# How to find the smallest flabby sheaf containing a given sheaf ?

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 ?

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Since you already wrote latexy formulas, you cound wrap them in $ and have them rendered! :) – Mariano Suárez-Alvarez Jul 28 '10 at 15:27 add comment ## 1 Answer 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). - 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$F$? – ron l winger Jul 28 '10 at 14:52 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. – Daniel Litt Jul 28 '10 at 15:12 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 ? – Elemer E Rosinger Jul 29 '10 at 14:07 Sorry, I sent the previous comment from the office of Prof. Rosinger. ron l winger – Elemer E Rosinger Jul 29 '10 at 14:09 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. –  Daniel Litt Jul 29 '10 at 14:17
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