Category as a parent and higher categories
Hello,
I'm quite new to Sage and until now I have read some basic tutorials on the categorical framework and on algebraic structures.
As I understood, categories are treated as an abstract class, to which one can add more concrete instances. I was wondering if there was a way to treat categories as more concrete objects, where you could explicitly define the objects and morphisms. I would expect that something like this could be implemented with a category of categories and specific parents. Along the same lines, one might ask how to handle the concepts of higher categories (or other properties to enrich a category).
In working with categorization, I would find a similar more concrete concept of category useful, in my case to implement the "Singular Coxeter Monoid" (insufficient karma to publish links, if you're interested look for "A singular Coxeter presentation"). I currently implement the object in question with a specific class that does not use predefined Parents.
So is there such a thing in Sage? Would it make sense for the Cat category to exist? Or are there fundamental problems that prevent a reasonable implementation of it?
This question looks more appropriate to Sage's development group: https://groups.google.com/g/sage-devel
I didn't know about that group, thanks!
In this very special case the category (in the sense of sage) will maybe not help to much. Definition 1.22 https://arxiv.org/pdf/2105.08563 needs rather a graph structure, where arrows can be composed. It is simpler to design a class that covers exactly this situation than to adapt the categories mechanism implemented in sage to the given relatively simple situation. The class should be designed so that the needed computations are easily coded.