Revisiting the definition of Riemann integral, Carla noticed that we can define partitions of an interval categorically: given a (nondegenerate) interval $[a,b]$ seen as an ordered set with top and bottom, partitions are just monotone functions that preserve tops and bottoms from $n$ to $[a,b]$, for $n\geq 2$.
Amusingly, morphisms of partitions seem to correspond to refinements of partitions. Has anyone noticed this before, and if yes, is there a reference? (If not, did Carla make a mistake?)
(Carla is allowing partitions like $(0,0,1)$ of $[0,1]$, just like Rudin does. It might be possible to disallow those by considering only strictly monotone morphisms).
Edit: Also, coproducts seem to correspond to common refinements.
Edit 2: A moment of thought shows this is stupid: the partitions $(0,1,3)$ and $(0,2,3)$ of $[0,3]$ are isomorphic, which shows this is the wrong way to think about partitions of an interval.