So category theory is really the theory of composition of mappings. I conjecture that all programming can be seen as just the composition of mappings. If this is correct then category theory is a theory of programming.
So category theory is really the theory of composition of mappings. I conjecture that all programming can be seen as just the composition of mappings. If this is correct then category theory is a theory of programming.
It can very much be. Here’s one example of this phenomenon (there are many others but this is the most famous): https://wiki.haskell.org/Curry-Howard-Lambek_correspondence
You don't need category theory to connect dots with arrows, graph theory is enough for this.
Category theory is actually a ‘simplified’ graph theory, i.e. you can see categories as a restricted class of graphs. E.G. ‘Category Theory for Computing Science’ introduces categories this way (a category is a directed graph with associative composition and identity; the free category on a graph is the graph with all identities and compositions filled in). But the restrictions (associative composition and identity) are harmless and natural for programming applications where there's always a notion of ‘do nothing’ or ‘do one thing after another’, and unlock a lot of higher structure.
But what's the utility of this definition? Does it help solve or prove something?
It helps you build an intuition for categories, if you're used to graphs :)
If you have a working intuition for categories then in most cases the specific formulation you choose as a foundation doesn't matter, just as most mathematicians work nominally in set theory without worrying about the subtleties of ZFC.
IMO the right intuition about a tool comes from applying it in the context where it provides a real leverage. In case of Category Theory that would be advanced algebraic topology (not re-phrasing basic things which are easier to understand without CT).
If you allowed infinite graphs maybe. How would you define a functor or natural transformation in graph theory? Seems like you would need to construct a conceptual system that is just equivalent to category theory
No, but if you want to talk about composing those arrows (and a sensible notion of composition should probably be associative, and perhaps have a unit) you eventually end up reinventing category theory.