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?
Why are sheaves called sheaves
algebraic-geometrymath-historyterminology
Related Solutions
There is a tradition on early modern mathematics regarding the usage of the term analysis :
François Viète, Isagoge in Artem Analyticem (Introduction to the Analytic Art), Tours, 1591 (several successive editions and translations);
Thomas Harriot, Artis Analyticae Praxis, London, 1631.
The background is the "rediscovery" of ancient Greek mathematics and, in particular of Pappus of Alexandria, (c.A.D. 290 – c.350) and his main work in eight books titled Synagoge or Collection, which Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.
See Henk Bos, Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction (2001), page :
Two kinds of analysis were distinguished in early modern geometry: the classical and the algebraic. The former method was known from examples in classical mathematical texts in which the constructions of problems were preceded by an argument referred to as "analysis;" in those cases the constructions were called "synthesis".
Reference to Pappus' problems is also found into René Descartes' La Géométrie (1637).
The two main line to be understood are :
analysis as a "method" to solve problem
analysis as the technique of treating geometrical problems with algebraic methods.
Both, I think, are "involved" into the use of analysis to name the new method introduced by Newton and Leibniz.
You can see :
Jaakko Hintikka & U.Remes, The Method of Analysis: Its Geometrical Origin and Its General Significance (1974)
and :
Michael Otte & Marco Panza, Analysis and Synthesis in Mathematics: History and Philosophy (1997).
All Serre needed to use was quasicoherent sheaves. For these it's a fundamental theorem that Cech cohomology is the "correct" cohomology. That means that it agrees with the cohomology defined abstractly, via derived functors. Since derived functors give rise to long exact sequences, the fundamental theorem implies that short exact sequences of (quasi)coherent sheaves yield long exact sequences in Cech cohomology. This just isn't true for coarser abelian categories like arbitrary sheaves or presheaves.
For an idea of why the theorem is true for quasicoherent sheaves, it's generally true that Cech cohomology becomes the right cohomology when the space has a "good cover," which in this case means a cover whose elements have no higher cohomology for whatever sheaf we're investigating. This is always true for quasicoherent sheaves by a reduction to affine varieties and then commutative algebra, which isn't possible for general sheaves since there may be no cover on which the sheaf is that associated to a module over a ring.
Best Answer
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).
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