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topos theory

# Contents

## 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 $F$.

## 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

Work relating flabby sheaves to the internal logic of a topos include:

category: sheaf theory

Last revised on April 3, 2020 at 22:51:27. See the history of this page for a list of all contributions to it.