I'm beginning learning category theory and seeing some examples I wonder whether (or why) is necessary that arrows preserve some structure of objects. I know that there are functors that "forget" some information of a category but my point is to do this inside the category. For example can I define a category $Top^*$ with the class of all topological spaces as objects and functions as arrows, or a category $Rng^*$ with rings as objects and group homomorphisms as arrows?. I know I doesn't make much sense to study this ('cause what's the point of objects in $Top^*$ have topologies), however my the question is if category definitions (or axioms) allow this classes.
The definition of category does not require that objects have any kind of structure or that arrows preserve structure. You can define a category by drawing some dots and putting some arrows between dots to satisfy the requirements for composition, existence of identities, and associativity.
The definition of category is abstract in the sense that it says a category has objects and arrows and imposes axioms on the objects and arrows.