If you want to get some surface-level mind-blown experiences, then there are some cool pages out there, in addition to the papers I already linked, which have great diagrams. [0] shows a periodic table which relates categories to truth values and sets, which is helpful for understanding what folks mean by "category theory is just as good as set theory for foundational work".
More mind-blowingly, category theory is "formally formal", which means that we can use category theory to formalize its own concepts. This leads to n-category theory. Because of this, we have tables like [1] which relate category theory's own building blocks to type theory and logic.
Category theory was originally the study of natural transformations. Any time that you find enough data to define a natural transformation, then that's interesting; the component categories are then also interesting objects of study. That's still what motivates its presence in physics. There are at least two other perspectives though, a logical perspective (a category is a deductive system for a formal logic) and a spatial perspective (a category is a topological space), and they're both rich as well. But the richest perspective is going to come after you've explored all three.
[0] https://ncatlab.org/nlab/show/periodic+table
[1] https://ncatlab.org/nlab/show/computational+trinitarianism
Study the flatMap function if you would like to apply category theory to life. A monad is just an object with a sensible flatMap method.
So you're normal (half joking). Humor is a good medication for depression, and at least you can joke about it.
Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular.
On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations.
[0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu...
[1] https://en.wikipedia.org/wiki/Yoneda_lemma
[2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf
Also, EETC/HoTT are basically junk, as explained in detail by Harvey Friedman in various postings on the foundations of math mailing list. We already have a perfectly good foundation for mathematics and EETC/HoTT don't improve on it in any way (and in fact are worse in many respects).
I had to go find a blow-by-blow of the drama because it's good. Simpson cannot imagine topoi which don't implement standard set theory [3]. Friedman demonstrates a total misunderstanding of sets vs. categories [4]. McLarty and Feferman claim that categorical FOM make sense once one is used to categories [1]. Finally, the truth is laid bare: Friedman and Simpson simply don't agree with us on whether categorical logic is philosophically valid [2].
For more on this perspective, check out Pratt's take on Yoneda's lemma [0].
(An interesting aside: Simpson is an Objectivist who hates postmodernism! I wonder if this is part of what causes them to reject topos theory, where there are many different logics and collections, with a single barren plain Boolean set theory?)
[0] http://boole.stanford.edu/pub/yon.pdf
[1] https://cs.nyu.edu/pipermail/fom/1998-March/001309.html
[2] https://cs.nyu.edu/pipermail/fom/1998-March/001467.html
[3] https://cs.nyu.edu/pipermail/fom/1998-January/000782.html
[4] https://cs.nyu.edu/pipermail/fom/1998-January/000835.html
Moreover, you omit some of Friedman's best postings on this topic. The challenge is to show what advantages this proposed theory has over the standard foundations, or to demonstrate some interesting problem or conceptual issue that it resolves. I did not see any serious replies to this.
(Incidentally, the same challenge is my reply to the other comment to my previous post, about the ZX-calculus.)
Here's an analogy. Why do mathematicians consider groups interesting, but general Moufang loops not interesting? I propose the answer is that the additional generality provided by Moufang loops doesn't aid in addressing any interesting questions external to their theory. On the other hand, the utility of the group concept is obvious to anyone who studies a subject with connections to algebra.
This reasoning is roughly why the vast majority of mathematicians don't give two hoots about category theory, beyond picking up the few tricks like Yoneda's lemma that are actually useful. You can – as certain members of the category theory community have shown – take any mathematical concept, consider some generalization, and play various definition pushing games. But that's pointless without a good motivation (e.g. a concrete problem).
The previous two paragraphs concern "actual" mathematics, but similar remarks apply to foundations of mathematics. Friedman is pointing out that we already have a perfectly good foundational theory of mathematics and asking what gain we get by introducing categorical concepts.
Friedman repeatedly demonstrates [1][2] that he isn't interested in grokking why category theory is even a thing; he doesn't see e.g. Pratt's careful explanation that category theory is motivated by studying (natural) transformations. Simpson comes across as a spoiled brat [3] and Friedman comes across as a narcissist who needs to fuel himself by being the bastion of FOM [4]. Seriously, in [5], he has the audacity to simultaneously claim that Lawvere's foundations are the same as Friedman's set-theoretic foundations, and also to ask what a Lawvere theory/sketch is! Unbelievably rude.
I read the entire three-month slapfight again, just to double-check that I hadn't mis-remembered the general outline. Simpson wastes message after message being wrong about Boolean algebras vs. Boolean rings (they're the same picture) and doesn't understand how categorical dualities like Stone duality lead to equivalences. At no point do either of them manage to fully grok a 2-category or how ETCC could be a practical foundations. Tragesser says it well in [5] when he analogizes the entire affair to the sheep and their shop [6], going around and around and always changing the framing but never actually getting to the philosophical meat of the inquiry.
[0] https://ncatlab.org/nlab/show/periodic+table
[1] https://cs.nyu.edu/pipermail/fom/1998-February/001182.html
[2] https://cs.nyu.edu/pipermail/fom/1998-March/001303.html
[3] https://cs.nyu.edu/pipermail/fom/1998-February/001228.html
[4] https://cs.nyu.edu/pipermail/fom/1998-February/001234.html
[5] https://cs.nyu.edu/pipermail/fom/1998-February/001210.html
[6] Edit: I can link directly to the text! https://en.wikisource.org/wiki/Through_the_Looking-Glass,_an...
More to the point, what I asked for in my previous comment has also not been provided! What does the knowledge that Sets are "just 0-categories" buy me in concrete terms? Does it facilitate the proof of any theorems in set theory?
Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language. Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic).
There's certainly a community trying to bring a category-theoretic approach into other fields such as statistics, economics, or other areas of computer science, but it's too early to say it's been "quite a few years".
https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#...
[0] http://math.ucr.edu/home/baez/rosetta.pdf
[1] https://arxiv.org/abs/quant-ph/0404040