I have been studying category theory for the past couple of months and I'm struggling to understand what's the point?
I'm going to build up an intuition from what I know to understand if I am on the right path.
- Historically, Category Theory was created to answer some question(s) in the field of Topology - which deals with questions about properties preserved under deformations (to keep it simple). Hence, it's understandable to have function composability be a first class citizen in category theory. That is, it's intuitive (for me) to appreciate the reasoning behind the creation of category theory and why composability matters from a non-mathematical perspective.
- Assuming one understands the various constructs like functors, natural transformations etc., what does on do when viewing/modeling the problem from a category-theory POV?
- Let's say I correctly identified and created a category $C$ in my domain (which may be outside of mathematics). What's my next step? Is the idea to find a connection with another category $D$ that one may uncover - (either within my domain or outside)? By connection I mean the ability to transform $C \rightarrow D$. The "question" I then answer is the insight of this transformationary connection between possibly disparate concepts, which is illuminating by itself.
- Let's say I don't have (or find) a category $D$. Then what? Does one play around with category $C$ to see where things lead and thus develop further insight about it? The act of playing will surely involve transformationary thinking from above AFAIK.
Given the above my understanding seems to be this:
Category theory offers a framework for mathematical reasoning via the lens of transformations to prove (or disprove) whatever that we set out to do. Instead of "deductive logic" that one sees in elementary geometry (or most of math IMO) you use transformationary reasoning to allow you to go from $a$ to $b$. Thus, it's an alternative framework in which to do mathematical reasoning, however, at a higher level between other mathematical constructs vs. proving one of Euclid's theorems (say).
This is my source of confusion - I have no idea if I'm right or wrong and even thinking about it the right way or wildly off. All books/resources introduce category theory "bottom up" i.e., laying out all the concepts, terminologies, definitions and providing information on how to use the framework, for what exactly is completely lost IMHO.
What are some questions I can see answered via category theory, preferably in a step-by-step fashion to help appreciate the insight to be gained from this POV and how it's different (or perhaps better) than what we have? Bonus points if the question to be answered is laid out at the start and then we transform the problem into a categorical language and do some reasoning there and transform the insight back into the language of the problem domain. That would be wonderful!
I've read books by Spivak and I only see him apply constructions but haven't found anything that does what I'm looking for.
This is by no mean a satisfactory answer to your question as I am not an expert. I am just going write how I feel about category theory.
According to my understanding, category theory has not designed to answer questions in any particular subject, but to model the abstract mathematics itself. It give us a bird-eye view of all mathematical theories and show us previously unseen/uncleared connections between different mathematical theories. Sometimes it reveals hidden structures that we haven't seen before and simplify existing theories. For example: Groups, Lie groups, algebraic group, Abelian groups, simplicial group, $2$-groups, Hopf algebras, etc are group objects in different categories. This understanding doesn't help us to prove particular properties of each structure, but captures all these different notions into a single definition. This inspire us to think about previously unexplored group (and cogroup) objects in nice categories like group objects in category of Simple Graphs.
Moreover, category theory can think of as the theory of transporting data between different mathematical disciplines. For example homology, homotopy, linear representations of groups and many more and functors, and same is true for natural transformations (I am pretty sure that you have heard the story of origin of category theory). This is a nice platform to understand duality theorems (see here and here) and generalized spaces. Two of the well known duality results are the Stone duality, and Gelfand duality. Both ultimately led mathematicians to discover new generalizations for topological spaces called locales and noncommutative spaces. Affine schemes is a generalization of algebraic varieties and, smooth loci is a generalization of smooth manifolds both obtain via dualities. In fact, mathematicians are still exploring nice duality results between analytic spaces and algebraic spaces. Another interesting direction in the generalization theme is (co)completing categories by freely adding limits and colimits, which is called yoneda completion.
Beside above mentioned applications, category theory itself has few deep theorems like the general adjoint functor theorem, Yoneda lemma, Brown's representability theorem. And consequences of these theorems uncover deep results in all over mathematics.