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The peculiar agricultural terminology commonly used in algebraic geometry and category theory, "sheaf", "stalk", "germ", is quite well-known. A sheaf is pictured as something like a bundle of stalks, in which reside germs. Very roughly and intuitively, a germ is a localized datum capable of being developed or extended to a function.

It is commonly noted that Jean Leray introduced sheaves (and also spectral sequences) while he was interned as a prisoner of war (in Austria during World War II). In French, the terms "faisceau, fibre, germe" are used, with faisceau making its first appearance in Leray. The term "fiber" is another well-known term in mathematical English, as in "fiber bundle", etc. -- such terms are conceptual neighbors of the sheaf-theoretic terms above, although I suppose the need was felt to translate this sense of the French fibre into something other than "fiber" (for example, the fiber of a vector bundle over a point of the base space is something different from the stalk (over the same point) of the corresponding sheaf of modules). Hence the English "stalk".

I am interested in knowing more about the provenance and etymology of the mathematical terms. Here is the precise question:

Are there any sources that attest to why Leray chose faisceau? Relatedly, are there earlier sources (predating Leray) which use for example the term "germ" (or any of its cognates in other languages)?

(I must confess that I haven't looked at Leray's seminal papers. I'm pretty sure I'm not alone in this, as many mathematicians from that time commented and complained about how obscure Leray's presentations were, and everyone nowadays seems to learn sheaf theory and spectral sequences from other sources.)

I might as well share my hunch or private folk etymology which leads me to ask this question: it was "germ/germe" that came first, with other terms like "faisceau/sheaf" and "fibre/stalk" later being built around it, and that "germ" itself might be rooted in the theory of Riemann surfaces, with later carry-over into the modern foundations of differentiable manifolds and algebraic varieties. The idea is that starting with a germ of an analytic function, i.e., its local behavior in arbitrarily small neighborhoods of a point, there germinates by analytic continuation a development into a maximal connected Riemann surface in which the analytic function naturally "lives" (possibly as a multi-valued meromorphic function). I think I probably first got this idea from Chapter 8 of Complex Analysis by Ahlfors.

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    $\begingroup$ I suggest moving this interesting question to MO where it is more likely to be answered. $\endgroup$ – Alexandre Eremenko May 18 at 21:15
  • $\begingroup$ @AlexandreEremenko I'll certainly consider asking for migration, if there is no action in a few days. Thanks for the support. $\endgroup$ – Todd Trimble May 18 at 21:55
  • $\begingroup$ I was also always puzzled with this terminology, which seems somewhat counter-intuitive to me (especially the use of "sections" and "fiber"). I discussed it with several mathematicians but no one could explain its origin. What kind of plant did Leray really have in mind? $\endgroup$ – Alexandre Eremenko May 19 at 0:49
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Todd and Alexandre, I just want to offer a perspective to your discussion from a German language point of view. I wanted to put it as a comment, but it was getting too long so I am putting it as an answer (recognizing it does not exactly answer your question). But just wanted to offer an additional perspective.

If you translate sheaf-stalk-germ into German, you get as a rather general translation Bündel-Stiel-Keim.

But if you relate it more strictly to an agricultural setting, the words that are used there are actually Garbe-Halm-Korn. (by the way in German mathematics texts, sheaf is commonly translated as Garbe, i.e. really taking it from the agricultural setting)

Now here is the interesting thing, and maybe this helps a bit for your question. If you take these agricultural expressions Garbe-Halm-Korn and translate those into English, you get the following:

Garbe – sheaf, burst of fire

Halm –culm, blade (of grass), stalk

Korn – grain, seed, corn; also: grit

Maybe you find something there that connects well with the intuition behind mathematical sheaves? I am not an expert in that field, so cannot really judge, just wanted to offer.

“burst of fire” certainly does not seem to fit the agricultural setting, like grit, and obviously comes from a different context.

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    $\begingroup$ Thanks, Claus. "Garbe" and "Korn" and words I've seen (and come to think of it, "Garbe" looks similar to the French "gerbe" which is another mathematical term in the same general area). Interestingly in English, "sheaf" can also refer to a sheaf of paper consisting of many individual sheets, which could be fancifully likened to sections of a mathematical sheaf. As far as I know, this second meaning doesn't attach to the French term. $\endgroup$ – Todd Trimble May 22 at 12:01

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