According to the nlab, horizontal categorization is a process in which a concept is realized as equivalent to a certain type of category with a single object, then this concept is generalized to the same type of categories with an arbitrary number of objects. The prototypical example is the concept of group, which horizontally categorizes the concept of groupoid. It is well known that in some parts of algebraic topology, groupoids are much more practical than groups. Similarly, monoids are categorized into categories, rings into linear categories, etc.
I have two questions about this. Let C be a concept which horizontally categorizes a concept D.
1) In many examples, C and D have been known and developed independently of each other. Or at least, D was not introduced as a horizontal categorization of C, but rather this connection was made later. For example, I'm pretty sure that the k-linear categories weren't introduced as a horizontal categorization of k-algebras; instead, they were introduced due to the abundance of examples of (large) k-linear categories that appear in everyday mathematics. Although representation theory seems to be in a current advance from a generalization of k-algebras to small k-linear categories, the concept of a k-linear category was already there before that. Are there examples where D was developed for the purpose of classifying C, let's say to solve some problems which concern C but which cannot be solved with C alone? Maybe categories C * (categorize C * algebras) could provide such an example, but I don't know the history of this concept. And maybe there are other examples too?
2) In all the examples I know, it is trivial that C has a horizontal categorization and that it is D. Are there more interesting examples where, when you look at C, it doesn & # 39; is not even clear how to interpret C as a category type with an object? I would like to see examples where the connection between C and D is deep and surprising. These examples should illustrate why horizontal categorization is an important and useful concept in practice.
I could also ask a more provocative question: if a mathematician working outside of category theory reads the nlab article in its current form, why should he even care, because, after all, concepts C and D have already been there without the categorization process?