A sheaf $F$ of sets on (the category of open subsets of) a topological space $X$ is called flabby (or often: flasque, which is the original French term) if for any open subset $U \subset X$, the restriction morphism $F(X)\to F(U)$ is surjective; equivalently if for any opens $U\subset V\subset X$ the restriction $F(V)\to F(U)$ is surjective.
The concept generalizes in a straightforward manner to flabby sheaves on locales, and can be further generalized to flabby objects of any elementary topos.
Flabbiness is a local property: if $F|_U$ is flabby for every sufficiently small open subset, then $F$ is flabby.
Given a continuous map $f:X\to Y$ and a flabby sheaf $F$ on $X$, the direct image sheaf $f_* F : V\mapsto F(f^{-1}V)$ is also flabby.
Any exact sequence of sheaves of abelian groups $0\to F_1\to F_2\to F_3\to 0$ in which $F_1$ is flabby, is also an exact sequence in the category of presheaves (the exactness for stalks implies exactness for groups of sections over any fixed open set). As a corollary, if $F_1$ and $F_2$ are flabby, then $F_3$ is flabby; and if $F_1$ and $F_3$ are flabby, so is $F_2$.
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.
For a more detailed discussion, see Blechschmidt.
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 $F$.
An archetypal example of a flabby sheaf is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle $E\to X$: Since every sheaf over a topological space is the sheaf of sections of an étale space (see there), every sheaf can be embedded into a flabby sheaf $C^0(X,F)$ defined by
where $F_x$ denotes the stalk of $F$ at the point $x$.
Beware that this construction assumes the law of excluded middle and that all stalks $F_x$ are inhabited. In the absence of either, the refined construction
works, where $P_{\leq 1}(F_x)$ is the set of subsingletons of $F_x$.
flabby sheaf
Flabby sheaves were probably first studied in Tohoku, where flabby resolutions were also considered. A classical reference is
See also
Work relating flabby sheaves to the internal logic of a topos include:
Anders Kock, Algebras for the Partial Map Classifier Monad, in Category Theory. Proceedings of the International Conference held in Como, Italy, July 22–28, 1990, pdf
Ingo Blechschmidt, Flabby and injective objects in toposes, arXiv:1810.12708
Martin Escardo, Injectives types in univalent mathematics, arxiv:1903.01211
Last revised on August 6, 2022 at 06:53:56. See the history of this page for a list of all contributions to it.