Confused about the definition of a group as a groupoid with one object.

A groupoid is defined to be a category where every morphism is an isomorphism. So sometimes a group is said to just be a groupoid with one object.

When I try to make sense of this, I denote the single object as $G$. I view the morphisms as the analogue of "elements." The identity $1_G$ is the analogue of the usual identity $e$, we can compose any two morphisms since they are all arrows on $G$, and for any arrow $f$, we have some $f^{-1}$ such that $f\circ f^{-1}=f^{-1}\circ f=1_G$, so the idea of inverses is still there.

So I informally associate the elements of the group in the usual definition to be the arrows in the groupoid. But what does the sole object $G$ in the groupoid "correspond to" if I were to try to informally make sense of a group in the usual sense? Does it even correspond to anything?

Solution 1:

The single object in the groupoid corresponding to a group $G$ does not really correspond to anything in the group - but you can think of it as a thing which has as its group of symmetries the group $G$.

As an example, the category whose only object is the set $[n]=\{1,\dotsc,n\}$, and morphisms are bijections from this object onto itself. Then the symmetries of $[n]$ are given by the group $S_n$.

For a more subtle example, consider a topological space $X$, and fix a point $x\in X$. Consider the category with one object, namely the point $x$. A morphism from $x\to x$ is a homotopy-class of paths from $x$ to $x$. Composition of morphisms comes from concatenation of paths. The automorphism group of $x$ is the fundamental group $\pi_1(X,x)$, which is a group of symmetries of the based topological space $(X,x)$.

However, if you consider groupoids with many objects all of which are isomorphic to each other, then the different objects correspond to different realizations of the same group. Moreover, each isomorphism between different objects gives rise to an isomorphism between the corresponding realizations of the group.

To make this precise, suppose $\mathcal G$ is a groupoid where all objects are isomorphic. For each object $x$ of $\mathcal G$, let $G_x$ denote the group $\mathrm{Mor}_\mathcal G(x, x)$. A morphism $\phi:x\to y$ defines an isomorphism $G_y\to G_x$ by taking $g\mapsto \phi\circ g\circ \phi^{-1}$.

A nice example of this is the category $\mathrm{FB}_n$ whose objects are finite sets of cardinality $n$ and morphisms are bijections. Then the automorphism group of each object is isomorphic to $S_n$. Isomorphisms between objects determine isomorphisms of their automorphism groups.

Another example is the Fundamental Groupoid of a path connected topological space $X$. Its objects are the points of $X$. The set $\mathrm{Mor}(x,y)$ is the set of homotopy classes of paths from $x$ to $y$. In this groupoid, $G_x$ is the fundamental group $\pi_1(X,x)$ of $X$ based at the point $x$. Different base points result in isomorphic fundamental groups, and isomorphisms are determined by homotopy classes of paths between these points.

Solution 2:

Disappointing as the answer may be, it doesn't really correspond to anything.

All the group axioms refer to elements of the group; and these elements correspond with the morphisms in the category. We don't care what the object is: there's only one object, after all! So all we're bothered about is the morphisms.