nLab
flabby sheaf

Definition

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.

Properties

• 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 $F_2$.

Examples

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

$$U \mapsto \prod_{x\in U} F_x$$

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

$$U \mapsto \prod_{x\in U} P_{\leq 1}(F_x)$$

works, where $P_{\leq 1}(F_x)$ is the set of subsingletons of $F_x$.

Characterization using the internal language

Proposition

Let $F$ be a sheaf on a topological space (or locale) $X$. Then the following statements are equivalent.

1. $F$ is flabby.

2. 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 $s'|_U = s$).)

3. 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,

   $$Sh(X) \models \forall K \subseteq F. (\forall s,s':K. s = s') \Rightarrow \exists s:F. (K \text{ is inhabited} \Rightarrow s \in K).$$

4. The canonical map $F \to \mathcal{P}_{\leq 1}(F), s \mapsto \{s\}$ is final from the internal point of view, that is

   $$Sh(X) \models \forall K : \mathcal{P}_{\leq 1}(F). \exists s : F. K \subseteq \{s\}.$$

   Here $\mathcal{P}_{\leq 1}(F)$ is the object of subsingletons of $F$.

Proof

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.

Remark

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.

Remark

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 $F$.

Related concepts

• sheaf

• abelian sheaf cohomology

  • soft sheaf

  • fine sheaf

  • flabby sheaf

References

Flabby sheaves were probably first studied in Tohoku, where flabby resolutions were also considered. A classical reference is

• Roger GodementTopologie Algébrique et Théorie des Faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958.

See also

• Günter Tamme, section I 3.5 of Introduction to Étale Cohomology
• wikipedia
• EOM

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

category: sheaf theory