Recently I saw a video talking about the Yoneda lemma from category theory being used in neuroscience. It was my first introduction to category theory. In category theory we have objects and maps between the same objects called morphisms. For example we have one type of object called set and its morphism would be the function. A function is a map from one set to another. Another example would be the object vector field with its morphisms being linear transformations. This just looks like we are renaming the same stuff.
Is a vector space not just a type of set we construct to obey our desired axioms? If so then introducing linear transforms would be truly just renaming stuff. And if they are not sets, why not allow non linear transforms?