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topoi.
Last seen on: The Guardian – Cryptic Crossword No 28,583 – Oct 22 2021
Random information on the term “topoi”:
This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.[citation needed]
During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of étale cohomology.
With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems for a long time. The first was to do with its points: back in the days of projective geometry it was clear that the absence of ‘enough’ points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold). There was also the difficulty, that was clear as soon as topology took form in the first half of the twentieth century, that the topology of algebraic varieties had ‘too few’ open sets.