> I wondered if such non-mathematical uses of "category" could be given some category-theoretic modeling. Considering we already have a category Cat for all categories, might there also a category for all uses of the word "category"? Or is it simply part of Cat?
Sadly, I don't think so. *Category* is just a term that mathematicians made up. It's an overloaded term, however. Aristotle used it one way as you mention. [Categorical distributions](https://en.wikipedia.org/wiki/Categorical_distribution) in statistics refer to something else. [Kant's categorical imperative](https://en.wikipedia.org/wiki/Categorical_imperative) is something yet different still.
A similar confusing word thrown around is *monad*. It started as an idea in ancient Greek philosophy. Parmenides thought of his *monad* as the totality of all things. In Leibniz's [*Monadology* (1714)](https://en.wikipedia.org/wiki/Monadology) he was referring to sort of elementary, indivisible substance. Centuries later, the mathematician Abraham Robinson took inspiration from Leibniz's approach to calculus in devising [non-standard analysis](https://en.wikipedia.org/wiki/Non-standard_analysis). Robinson called one mathematical construct he invented a [*monad*][monad], but it has nothing to do with Leibniz's monads as far as I can tell. The term is so confusing that Goldblatt just calls them *halos* in his 1998 text *Lectures on the Hyperreals*. Saunders Maclane coined *monad* as it is commonly used in category theory about a decade later. It has nothing to do with Robinson's monad. In the late 80s Philip Wadler introduced monads to Haskell to model side effects and it's been baffling programmers ever since. And yet still there's [*monadic first order logic*](https://en.wikipedia.org/wiki/Monadic_predicate_calculus) where *monadic* is just Greek for every predicate can take at most one argument.
Keeping the terminology straight is hard. When a computer scientist talks about a network topology, she doesn't mean [general topology](https://en.wikipedia.org/wiki/General_topology). She *usually* means a graph structure. When a cryptographer talks about [lattice cryptography](https://en.wikipedia.org/wiki/Lattice-based_cryptography), they don't care about lattices from [lattice theory][lattice theory]. And λ-calculus isn't the calculus we learned in high school.
[lattice theory]: https://en.wikipedia.org/wiki/Lattice_(order)