# Why are sheaves called sheaves?

By "why" I mean: what were the stated intentions of the namers? More generally, how did the theme of agricultural terminology in algebraic geometry come about?

• Didn't Jean Leray invent this? – Lord Shark the Unknown Sep 20 '18 at 4:15
• The term was introduced by Leray in French as "faisceau." I don't know an etymology more convincing than a sheaf being a collection of stalks. – Qiaochu Yuan Sep 20 '18 at 4:33
• If you google image search for "sheaf of grain", you can see that the object in question is a bunch of stalks of grain held together. Suggestive, no? – KReiser Sep 20 '18 at 4:44
• @KReiser But in English sheaf could also be laying things down horizontally one on top of another, like sheaf of papers, so both the skyscraper sheaf and the constant sheaf are there. – user10354138 Sep 20 '18 at 6:19
• Would you prefer "shivs"? – Asaf Karagila Sep 20 '18 at 6:25

Tom Lovering has some great notes about sheaves where he explains this. It's essentially what they've told you above. You can find them here: https://tlovering.wordpress.com/2011/04/17/sheaf-theory-essay/

[banter]

Let me make a few additional comments about some ideas I find rather amusing. The words bundle, as in vector/principal/fibre bundle, and sheaf are both used for the same reason: they denote some sort of packing of stuff. You could argue that this is rather unfortunate, since every bundle has an associated sheaf of sections, and sometimes you can build bundles from sheaves.

This confusion is not exclusive to English. In Spanish, and more particularly in some places of South America (AFAIK), they use the same word ("haz") to refer to bundles and sheaves! Just imagine! Everywhere else I can think about the word "haz" is reserved for bundles and the word "gavilla" is used for sheaves. Both these words are of agricultural origin and denote the same concept as in English: a grouping or packing of things.

One last remark: Funnily enough the Spanish word fasces seems to be directly related to the French faisceaux and the Portuguese faisceau. However, I've never seen it used in a mathematical context. However, if my hypothesis is true and all these words have the same origin, all of them would be directly related to a certain "form of radical authoritarian ultranationalism". (Wikipedia quote).

[/banter]

• "Faisceaux" and the likes of it come from.the latin word "fascis", which is a sort of weapon that was specifically reserved to lictors, who were, in Ancient Rome, gards of high magistrates; when Mussolini used similar symbols, it was associated to his "movement" and so the word fascis was associated to it, which led to "facism", which is indeed a "form of radical authoritarain ultranationalism" – Max Sep 20 '18 at 7:33
• I think it's the other way around, Max. The word fascis comes from bundle, and the lictors used a fasces (see Wikipedia) as their symbol. – user347489 Sep 20 '18 at 7:50
• In the French wikipedia (fr.m.wiktionary.org/wiki/fascis#la) they seem to say that lictors indeed used a fascis (whose plural is fasces) – Max Sep 20 '18 at 7:53