Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So I can't just say let's define database tables as objects in the category of...Or lets define data types as objects in the category of...and so on, because it's wrong. I have to proof in some way that some thing is really an object in the category. So as I understood there are some laws or criterias that every
object in any category should satisfy or we can't call them "objects".
But I couldn't find any really formal definition of an
object in the category. I always thought that category theory is very abstract and we can call objects not only mathematical structures but whatever we want just declaratively. I agree with the point that if I say:"Let objects of the category be magmas. Than - yes, I should proof this". But can I call an object every thing I want without formal proof?
So I' interested are there really some laws that
object in the category should satisfy and we define some kind of category we should proof that some element is really an object.