This is also called the pair groupoid of and sometimes the chaotic groupoid on .
For set, the codiscrete groupoid of is the groupoid with
This definition makes sense also internally, for an object in any category with finite limits. (In fact, this is one of those cases where the category can easily be defined with some limits lacking; we need only finite products of .)
Remark The codiscrete groupoid on is also sometimes called the chaotic groupoid on . The intuition is probably that “everything being connected with everything else sounds pretty chaotic”, but one can argue that the term “chaotic groupoid” exactly misses the true intrinsic nature of codiscrete groupoids: since these are all just “puffed up versions of the point” they are “maximally homogenous” things. Which space would be less chaotic than the point?
For a finite set of cardinality , the category algebra of is the algebra of matrices. The contractibility of is reflected in the fact that this algebra is Morita equivalent to the ground ring, which is the category algebra of the point.
This maybe serves to illustrate: even though codiscrete groupoids are pretty trivial, they are not too trivial to be entirely without interest. Often it is useful to have big puffed-up versions of the point available.