Horizontal categorification: Two questions

Question

According to the nlab, horizontal categorification is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized to the same type of categories with an arbitrary number of objects. The prototypical example is the concept of a group, which horizontally categorifies to the concept of a groupoid. It is well-known that in certain parts of algebraic topology groupoids are much more convenient than groups. Similarly, monoids categorify to categories, rings to linear categories, etc.

I have two questions about this. Let C be a concept which horizontally categorifies to a concept D.

1) In many examples, C and D have been known and developed independently from each other. Or at least, D was not introduced as the horizontal categorification of C, but rather this connection was realized afterwards. For example, I am pretty sure that k-linear categories were not introduced as a horizontal categorificiation of k-algebras; instead they were introduced because of the abundance of examples of (large) k-linear categories which appear in everyday mathematics. Although representation theory seems to be in a current progress of a generalization from 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 with the purpose to categorify C, say in order to solve some problems which deal with C but which are not solvable with C alone? Perhaps C*-categories (categorifying C*-algebras) could provide such an example, but I am not familiar with the history of this concept. And maybe there are other examples as well?

2) In all the examples I know of, it is trivial that C has a horizontal categorification and that it is D. Are there any more interesting examples where, when you look at C, it is not even clear how to interpret C as a type of category with one object? I would like to see examples where the connection between C and D is deep and surprising. These examples should illustrate why horizontal categorification is an important and useful concept in practice.

I could also ask a more provocative question: if any mathematician working outside of category theory reads the nlab article in its current form, why should he/she even care, since, after all, both concepts C and D have been there already without the categorification process?

Answer 1

1. Often when a horizontal categorification is introduced, the author has intended examples, which the author gives as the motivation. It thus makes it a little tricky to determine when a structure has been motivated explicitly by horizontal categorification. However, here are a few examples where (1) the concept had not previously appeared in the literature, and (2) the authors make it clear horizontal categorification was intended (though this precise language may not appear).
   
   In Street's Elementary cosmoi I, the concept of extension system is introduced, which is explicitly intended to be to closed categories what bicategories are to monoidal categories. Intuitively, the analogue of closed structure is proved by right-extensions.
   
   Two more examples are the multi- and poly-bicategories of Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, which are horizontal categorifications of multicategories and polycategories respectively.

2. My favourite example of a surprising horizontal categorification is that of symmetric monoidal categories. A symmetric bicategory (not to be confused with a symmetric monoidal bicategory) is a bicategory with an involution. Explicitly, a bicategory $\mathcal K$ is symmetric when equipped with a biequivalence $\mathcal K \simeq \mathcal K^{\mathrm{op}}$ that is a bijection on objects. A symmetric monoidal category is precisely a one-object symmetric bicategory for which the involution is the identity. This is interesting in two ways: the generalisation from symmetry to involution is a highly non-obvious step; and one is forced to make a generalisation to the original structure in order to horizontally categorify. There is an analogous concept of closed symmetric bicategory, which is a symmetric bicategory with left- and right-extensions. (Closed) symmetric bicategories are defined and studied in May–Sigurdsson's Parametrized Homotopy Theory.

I think these examples give some indication that studying horizontal categorifications in their own right can lead to interesting structures that may then be motivated by concrete examples.