Category theory for a set/model theorist

manpreet Best Answer 2 years ago

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 assumptionson 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.