nLab 0-groupoid (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

A 0-groupoid is a set. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, groupoids, 2-groupoids, 3-groupoids, etc. Doing so reveals patterns such as the periodic table. (It also sheds light on the theory of homotopy groups and n-stuff.)

For example, there should be a $1$-groupoid of $0$-groupoids; this is the underlying groupoid of the category of sets. Then a groupoid enriched over this is a $1$-groupoid (more precisely, a locally small groupoid). Furthermore, an enriched category is a category (or $1$-category), so a $0$-groupoid is the same as a 0-category.

One can continue to define a (-1)-groupoid(−1)-groupoid? to be a truth value and a (-2)-groupoid(−2)-groupoid? to be a triviality (that is, there is exactly one).