Category theory for a set/model theorist

Question

I am looking for a book or other reference which develops category theory 'from the ground up' assuming a healthy background in set and model theory, not one in homological algebra or Galois theory etc.

Up until now I have been satisfied by the heuristic that most of what takes place in category theory can be translated into model theoretic terms and vice-verse, and because my research is not directly related to either and model theory has always felt like breathing to me I have always just studied model theory.

This casual perspective is no longer sufficient; my current research is into field extensions, specifically how to canonically and 'internally' extend the field of fractions of the Grothendieck ring of the ordinals into the Surreal numbers. To this end I have purchased 'Galois theories' by Francis Borceux and George Janelidze, and from the preface it looks like chapter seven is what I need since none of the extensions I care about are Galois (they fail to be algebraic or normal), and they promise a Galois theorem in the general context of descent theory for field extensions without any Galois assumptions on the extensions.

I believe I still have loads of descent happening so this is right up my alley, but they give the comment that "The price to pay [for dropping these assumptions] is that the Galois group or the Galois groupoid must now be replaced by the more general notion of a precategory". This, along with other comments like 'pulling back along this morphism in the dual category of rings yields a monadic functor between the corresponding slice categories' and many others which clearly have some precise meaning that is completely hidden to me have convinced me that it's time to start seriously learning some category theory.

Are there any good references on 'category theory in the large' for someone with a decent background in set/model theory? I enjoy thinking about 'large scale' mathematics and have always sensed that 'pure' category theory was exactly this, so a reference with an eventual eye towards higher category theory would be appreciated.

Answer 1

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.

Answer 2

Tim gave a very good answer. I shall add two references.

1. Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation.
2. Suggested Readings.