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Quoting Wikipedia:

The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations.

It is clear now that Algebraic Geometry has benefitted a lot from the language provided by Category Theory, but, historically, what were the complications when people tried to formulate A.G. over the language of set theory?

Why did we require Category Theory?

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    $\begingroup$ This is deeply misleading. The modern formulations of algebraic geometry are in set theory, at least as much as any other part of math is. So is category theory, for that matter. The historical problems with formulating algebraic geometry have nothing to do with trying to do it in set theory. $\endgroup$ – Eric Wofsey Aug 15 '17 at 6:34
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    $\begingroup$ It's not so much the use of sets, but the methods and notions that come along with it. For example, set theory teaches us that the notion "X is a subset of Y" is a yes-no proposition, whereas category theory teaches us the notion is about monomorphisms, and isn't very propositional at all in nature (except in special cases). $\endgroup$ – Hurkyl Aug 15 '17 at 10:04
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    $\begingroup$ Wait, are you quoting en.wikipedia.org/wiki/Category_theory#Historical_notes? The current text on that page has a full stop after "algebraic geometry". There is nothing about "the field most resistant..." Did someone in this comment thread take that part out? $\endgroup$ – Nefertiti Aug 15 '17 at 11:17
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    $\begingroup$ @Nefertiti: Here is a link to the previous revision which has the quoted text. It was removed in an anonymous edit with the comment "remove unsourced claim of dubious relevance". $\endgroup$ – Hurkyl Aug 15 '17 at 11:41
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    $\begingroup$ I think the point that this writer is trying to get at is that algebraic geometry as formulated rigorously in Weil's Foundations, which I believe is precategorical, is a tremendous pain to wrap ones head around. Just having access to words like "category" and "functor" allowed Serre, Grothendieck, and the rest to say things much more cleanly. In other words, it's a psychological thing. $\endgroup$ – Tabes Bridges Aug 15 '17 at 18:58

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