What you're describing sounds more like homology theory, where there is an abstract set of concepts (short exact sequences, etc.) which become different theorem

What you're describing sounds more like homology theory, where there is an abstract set of concepts (short exact sequences, etc.) which become different theorems in different contexts. But, unless I'm mistaken, most of the time these are closer to unifying foundations, or starting points, from which the special features of each model allow you to say more beyond the abstract framework. So sure you can get some general category-theoretic theorems, but saying the specific model is useless (as the Language folks say about Turing machines) is ludicrous.