Category theory for a set/model theorist

See here for a more comprehensive list with a slightly different audience in mind.

• Books aimed at various sorts of students:

Awodey, Leinster, Riehl, van Oosten. Many others, I think.

Of these, Awodey has the most logical bent.

• Books with a structural bent:

• The Joy of Cats (if you're looking to see category theory as an organizing principle. Just don't get too hung up on some of the idiosyncratic terminology in there. The technical tools developed here are not very standard. Should be supplemented with a more standard text.)

• If you really want to go hard-core, though, you might eventually want to read Makkai and Pare's Accessible Categories: The Foundations of Categorical Model Theory or Adamek and Rosicky's Locally Presentable and Accessible Categories. That's where the study of category theory in the large really has some depth to it. But you probably want to go through the basics with another book first.

• Parts of SGA 4 can be read as a category theory text, which covers some really good material. Probably better for a second book, though.

• The classics:

• Mac Lane's Categories for the Working Mathematician is specifically aimed at mature mathematicians who are familiar with examples from different parts of mathematics. So as far as combining the aims of learning basic category theory and doing category theory "in the large", this may be your best bet.

• Borceux's Handbook of Categorical Algebra would probably not be a bad place to learn from. But you shouldn't go straight through -- see the syllabus below for a guide to how to skip around.

Then there are books of a more specifically categorical - logic bent, but I suspect that's not what you're interested in.

Syllabus: I think most would agree that a basic introduction to category theory should discuss categories, functors, natural transformations, equivalence of categories, limits and colimits, adjunctions, and monads. The main theorems which should be covered are the Yoneda lemma, the adjoint functor theorem, and the Beck monadicity theorem. Optionally, one should learn something about Grothendieck fibrations, and of course there are lots of other topics which could be added here.

After that, to me the most important part of category theory in the large is the study of locally presentable and accessible categories. As a set / model theorist, you might be particularly interested with the connections to AECs.