I'm looking for some pointers on proper mathematical (FP?, category-theory?) terminology.
My apologies if the below is somewhat imprecise; I suppose the precision is precisely what I'm looking for in the answer.
Say you have a particular Abstract Syntax Tree. Now you also have some editor that allows you to annotate arbitrary nodes in the tree by hand. For example to either highlight nodes, or tie particular textual annotations to nodes ("comments").
The datatypes that allow you to store the information as produced by this editor are somehow "embellished versions" of the original datatype for the AST. E.g. for each node we now have a node that also knows whether it is highlighted or not, and recursively contains such nodes. Values from such datatypes have at least the property that you can throw away the annotations and end up with the original tree.
There is also some notion of "coupling" between the embellished AST's nodes and the "underlying AST"'s nodes, i.e. we can always point at the "source" for any embellished node. This is also to say: if there are two ways of embellishing, we can connect them.
What is the proper framework to (generally) think about this?
