I would recommend not learning category theory until you’ve seen enough concrete examples to be able motivate its study properly — at the very least one course in group theory, one in linear algebra, and one in general point-set topology.
The professor of the short algebraic topology course I took during my second year refused to give us the definition of functor, saying that it is better to start using them and building up interesting examples before looking at the abstract definitions. I remember that at the time I tough that this was quite stupid, but now I actually agree with him. I was happy to learn the basics of category theory the following year having more examples in mind. Most of the concepts of category theory are extremely natural but to realize that you need a good background. Otherwise you will most likely struggle against the abstractness with few chances to understand what’s truly going on.
Well, yeah, any run-of-the-mill category theory textbook will of course load you down with examples. That doesn’t mean they’ll give you the background instruction necessary to understand those examples. It’s all very well being told that the classic example of a non-concretizable category is the category of topological spaces and homotopy classes of continuous maps between them—if you’ve never taken a topology course, you won’t have any idea what that means, and the book isn’t going to include a beginner’s topology textbook as a footnote.
An example isn’t being told something like that, it’s being shown something like that, with diagrams. A beginner’s topology course is not required, the diagrams are.
I’m probably at the mathematically naive level that the linked post warns against, and after looking at many of the examples with diagrams in the various category theory textbooks, I still have basically no idea what CT brings to the table or how I should use it. It unifies formal proofs, topological computations and quantum mechanical systems? Great! Except I don’t know how to grind proofs, derive topology or compute quantum mechanics by hand, so I have little idea what that means in practice.
A lot of people seem to describe learning Haskell monads as a similar experience. Most of the examples are basically incomprehensible until you just work with the raw formalism from sufficiently many angles that you start to build the necessary headspace to work it into something useful. Maybe studying advanced topology and abstract algebra will get you familiar with working with sufficiently similar formal structures that you can actually get significant bits of category theory by analogy from something as short as a textbook example
This comment makes a fair point that people might be missing at first glace. It’s true, as IainM said, that many of the examples will be of the form “Remember that concept from Abstract Algebra of a free group generated by a set? Remember also the concept of the underlying set of elements in a group? Turns out that those concepts are an example of an adjoint pair of functors.” If you don’t have any recollection of the motivating concepts to draw upon, those examples won’t be of much help to you.
However, any category theory text will give you some example without assuming any prior familiarity. For example, preorders and posets, groups (defined as single-object categories in which all morphisms are invertible) and groupoids, plus the constructions that can be built out of given categories, such as comma categories, functor categories, and products of categories.
That said, if you don’t have some prior familiarity with Set Theory and the set-theoretical definition of functions, you will have a rough time. Most authors I’ve seen want to be able to invoke Set as an example of a category without having to explain to you what a set is or what functions between sets are. (Though, Goldblatt is an exception. He tries to teach the reader set theory before moving on to the category theory.)
This seems like exactly the sort of thing I’d expect to be in a good category theory textbook. If I flip open, say, a physics textbook, I don’t see unmotivated derivation of laws, I see “here’s what a pendulum, a buoy in rough waters, and a mass on a spring have in common,” for example.
Part of the point of learning from a textbook, instead of the original papers, is so you can get that motivation and learn those concepts in a natural way. That’s what a good textbook means, I’d say.
Mathematics StackExchange on learning category theory:
Yes, it’s not as if the textbooks will give you any examples.
Well, yeah, any run-of-the-mill category theory textbook will of course load you down with examples. That doesn’t mean they’ll give you the background instruction necessary to understand those examples. It’s all very well being told that the classic example of a non-concretizable category is the category of topological spaces and homotopy classes of continuous maps between them—if you’ve never taken a topology course, you won’t have any idea what that means, and the book isn’t going to include a beginner’s topology textbook as a footnote.
An example isn’t being told something like that, it’s being shown something like that, with diagrams. A beginner’s topology course is not required, the diagrams are.
I’m probably at the mathematically naive level that the linked post warns against, and after looking at many of the examples with diagrams in the various category theory textbooks, I still have basically no idea what CT brings to the table or how I should use it. It unifies formal proofs, topological computations and quantum mechanical systems? Great! Except I don’t know how to grind proofs, derive topology or compute quantum mechanics by hand, so I have little idea what that means in practice.
A lot of people seem to describe learning Haskell monads as a similar experience. Most of the examples are basically incomprehensible until you just work with the raw formalism from sufficiently many angles that you start to build the necessary headspace to work it into something useful. Maybe studying advanced topology and abstract algebra will get you familiar with working with sufficiently similar formal structures that you can actually get significant bits of category theory by analogy from something as short as a textbook example
Sarcasm is nearly always inadvisable!
This comment makes a fair point that people might be missing at first glace. It’s true, as IainM said, that many of the examples will be of the form “Remember that concept from Abstract Algebra of a free group generated by a set? Remember also the concept of the underlying set of elements in a group? Turns out that those concepts are an example of an adjoint pair of functors.” If you don’t have any recollection of the motivating concepts to draw upon, those examples won’t be of much help to you.
However, any category theory text will give you some example without assuming any prior familiarity. For example, preorders and posets, groups (defined as single-object categories in which all morphisms are invertible) and groupoids, plus the constructions that can be built out of given categories, such as comma categories, functor categories, and products of categories.
That said, if you don’t have some prior familiarity with Set Theory and the set-theoretical definition of functions, you will have a rough time. Most authors I’ve seen want to be able to invoke Set as an example of a category without having to explain to you what a set is or what functions between sets are. (Though, Goldblatt is an exception. He tries to teach the reader set theory before moving on to the category theory.)
This seems like exactly the sort of thing I’d expect to be in a good category theory textbook. If I flip open, say, a physics textbook, I don’t see unmotivated derivation of laws, I see “here’s what a pendulum, a buoy in rough waters, and a mass on a spring have in common,” for example.
Part of the point of learning from a textbook, instead of the original papers, is so you can get that motivation and learn those concepts in a natural way. That’s what a good textbook means, I’d say.