Here is what confuses me: from before, I thought morphisms were “just” arrows between objects, with a specific identity.
But in the case of functions, we have to smuggle in the set of ordered pairs that define them. Do you simply equate the identity of a function with this set definition?
That might be fine, but it means there needs to be some kind of … semantics? that gives us the “meaning” (~ implementation) of composition based on the “meaning” (the set of ordered pairs) of the composed morphisms.
You raise a good point. Think of category theory as a language for expressing, in this case, the logic of sets and functions. You still need to know what that logic is. Then you can use category theory to work efficiently with that logic owing to its general-abstract nature.
Here is what confuses me: from before, I thought morphisms were “just” arrows between objects, with a specific identity.
But in the case of functions, we have to smuggle in the set of ordered pairs that define them. Do you simply equate the identity of a function with this set definition?
That might be fine, but it means there needs to be some kind of … semantics? that gives us the “meaning” (~ implementation) of composition based on the “meaning” (the set of ordered pairs) of the composed morphisms.
Am I right here?
You raise a good point. Think of category theory as a language for expressing, in this case, the logic of sets and functions. You still need to know what that logic is. Then you can use category theory to work efficiently with that logic owing to its general-abstract nature.
That would the ambush part?