Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft introduction to algebraic topology. But I don't know any references in category theory. Can someone give some idea for what book or lecture notes I can get a introduction on this topic?

Thank you so much in any advance!

Solution 1:

I found Category Theory by Steve Awodey to be a really good introduction to the subject. It doesn't lead directly into algebraic geometry (or, indeed, in any particular direction), but gives a readable introduction to the subject with plenty of examples and problems. When you've worked through this you could move on to a more advanced text or one which is more specifically geared towards algebraic geometry.

On the side, I'll mention Ravi Vakil's epic notes on algebraic geometry which kick off with category theory before moving on.

Solution 2:

I think the best way to learn the basics of category theory is to see it in action. For this reason, I recommend Rotman's An Introduction to Homological Algebra, but you have to be algebra-inclined.

Solution 3:

You can also take a look at this wonderful set of video lectures by Eugenia Cheng

Solution 4:

You could try Gelfand and Manin's Methods of Homological Algebra - it introduces a lot of category theory (the whole thing is really an intro to derived categories) from scratch, and homological algebra is a big part of algebraic geometry. Their examples are more topological than geometric though.