nLab
codiscrete groupoid

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.

This is also called the pair groupoid of $X$ and sometimes the chaotic groupoid on $X$.

Definition

For $X$ set, the codiscrete groupoid of $X$ is the groupoid $Codisc(X)$ with

• $Obj(X) = X$;

• $Mor(X) = X \times X$.

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 $X$.)

Remark 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?

Properties

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

• For $X$ a finite set of cardinality $n \gt 0$, the category algebra of $Codisc(X)$ is the algebra of $n\times n$ matrices. The contractibility of $Codisc(X)$ 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.

• The underlying quiver of a codiscrete groupoid is a complete graph? (in that there is one and only one edge between any ordered pair of vertices).

• The nerves of codiscrete groupoids are precisely the codiscrete objects in sSet, regarded as a cohesive topos.

Related concepts

• discrete groupoid