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I just started working with flasque, or flabby sheaves, that is sheaves whose restriction maps are surjective for any two open set of the space.

I wonder about the etymology of the term. In French, flasque can mean both "flabby, without consistence" and "bottle". It seems that the English term comes from the first of those meanings, but in French and Italian (where the two meanings are distinguished by pronunciation) the "bottle" meaning is used. This is justified by the fact that a flasque sheaf has its section "falling down" from the global sections as if they were poured from a bottle.

So I wonder: is the English translation a misinterpretation? If not, is there a motivation for it?

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    $\begingroup$ No, flabby is a totally accurate translation of flasque. $\endgroup$ Oct 25, 2014 at 2:03
  • $\begingroup$ I prefer flaccid as a translation of flasque. It has the advantage of sharing the Latin root. $\endgroup$ Feb 5, 2015 at 4:08

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In French the meaning "bottle" is definitely never used, in particular because it is syntactically impossible!
The meaning in French of flasque is exactly flabby and the terminology is very appropriate: any section on an open subset of a flabby sheaf can be extended to the whole space.
A tougher sheaf would never tolerate that: just try with the sheaf $\mathcal O$ of holomorphic functions on $\mathbb C$, where some functions on the unit disk cannot be extended through a single point of the boundary circle : there's a very unflabby sheaf for you!
Note that in the same register wee ze French also speak of faisceaux mous (= soft sheaves) and faisceaux fins (= fine sheaves) .

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  • $\begingroup$ I tought flasque is the French term for the english flask, or for the italian "fiasco".. $\endgroup$
    – Franco
    Oct 25, 2014 at 2:52
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Flasque, when used as an adjective, say describing sheaves, means flabby, flaccid, soft, easily deformed in an algebraic geometry context. Only when used as a noun (feminine) is flasque the French term for English flask. That is NOT the usage here though.

Even the very brief definition for flasque in English language Wiktionary provides the most common usage, as an adjective meaning,

flabby, not firm

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