Do you really need to introduce category theory for that?
Seems like overkill, abstract algebra seems sufficient to categorize both boolean logic and integer operations as having the common structure of a ring.
Of course you don't "need" to introduce category theory for that, which is why I saved it for fun at the end. I just think it is neat. It's also one of those bridges to "category theory is simpler than it sounds", which is also why I disagree with it being "overkill" in general in part because that keeps category theory in the "too complex for real needs" box, which I think is the wrong box. Which, case in point:
> […] abstract algebra seems sufficient to categorize both boolean logic and integer operations as having the common structure of a ring.
I don't think Ring Theory is any easier than Category Theory to learn/teach, I rather think that Category Theory is a subset of some of best parts of abstract algebra, especially Group Theory, boiled down to the sufficient parts to describe (among other things) practical function composition tools for computing.
> […] abstract algebra seems sufficient to categorize both boolean logic and integer operations as having the common structure of a ring.
I don't think Ring Theory is any easier than Category Theory to learn/teach, I rather think that Category Theory is a subset of some of best parts of abstract algebra, especially Group Theory, boiled down to the sufficient parts to describe (among other things) practical function composition tools for computing.