nLab
codiscrete groupoid

Urs: after writing this I remembered that there is already indiscrete category. Will merge both entries…

Idea

The codiscrete groupoid on a set is the groupoid whose objects are the elements of the set and which has a unique morphism for every ordered pair of objects.

Definition

For $X$ set, the codiscrete groupoid of $X$ is the groupoid $\mathrm{Codisc}(X)$ with

• $\mathrm{Obj}(X)=X$;

• $\mathrm{Mor}(X)=X\times X$.

This definition makes sense also internally, for $X$ 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 $X$.)

Eric: If true, is it worth pointing out that the underlying directed graph is complete?

Todd: I know what you’re saying, and it’s the correct picture, but ‘complete’ here refers to a definition of directed graph which is not the one used here, where there can be multiple edges from a given source to given target. Graph theorists who speak of ‘complete’ intend digraph to mean at most one edge from given source to given target. In different language, given by an incidence matrix with 0’s and 1’s. Equivalently, given by a binary relation on the set of nodes.

Toby: Being a complete directed graph (in the graph theorists' usual sense) is still a reasonable property of a directed pseudomultigraph (our kind). And since there's no way that a multigraph can be complete in the sense of having as many edges as possible, I would still use the term ‘complete’ here. It's a sort of existence and uniqueness condition, like a universal property.

Remarks

• Every codiscrete groupoid on an inhabited set is contractible: equivalent to the point. More generally, any codiscrete groupoid is equivalent to a truth value.

• The codiscrete groupoid on $X$ is also sometimes called the chaotic groupoid on $X$. 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 $X$ a finite set of cardinality $n>0$, the category algebra of $\mathrm{Codisc}(X)$ is the algebra of $n\times n$ matrices. The contractibility of $\mathrm{Codisc}(X)$ 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.