nLab flabby sheaf

Contents

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 called flabby (or often: flasque, which is the original French term) if for any open subset UXU \subset X, the restriction morphism F(X)F(U)F(X)\to F(U) is surjective; equivalently if for any opens UVXU\subset V\subset X the restriction F(V)F(U)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.

Properties

General

  • Flabbiness is a local property: if F| UF|_U is flabby for every sufficiently small open subset, then FF is flabby.

  • Given a continuous map f:XYf:X\to Y and a flabby sheaf FF on XX, the direct image sheaf f *F:VF(f 1V)f_* F : V\mapsto F(f^{-1}V) is also flabby.

  • Any exact sequence of sheaves of abelian groups 0F 1F 2F 300\to F_1\to F_2\to F_3\to 0 in which F 1F_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 1F_1 and F 2F_2 are flabby, then F 3F_3 is flabby; and if F 1F_1 and F 3F_3 are flabby, so is F 2F_2.

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.

Examples

An archetypal example of a flabby sheaf is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle EXE\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)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 the point xx.

Beware that this construction assumes the law of excluded middle and that all stalks F xF_x are inhabited. 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.

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 August 6, 2022 at 02:53:56. See the history of this page for a list of all contributions to it.