I’m really not convinced by this framing in terms of “objects doing things to other objects”.
Let’s take a typical example of a morphism: let’s say f:Z>0→R (note for non-mathematicians: that is, f is a function that takes a positive integer and gives you a real number) given by f(n)=√n. How is it helpful to think about this as Z>0 doing something to R? How is it even slightly like “Alice pushes Bob”? You say “Every model is ultimately found in how one object changes another object”—are you saying here that the integers change the real numbers? Or vice versa? (After that’s done, what have the integers or the real numbers become?)
The only thing here that looks to me like something changing something else is that f (the morphism, not either of the objects) kinda-sorta “changes” an individual positive integer to which it’s applied (an element of one of the objects, again not either of the objects) by replacing it with its square root.
But even that much isn’t true for many morphisms, because they aren’t all functions and the objects of a category don’t always have elements to “change”. For instance, there’s a category whose objects are the positive integers and which has a single morphism from x to y if and only if x≤y; when we observe that 5≤9, is 5 changing 9? or 9 changing 5? No, nothing is changing anything else here.
So far as I can see, the only actual analogy here is with the bare syntactic structure: you can take “A pushes B” and “A has a morphism f to B” and match the pieces up. But the match isn’t very good—the second of those is a really unnatural way of writing it, and really you’d say “f is a morphism from A to B”, and the things you can do with morphisms and the things you can do with sentences don’t have much to do with one another. (You can say “A pushes B with a stick”, and “A will push B”, and so forth, and there are no obvious category-theoretic analogues of these; there’s nothing grammatical that really corresponds to composition of morphisms; if A pushes B and B eats C, there really isn’t any way other than that to describe the relationship between A and C, and indeed most of us wouldn’t consider there to be any relationship worth mentioning between A and C in this situation.)
I’m really not convinced by this framing in terms of “objects doing things to other objects”.
Let’s take a typical example of a morphism: let’s say f:Z>0→R (note for non-mathematicians: that is, f is a function that takes a positive integer and gives you a real number) given by f(n)=√n. How is it helpful to think about this as Z>0 doing something to R? How is it even slightly like “Alice pushes Bob”? You say “Every model is ultimately found in how one object changes another object”—are you saying here that the integers change the real numbers? Or vice versa? (After that’s done, what have the integers or the real numbers become?)
The only thing here that looks to me like something changing something else is that f (the morphism, not either of the objects) kinda-sorta “changes” an individual positive integer to which it’s applied (an element of one of the objects, again not either of the objects) by replacing it with its square root.
But even that much isn’t true for many morphisms, because they aren’t all functions and the objects of a category don’t always have elements to “change”. For instance, there’s a category whose objects are the positive integers and which has a single morphism from x to y if and only if x≤y; when we observe that 5≤9, is 5 changing 9? or 9 changing 5? No, nothing is changing anything else here.
So far as I can see, the only actual analogy here is with the bare syntactic structure: you can take “A pushes B” and “A has a morphism f to B” and match the pieces up. But the match isn’t very good—the second of those is a really unnatural way of writing it, and really you’d say “f is a morphism from A to B”, and the things you can do with morphisms and the things you can do with sentences don’t have much to do with one another. (You can say “A pushes B with a stick”, and “A will push B”, and so forth, and there are no obvious category-theoretic analogues of these; there’s nothing grammatical that really corresponds to composition of morphisms; if A pushes B and B eats C, there really isn’t any way other than that to describe the relationship between A and C, and indeed most of us wouldn’t consider there to be any relationship worth mentioning between A and C in this situation.)