The main reason I like the paths formulation is that I can look around at the real world and immediately pick out systems which might make sense to model as categories. “Paths in graphs” is something I can recognize at a glance. Thinking about links on the internet? The morphisms are paths of links. Friend connections on facebook? The morphisms are friend-of-friend paths. Plane flights? The morphisms are travel plans. Etc.
I expect that the big applications of category theory will eventually come from something besides sets and functions, and I want to be able to recognize such applications at a glance when opportunity comes knocking.
The cost is needing to build some intuition for path equivalence. I’m still building that intuition, and that is indeed what tripped me up. It will come with practice.
The main reason I like the paths formulation is that I can look around at the real world and immediately pick out systems which might make sense to model as categories. “Paths in graphs” is something I can recognize at a glance. Thinking about links on the internet? The morphisms are paths of links. Friend connections on facebook? The morphisms are friend-of-friend paths. Plane flights? The morphisms are travel plans. Etc.
I expect that the big applications of category theory will eventually come from something besides sets and functions, and I want to be able to recognize such applications at a glance when opportunity comes knocking.
The cost is needing to build some intuition for path equivalence. I’m still building that intuition, and that is indeed what tripped me up. It will come with practice.