nLab
flabby sheaf

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

A sheaf FF of sets on (the category of open subsets of) a topological space XX is flabby (flasque) if for any open subset UXU\subset X, the restriction morphism F(X)F(U)F(X)\to F(U) is onto. Equivalently, for any open UVXU\subset V\subset X the restriction F(V)F(U)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

Examples

An archetypal example is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle EXE\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)C^0(X,F) defined by

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

where F xF_x denotes the stalk of FF at point xx. This construction assumes that all stalks F xF_x are inhabited, and also that the law of excluded middle is available. In the absence of either, the refined construction

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

works, where P 1(F x)P_{\leq 1}(F_x) is the set of subsingletons of F xF_x.

Characterization using the internal language

Proposition

Let FF be a sheaf on a topological space (or locale) XX. Then the following statements are equivalent.

  1. FF is flabby.

  2. For any open subset UXU \subseteq X and any section sF(U)s \in F(U) there is an open covering X= iV iX = \bigcup_i V_i such that, for each ii, there is an extension of ss to UV iU \cup V_i (that is, a section sF(UV i)s' \in F(U \cup V_i) such that s| U=ss'|_U = s). (If XX is a space instead of a locale, this can be equivalently formulated as follows: For any open subset UXU \subseteq X, any section sF(U)s \in F(U), and any point xXx \in X, there is an open neighbourhood VV of xx and an extension of ss to UVU \cup V (that is, a section sF(UV)s' \in F(U \cup V) such that s| U=ss'|_U = s).)

  3. From the point of view of the internal language of the topos of sheaves over XX, for any subsingleton KFK \subseteq F there exists an element s:Fs : F such that sKs \in K if KK is inhabited. More precisely,

    Sh(X)KF.(s,s:K.s=s)s:F.(K is inhabitedsK). 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 F𝒫 1(F),s{s}F \to \mathcal{P}_{\leq 1}(F), s \mapsto \{s\} is final from the internal point of view, that is

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

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

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 “232 \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 𝒫 1(F)\mathcal{P}_{\leq 1}(F) of subsingletons of FF can be interpreted as the object of "partially-defined elements" of FF. The sheaf FF is flabby if and only if any such partially-defined element can be refined to an honest element of FF.

References

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:

category: sheaf theory