[tristramb]

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.

[ihm]

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

[sesm]

You don't need category theory to connect dots with arrows, graph theory is enough for this.

[Twey]

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.

[griffzhowl]

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

[xanderlewis]

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.