletons of $F$.
The implication “1 $\Rightarrow$ 2” is trivial. The converse direction uses a typical argument with Zorn's lemma, considering a maximal extension. The equivalence “$2 \Leftrightarrow 3$” is routine, using the Kripke-Joyal semantics to interpret the internal statement. We omit details for the time being. Condition 4 is a straightforward reformulation of condition 3.
Condition 2 of the proposition is, unlike the standard definition of flabbiness given at the top of the article, manifestly local. Also the equivalence with condition 3 and condition 4 is constructively valid. Therefore one could consider to adopt condition 2 as the definition of flabbiness.
The object $\mathcal{P}_{\leq 1}(F)$ of subsingletons of $F$ can be interpreted as the object of "partially-defined elements" of $F$. The sheaf $F$ is flabby if and only if any such partially-defined element can be refined to an honest element of $F$.
flabby sheaf
Flabby sheaves were probably first studied in Tohoku, where flabby resolutions were also considered. A classical reference is Godement’s monograph.
Last revised on June 13, 2018 at 13:14:37. See the history of this page for a list of all contributions to it.