As we study mathematics, we come across objects like vector spaces, groups, rings, fields, metric spaces and topological spaces. We may see many seemingly different examples of the objects, but we can classify them up to isomorphism/homeomorphism. Then I have the following question in mind:
How do we know that two classes of objects, for example, groups and topological spaces, are essentially different, that there does not exist an "isomorphism" between them?
This seems to be related to category theory, but I have never taken any course on that.