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Motivation: In any history, there is a cause-and-effect relationship. So I became curious about the situation in which the sheaf theory came to appear. In other words, I'm curious about what problem was left unsolved and led to the emergence of sheaf as a solution. Or I wonder if sheaf emerged as a more minor concept in the progress of researching a certain concept.

Addition: I know that "Cousin problem" that Kiyoshi Oka contributed is related to sheaf theory. This thinking is right?

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    $\begingroup$ You are half-right. Sheaf theory had two roots, Cartan's 1940-44 work on the Cousin problems (anticipated by Oka), and Leray's 1946 cohomological approach to PDE. As Chorlay puts it in Emergence of the Sheaf Concept:"In spite of Cartan's involvement, the sheaf cohomology theory of 1945-1950 was developed independently of Cartan's 1940-1944 structural approach... it can be argued that the unexpected merger of the two research lines, in the work of Cartan and Serre in the early 1950s, would be the real take-off point for sheaf theory as we know it." $\endgroup$
    – Conifold
    Dec 26, 2023 at 5:32
  • $\begingroup$ @Conifold Thanks to you, my scattered concepts came together well! $\endgroup$
    – pokssin
    Dec 26, 2023 at 7:36

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Sheaves were created by Leray to avoid doing anything resembling applied mathematics while he was in a prisoner of war camp during World War II, as he was an expert on PDEs and did not want this to become known. Hence he turned his attention to algebraic topology.

For more details, see the notes by Haynes Miller here.

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  • $\begingroup$ And, although perhaps not directly causally connected (due to the low connectivity of the world in those years!), yes, Oka's "ideals of indefinite domain", and Cousin problem work, and other several-complex-variable stuff, could/can be nicely discussed in terms of sheaves. $\endgroup$ Dec 22, 2023 at 23:52
  • $\begingroup$ Looking at the paper in your answer, the question was solved. Thanks for your help! $\endgroup$
    – pokssin
    Dec 23, 2023 at 0:53
  • $\begingroup$ @paulgarrett Thanks! Your answer was helpful to me! $\endgroup$
    – pokssin
    Dec 23, 2023 at 0:54
  • $\begingroup$ @pokssin ... :) Good. :) $\endgroup$ Dec 23, 2023 at 2:41

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