Earlier quoted context omitted.
What you need is that functions are values in your language. Lambdas are just a notation for function values. Typed lambda calculus is the internal language of cartesian closed categories and function values are then called internal morphisms. The composition above is then the internal composition of internal morphisms. It would be possible for external composition to be already defined by the language, take the unix…
> What you need is that functions are values in your language. Well yeah, that's what I meant by higher order functions . > Lambdas are just a notation for function values. But regular (named) functions can still be used as function values. So this doesn't explain why you need things like lambdas in order to implement function composition. > Typed lambda calculus is the internal language of cartesian closed categorie…
Interesting. I might say that partial application is a kind of closure. Certainly, it winds up the same - "function carrying some data that it uses internally". I think you are correct that compose and apply does not require closures of any sort.