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
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