Flasque sheaf

Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space, Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories). For open subsets $U\subseteq V\subseteq X$ write ${\rho }_U^V:\mathcal{F}(V)⟶\mathcal{F}(U)$ for the restriction map of $\mathcal{F}$ (Sections, restrictions, and global sections of a presheaf).

Then $\mathcal{F}$ is flasque, also called flabby, when each of these restriction maps is surjective: for all open $U\subseteq V\subseteq X$ the map ${\rho }_U^V$ is onto. Equivalently, $\mathcal{F}$ is flasque when every section of $\mathcal{F}$ over an open subset $U$ extends to every larger open subset $V$, that is, when for every $s\in \mathcal{F}(U)$ there is $t\in \mathcal{F}(V)$ with $t{\mathrm{\mid }}_U=s$.

The same condition on restriction maps defines flasque sheaves of sets and flasque sheaves of modules over a sheaf of rings; on this page the notion is used for sheaves of abelian groups.