Horizontal categorification

There are many concepts (especially algebraic structures) that can be reformulated as properties of one-object categories. For example, a group can be considered "equivalent to" a one-object category in which every arrow is invertible. It is then natural to consider many-object categories with similar properties. For example, in the case of groups we might naturally consider categories in which every arrow is invertible. Such categories in this case are called groupoids.

In most (but not all!) examples, the type of category generalizing the original concept/structure is called the oididication (or horizontal categorification) of the original concept/structure.

This is all somewhat ad hoc, but it seems Tom Leinster and Eugenia Cheng have worked out some formalization of the procedure, in terms of operads and so-called suspensions.

Here's a quick table of some examples. See nLab for a bunch more.