None of the spaces C^k (\mathbb{R}^n), with 0 \leq k \leq \infty, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves C^k_{nd} (\mathbb{R}^n) of functions f : \mathbb{R}^n \longrightarrow \mathbb{R} for which there exist \Gamma \subset \mathbb{R}^n, \Gamma closed and nowhere dense, such that f restricted to \mathbb{R}^n \setminus \Gamma is C^k-smooth.

Is any similar way known to construct flabby sheaves which contain given sheaves of functions defined on some topological space X ?

  • $\begingroup$ Since you already wrote latexy formulas, you cound wrap them in $ and have them rendered! :) $\endgroup$ – Mariano Suárez-Álvarez Jul 28 '10 at 15:27

There is the "flabbification" or "flasquification" functor used in the Godement resolution. Namely, given a sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_x$ be the stalk at a point $x$. Then we define the sheaf $\Phi(\mathcal{F})$ to have sections $$\Gamma(U, \Phi(\mathcal{F}))=\prod_{x\in U} \mathcal{F}_x$$ with the obvious restriction maps. (This is the same as endowing the etale space of $\mathcal{F}$ with the trivial topology and considering the sheaf of sections.) There is a natural injection $\mathcal{F}\to \Phi(\mathcal{F})$ sending a section of $\mathcal{F}$ to its stalks. Note that $\Phi$ is functorial; indeed, it is the right adjoint to the natural inclusion of the full subcategory (Flasque sheaves on $X$) $\hookrightarrow$ (Sheaves on $X$).

See for example this Wikipedia article or Godement's book on sheaf theory.

Edit: One can also mimic your construction of $C^k_{nd}$ as follows. Let $\mathcal{F}_{nd}$ have global sections given by $\bigcup \Gamma(X-\partial U, \mathcal{F})/\sim$ where the union is taken over all open sets $U$, and we say $(f, X-\partial U)\sim (f', X-\partial U')$ if $f=f'$ when restricted to $X-(\partial U\cup \partial U')$. Then local sections will be restrictions of these global sections. This will always be flabby, but will not necessarily have a morphism $\mathcal{F}\to \mathcal{F}_{nd}$ unless $\mathcal{F}$ has enough sections (for example, if $\mathcal{F}$ is fine, as in your example).

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  • $\begingroup$ Thank you for the answer. It seems however that the Godement flabbification is not the one which goes from $C^k$ to $C^k_{nd}$. And then, what is the difference ? More precisely, given a topological space $X$ and a sheaf $F$ of functions from open subsets $U \subseteq X$ to some set $E$, can one similar to the above, and not to Godement, construct some smallest flabby sheaf $F_{nd}$ of fucntions from open subsets $U \subseteq X$ to $E$, so that $F_{nd}$ extends $F$ ? $\endgroup$ – ron l winger Jul 28 '10 at 14:52
  • $\begingroup$ I've added another construction to address your concerns. I'm not convinced it's useful though--partly because I'm not sure what you mean by "smallest." If you can come up with some functorial property you're looking for, that would help. $\endgroup$ – Daniel Litt Jul 28 '10 at 15:12
  • $\begingroup$ What I mean by "smallest" seems to happen with $C^k_{nd}$ when compared with $C^k$. Indeed, it seems that one cannot squeeze a flabby sheaf between the two of them, unless it is $C^k_{nd}$ itslef. But then, perhaps, instead of the "smallest", one should rather talk about a "minimal" one ? $\endgroup$ – Elemer E Rosinger Jul 29 '10 at 14:07
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    $\begingroup$ Sorry, I sent the previous comment from the office of Prof. Rosinger. ron l winger $\endgroup$ – Elemer E Rosinger Jul 29 '10 at 14:09
  • $\begingroup$ Ah -- you claim that $C^k_{nd}$ has no flabby subsheaf containing $C^k$. Can you prove this? I think it's false. For example, one can require that the $f|_\Gamma$ (where $\Gamma$ is your closed nowhere dense set) equals zero; this gives a smaller flabby sheaf containing $C^k$, unless I'm mistaken. $\endgroup$ – Daniel Litt Jul 29 '10 at 14:17

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