I find the Monoid/Semigroup typeclass pretty concisely captures what is generally meant by "composition" in the minimal sense.

> As a rule, I've always viewed "composition" as "gluing together things that don't know necessarily know about each other"

The extension to this definition given the context of Monoids would be "combining two things of the same type such that they produce a new thing of the same type". The most trivial example of this is adding integers, but a more practical example is function composition where two functions can be combined to create a new function. You can also think of an abstraction that let's you combine two web components to create a new one, combining two AI agents to make a new one, etc.

> "inheritance is vertical, composition is horizontal", but of course that doesn't really mean anything.

This can actually be clearly defined, what you're hinting at is the distinction between sum types and product types. The latter of which describes inheritance. The problem with restricting yourself to only product types is that you can only add things to an existing thing, but in real life that rarely makes sense, and you will find yourself backed into a corner. Sum types let you have much more flexibility, which in turn make it easier to implement truly composable systems.

I actually knew most of that (I've done a lot of Haskell). I don't really disagree with what you said, but I feel like like you eliminate a lot of stuff that people would consider "composition" but aren't as easily classified in happy categories.

For example, a channel-based system like what Go or Clojure has; to me that is pretty clearly "composition", but I'm not 100% sure how you'd fully express something like that with categories; you could use something like a continuation monad but I think that loses a bit because the actual "channel" object has separate intrinsic value.

In Clojure, there's a "compose" function `comp` [1], which is regular `f(g(x))` composition, but lets suppose instead I had functions `f` and `g` running in separate threads and they synchronize on a channel (using core.async)? Is that still composition? There are two different things that can result in a very similar output, and both of which are considered by some to be composition. So which one of these should I "prefer" instead of inheritance?

Of course this is the realm of Pi Calculus or CSP if you want to go into theory, but I'm saying that I don't think that there's a "one definition to rule them all" for composition.

[1] https://clojuredocs.org/clojure.core/comp