Horizontal categorification or Oidification describes the process by which
a concept is realized to be equivalent to a certain type of category with a single object;
and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.
This is to be contrasted with vertical categorification.
It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.
It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.
The horizontal categorification of groups are groupoids: categories in which every morphism is invertible.
A horizontal categorification of algebras are algebroids: internal categories in the category of vector spaces.
David Roberts: How do Lie algebroids fit into this framework?
Urs Schreiber: at a rough level it is clear that the base space of a Lie algebroid has to be regardedas the “space of objects”. Certainly a Lie algebroid over a point is precisely a Lie algebra.
But for a more precise statement one needs a more conceptual way to think of Lie algebroids. I am claiming at ∞-Lie algebroid that there is a way to regard Lie algebroids precisely as certain kinds of synthetically smooth groupoids, namely those all whose morphisms have “infinitesimal extension” in some sense. In such a picture Lie algebroids are on the same footing as Lie groupoids and are precisely the many-object version of Lie algebras = infinitesimal Lie groups.
A horizontal categorification of rings are ringoids: enriched categories over the category of abelian groups. (blog)
A horizontal categorification of $C^*$-algebras hence ought to be known as $C^*$–algebroids but is usually known as C*-categories.
Since, by the Gelfand-Naimark theorem, C-star algebras are dual to topological spaces, Paolo Bertozzini et. al proposed to define spaceoids to be entities dual to $C^*$-categories (blog).
And finally the exception from the rule: a many-object monoid is not called a monoidoid – but is called a category! :-)
Mike: Is there (and do we want there to be) a general rule about whether an X-oid means an internal category whose one-object version is an X, or an enriched category whose one-object version is an X? The examples above seem to be taking the “internal” side, but the Cafe discussion on “ringoids” was about the “enriched” version; the two are very different! And “dg-algebroid” was suggested for dg-category, which is an enriched oidification of a dg-algebra, but a Hopf algebroid is an internal oidification of a Hopf algebra.
Urs: good point. We should say this at the beginning and split the list of examples in two sorts
Related $n$-Café-discussion is in