A group $G$ is a category with exactly one object and all invertible morphisms, which are tue usual elements of the set theoretic group which may be denoted by $|G|$.
Now, for a $G$-set $S$, $\textit{i.e.}$ a functor $G\to\mathrm{Set}$, we can define the translation groupoid $G\ltimes S$ to be the groupoid having the set $S$ (the image of the only object in $G$ through the functor) as objects, and $\mathrm{Hom}(x,y)$ are pairs $(x,g)$ such that $gx=y$. That means hom-sets are either empty if the elements are not in the same orbit, or consisting of just one element. A very special case of this construction happens when $S=|G|$, so that we end up with a contractible (all hom-sets are one-point) category $EG$ whose objects are the elements of the group, which is also known as total space of $G$, because of the strict analogy with universal covers.
Are there known analogous constructions of $EG$ for a (let's say connected) groupoid $H$ instead of a group $G$?
The nLab page on the action groupoid says that $EG=G\ltimes |G|$ arises as $2$-colimit of the diagram $G\to\mathrm{Set}$ which picks the set $|G|$ and acts by evident multiplication, but it doesn't make that sense to me since this colimit just gives a set of course, not a groupoid. However, a generalization in this sense seems to be reasonable by considering the functor $H\to \mathrm{Set}$ (or maybe $\mathrm{Cat}$?) mapping an object $p$ to the arrows in $p$ and then acting by postcomposition of morphisms. What should be the right environment in order to define $EH$ as the colimit of this diagram?
Thanks in advance, I hope the question is not too vague.
Best Answer
A universal cover of a connected groupoid $B$ is a discrete fibration $E \to B$ where $E$ is a simply connected groupoid. As it turns out, there is a very easy construction. Choose a point $b$ of $B$ and take $E$ to be the slice $B_{/ b}$. $E$ has a terminal object, so it is contractible, hence simply connected a fortiori. The projection $E \to B$ is a fibration and it is easy to check that the fibres are discrete. Hence we have the desired universal cover of $B$.
Incidentally, if you do this with $\infty$-groupoids instead of 1-groupoids you end up with a (not necessarily discrete) fibration $E \to B$ where $E$ is contractible, so in some sense I have cheated and exploited the 1-dimensionality of $B$ in the above.