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\begin{document}

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\section*{flabby sheaf}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{characterization_using_the_internal_language}{Characterization using the internal language}\dotfill \pageref*{characterization_using_the_internal_language} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A [[sheaf]] $F$ of [[set]]s on (the [[category of open subsets]] of) a [[topological space]] $X$ is \textbf{flabby} (flasque) if for any open subset $U\subset X$, the restriction morphism $F(X)\to F(U)$ is [[surjection|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 \textbf{flasque} is still often used instead of flabby here.

The concept generalizes in a straightforward manner to flabby sheaves on [[locales]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item Flabbiness is a local property: if $F|_U$ is flabby for every sufficiently small open subset, then $F$ is flabby.
\item 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.
\item 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 $F_2$.

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{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

\begin{displaymath}
U \mapsto \prod_{x\in U} F_x
\end{displaymath}
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

\begin{displaymath}
U \mapsto \prod_{x\in U} P_{\leq 1}(F_x)
\end{displaymath}
works, where $P_{\leq 1}(F_x)$ is the set of subsingletons of $F_x$.

\hypertarget{characterization_using_the_internal_language}{}\subsection*{{Characterization using the internal language}}\label{characterization_using_the_internal_language}

\begin{prop}
\label{}\hypertarget{}{}
Let $F$ be a sheaf on a topological space (or [[locale]]) $X$. Then the following statements are equivalent.

\begin{enumerate}%
\item $F$ is flabby.


\item 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 $s'|_U = s$).)


\item 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,

\begin{displaymath}
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).
\end{displaymath}

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

\begin{displaymath}
Sh(X) \models
\forall K : \mathcal{P}_{\leq 1}(F).
\exists s : F.
K \subseteq \{s\}.
\end{displaymath}
Here $\mathcal{P}_{\leq 1}(F)$ is the object of subsingletons of $F$.



\end{enumerate}
\end{prop}
\begin{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.

\end{proof}
For a more detailed discussion, see \hyperlink{Blechschmidt18}{Blechschmidt}.

\begin{remark}
\label{}\hypertarget{}{}
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 [[intuitionistic logic|constructively valid]]. Therefore one could consider to adopt condition 2 as the definition of flabbiness.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The object $\mathcal{P}_{\leq 1}(F)$ of subsingletons of $F$ can be interpreted as the object of [[partial map classifier|``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 $F$.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sheaf]]


\item [[abelian sheaf cohomology]]

\begin{itemize}%
\item [[soft sheaf]]


\item [[fine sheaf]]


\item \textbf{flabby sheaf}



\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Flabby sheaves were probably first studied in [[Tohoku]], where flabby [[resolution]]s were also considered. A classical reference is

\begin{itemize}%
\item [[Roger Godement]] \emph{Topologie Algébrique et Théorie des Faisceaux}. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958.

\end{itemize}
See also

\begin{itemize}%
\item [[Günter Tamme]], section I 3.5 of \emph{[[Introduction to Étale Cohomology]]}
\item \href{https://en.wikipedia.org/wiki/Injective_sheaf#Flasque_or_flabby_sheaves}{wikipedia}
\item \href{https://www.encyclopediaofmath.org/index.php/Flabby_sheaf}{EOM}

\end{itemize}
Work relating flabby sheaves to the internal logic of a topos include:

\begin{itemize}%
\item [[Anders Kock]], \emph{Algebras for the Partial Map Classifier Monad}, in Category Theory. Proceedings of the International Conference held in Como, Italy, July 22–28, 1990, \href{http://home.math.au.dk/kock/jonna5.pdf}{pdf}


\item [[Ingo Blechschmidt]], \emph{Flabby and injective objects in toposes}, \href{https://arxiv.org/abs/1810.12708}{arXiv:1810.12708}


\item [[Martin Escardo]], \emph{Injectives types in univalent mathematics}, \href{https://arxiv.org/abs/1903.01211}{arxiv:1903.01211}



\end{itemize}
category: sheaf theory

[[!redirects flasque sheaf]] [[!redirects flabby sheaves]] [[!redirects flasque sheaves]]



\end{document}