This entry is about ( a fragment of) a paper by Anders Kock
Let $G:=\{G_0\to G_1\}$ a groupoid, $A,B\subset G_0$ subsets.
Then the object $G(A,B)$ defined to be the set of arrows in $G_1$ whose source is in $A$ and whose target is in $B$ equipped with the two evident structure maps
is a pregroupoid. If $A=B$ this is a full subgroupoid of $G$.
Let $p:P\to A$ be a principal right $G$-bundle for a group $G$. Then the span $A\xleftarrow{p}P\xrightarrow{!} *$ is a pregroupoid.
There is a category $\mathbf{PGrpd}$ with pregroupoids as objects and triples $(c_0,c,c_1)$ of morphisms making
commute.
There is a forgetful functor
There are three candidates of a groupoid associated to a pregroupoid: $PP^{-1}$, $P^{-1}P$ and $E(P)$. $PP^{-1}$ and $P^{-1}P$ are called edges of the pregroupoid $P$, since they appear as edges of some bisimplicial set. $E(P)$ is called enveloping groupoid for the pregroupoid $P$. $E(P)$ contains the edges of $P$ as subgroupoids.
For a pregroupoid $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ the Ehresmann groupoid $PP^{-1}$ is defined by $PP^{-1}_0:=A$ and
where $(x,y)\sim(u,z)$ iff $u=xy^{-1}z$
For a pregroupoid $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ the ”inverse Ehresmann” groupoid $P^{-1}P$ is defined by $P^{-1}P_0:=B$ and
where $(y,z)\sim(x,u)$ iff $u=xy^{-1}z$
If $P$ is a principal $G$-bundle, the groupoid $P^{-1}P$ is canonically isomorphic to the one object groupoid $G$ by
Let $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ be a pregroupoid.
The enveloping groupoid $E(P)$ for $P$ is defined by $E(P)_0:=A\coprod B$ and $E(P)_1:=PP^{-1}\coprod P^{-1}P\coprod P\coprod P^{-1}$, where $P^{-1}$ denotes the pregroupoid $P$ with $\alpha$ and $\beta$ interchanged. And calculations (see arxiv p.9) show that this is a groupoid.
The edges of $P$ act on $P$ principally on the left and the right, respectively and the actions commute with each other.
Let $p:P\to A$ be a principal $G$ bundle. Then there is a groupoid $E$ with $E_0=A\coprod *$ and $E_1=P$. In this cases $G=E(*,*)$
The functor $E(-):PGrdp\to Grpd$ is a faithful left adjoint to the forgetful functor $V$, the unit for the adjunction is injective.
Anders Kock: