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Sheafs are often denoted by the letter $\mathcal O$. What does this O stand for? To me it seems that more natural choices of symbols for sheaves would be $\mathcal S$ or $\mathcal F$ (for the french faisceau).

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My guess. From the symbol for holomorphic functions. –  ABC Jul 4 '13 at 13:41
    
And where does that come from? –  Dominik Jul 4 '13 at 13:44
    
See mathoverflow.net/questions/92135/… –  Brad Jul 4 '13 at 14:02
    
@Brad I think this answers my question. Could you turn that into an answer? –  Dominik Jul 4 '13 at 14:10

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up vote 3 down vote accepted

Quoting the historical footnote in Grauert/Remmert, Coherent Analytic Sheaves, concerning the origin of the notation $\mathcal{O}$ for the rings of holomorphic functions (and their associated sheaves):

Some people think the symbol $\mathcal{O}$ was chosen in honor of Oka, sometimes it is even said that $\mathcal{O}$ reflects the French pronunciation of holomorphe. The truth is that the symbol was chosen accidentally. In a letter to the authors from March 22, 1982, H. Cartan writes: "Je m'étais simplement inspiré d'une notation utilisée par van der Waerden dans son classique traité 'Moderne Algebra' (cf. par exemple §16 de la 2e édition allemande, p.52)"

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And why did van der Waerden use it? –  Dominik Jul 4 '13 at 14:18
    
Presumably because Dedekind used $\mathfrak{o}$ for an order ("Ordnung" in German). But that may also be idle speculation, I don't know why. –  Daniel Fischer Jul 4 '13 at 14:21

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