Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here: http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf Allyn Jackson writes about how Mumford was profoundly impressed: Mumford ...