Nonetheless one should learn the language of topos: Grothendieck... - Colin McLarty [2018]

Grothendieck's 1973 topos lectures
Colin McLarty
3 mai 2018

In the summer of 1973 Grothendieck lectured on several subjects in Buffalo NY, and these lectures were recorded, including 33 hours on topos theory. The topos lectures were by far the most informal of the series, with the most significant audience discussion, and Grothendieck says they are the only ones for which he developed new ideas. While they do not contradict any of his published accounts (including Récoltes et semailles) they provide important new insights. For one, they reveal more of how he connected his idea of topos to his Tohoku paper on homological algebra; and also to Serre's isotrivial covers. The beginning lectures show Grothendieck's choice of what to emphasize in the subject. As he warms up in the following lectures he goes beyond SGA by offering a more directly, or naively, geometrical conception of a topos as a generalized topological space than occurs in SGA. Much of the series is spent on extending the ideas of algebraic structure, and classifying topos, in the 2-categorical setting. The lectures illustrate several aspects of what Grothendieck called "building houses", and why, despite the resistance he met, he said "we advise the reader nonetheless to assimilate the language of topos" (printed in SGA VII). At the end of the last topos lecture (July 13, 1973) he discusses the news that Deligne had completed the proof of the Weil Conjectures. This is before he learned how Deligne did it. Here he discusses the role of universes, and what could be desirable alternatives to them.

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