What is a categorical definition of algorithm? Maybe there are several simple and equivalent definitions? Is it just arbitrary morphism in a category of sets and partially defined functions? What is the Turing machine then?
I've just a little bit lost in a big amount of different information from the first part of twenty century math books.
You may be interested in the effective topos or its many, many variations. However, I'd argue that category theory need not be "the" correct way to think about algorithms (although it's certainly a valuable perspective); the definition of a Turing machine is fully rigorous, and doesn't need to be grounded in category theory in order for computability theory to work.
I want to make it clear that I'm not saying category-theoretic approaches to computability theory are uninteresting or not useful, just that it sounds like you're starting from a point of "this is the right way to look at mathematical constructions," and - similarly to how set-theoretic formulations aren't always (or even frequently :P) the right way to go about things - that need not be the case.