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category theory

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I'd say category theory informs math in two main ways. First it is essentially the "language" in which many areas of advanced math are written. In algebraic geometry we could define sheaves without using terms from category theory but when we use category theory everything becomes much clearer. Similarly when talking about morphisms of sheaves or adjunctions, everything become clearer when we use the language of category theory. The same principle holds in algebraic topology, only more so. We couldn't really have done most of the past 60+ years of algebraic topology research without having category theory to do it with. Second, categories are interesting in their own right, and tie into other areas of math. For instance we can take the classifying space of a category to get a topological space, and this connection turns out to come up a lot in homotopy theory. For another example, the algebraic K-theory of a ring is actually about the category of finitely generated projective modules over that ring. So we can study algebraic K-theory of other categories, and do it using category-theoretic tools.
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In the Algebraic Geometry course I took, we borrowed heavily from category theory.
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Abelian categories (a special type of categories introduced by Buchsbaum and Grothendieck) allowed Grothendieck to reformulate many Homology/Cohomology theories (such as Groups and Sheaves H/C) in a unified manner.
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At the risk of starting a flame war, be careful.

Category theory is a language and its potential for expressiveness is immense, but it’s easy to lose sight of mathematical substance when you end up wrapped in categorical abstraction. People like to call this “abstract nonsense” but this is a strawman. In my opinion, a better term for this is “abstractophilia.”

To paraphrase a quote I heard secondhand due to a well known geometer who I won’t name: “you can’t just abstract away a difficult and subtle problem.”

It seems that many younger students of mathematics don’t appreciate how true this can be.

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