What branch of Algebra has the most category theory in it?

Question

I like both category theory and algebra so I am wondering in your opinion what subject maximizes their intersection.

Answer 1

Algebra is a broad term, so it is a bit difficult to answer that question. But let me try to connect some sub-subjects of category theory with the algebraic domain they describe (or even sometimes were created for).

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Abelian categories. They are the realm in which most of homological algebra and topological algebra takes place.

Grothendieck toposes. Those are the categories of sheaves appearing in algebraic geometry which, in its current form, heavily relies on category theory.

Lawvere theories. They are the categorical point of view on universal algebra. Going from there to full model theory will take you to elementary toposes.

Monoidal categories. They appears a lot in theoretical quantum physics, because it is a sort of generalized linear algebra. They support a very handy "visual language" (called string diagram) that mathematician-physicist are fan of because it takes the brac-cket notation to the next level. Keywords here are Hopf algebras, Froebenius algebras, etc.

$\infty$-categories. Those are categories with higher morphisms, meaning that you don't just have objects and morphisms between those, but also morphisms between morphisms, and morphisms between those, and morphisms between those ... all the way to the top ! This is where homotopical algebra happens. Category-keywords here are model categories, simplicial categories, etc.

Categories. Last but not least, categories themselves can be studied as algebras of a (essentially) algebraic theory ! Going further in that direction will eventually lead you to enriched category theory.

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Of course, this list is very far from exhaustive and is more an invitation for you to research a little those fields and see if they please you.