[tylerhou]

> I would like to learn category theory properly one day, at least to that kind of "advance undergraduate" level she mentions.

As someone who tried to learn category theory, and then did a mathematics degree, I think anyone who wants to properly learn category theory would benefit greatly from learning the surrounding mathematics first. The nontrivial examples in category theory come from group theory, ring theory, linear algebra, algebraic topology, etc.

For example, Set/Group/Ring have initial and final objects, but Field does not. Why? Really understanding requires at least some knowledge of ring/field theory.

What is an example of a nontrivial functor? The fundamental group is one. But appreciating the fundamental group requires ~3 semesters of math (analysis, topology, group theory, algebraic topology).

Why are opposite categories useful? They can greatly simplify arguments. For example, in linear algebra, it is easier to show that the row rank and column rank of a matrix are equal by showing that the dual/transpose operator is a functor from the opposite category.

[lambdas]

Agreed. In addition to yours, notions like limits/colimits, equalisers/coequalisers, kernels/cokernels, epi/monic will be very hard to grasp a motivation for without a breadth of mathematical experience in other areas.

Like learning a language by strictly the grammar and having 0 vocabulary.

[libraryofbabel]

I should have mentioned in my post that I have an applied math masters and a solid amount of analysis and linear algebra with some group theory, set theory, and a smattering of topology (although no algebraic topology). So, I'm not coming to this with nothing, although I don't have the very deep well of abstract algebra training that a pure mathematician coming to category theory would have.

Although, it feels like category theory _ought_ to be approachable without all those years of advanced training in those other areas of math. Set theory is, up to a point. But maybe that isn't true and you're restricted to trivial examples unless you know groups and rings and fields etc.?

[griffzhowl]

You could take a look at Topology: A Categorical Approach by Bradley, Bryson and Terilla.

It's a crisp, slim book, presenting topology categorically (so the title is appropriate). It both deepens the undergraduate-level understanding of topology and serves as an extended example of how category theory is actually used to clarify the conceptual structure of a mathematical field, so it's a way to see how the flesh is put on the bare bones of the categorical concepts.

It's also available for free online:

https://topology.mitpress.mit.edu/