# Contents

This entry is about ( a fragment of) a paper by Anders Kock

## Pregroupoids derived from a groupoid

Let $G:=\left\{{G}_{0}\to {G}_{1}\right\}$ a groupoid, $A,B\subset {G}_{0}$ subsets.

Then the object $G\left(A,B\right)$ 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

$s:G\left(A,B\right)\to A$s:G(A,B)\to A
$t:G\left(A,B\right)\to B$t:G(A,B)\to B

is a pregroupoid. If $A=B$ this is a full subgroupoid of $G$.

## Pregroupoid associated to a principal bundle

Let $p:P\to A$ be a principal right $G$-bundle for a group $G$. Then the span $A\stackrel{p}{←}P\stackrel{!}{\to }*$ is a pregroupoid.

## The category of pregroupoids

There is a category $\mathrm{PGrpd}$ with pregroupoids as objects and triples $\left({c}_{0},c,{c}_{1}\right)$ of morphisms making

$\begin{array}{ccccc}A& \stackrel{a}{←}& {}^{X}& \stackrel{b}{\to }& B\\ {}_{{c}_{0}}↓& & {}_{c}↓& & {}_{{c}_{1}}↓\\ {A}^{\prime }& \stackrel{{a}^{\prime }}{←}& {X}^{\prime }& \stackrel{{b}^{\prime }}{\to }& {B}^{\prime }\end{array}$\array{A&\xleftarrow{a}&^X&\xrightarrow{b}&B\\ {}_{c_0}\downarrow&&{}_c\downarrow&&{}_{c_1}\downarrow\\A^\prime&\xleftarrow{a^\prime}&X^\prime&\xrightarrow{b^\prime}&B^\prime}

commute.

There is a forgetful functor

$V:\left\{\begin{array}{l}\mathrm{Grpd}\to \mathrm{PGrpd}\\ G\to G\left({G}_{0},{G}_{0}\right)\end{array}$V:\begin{cases}Grpd\to PGrpd\\G\to G(G_0,G_0)\end{cases}

## Groupoids associated to a pregroupoid

There are three candidates of a groupoid associated to a pregroupoid: ${\mathrm{PP}}^{-1}$, ${P}^{-1}P$ and $E\left(P\right)$. ${\mathrm{PP}}^{-1}$ and ${P}^{-1}P$ are called edges of the pregroupoid $P$, since they appear as edges of some bisimplicial set. $E\left(P\right)$ is called enveloping groupoid for the pregroupoid $P$. $E\left(P\right)$ contains the edges of $P$ as subgroupoids.

### The Ehresmann groupoid ${\mathrm{PP}}^{-1}$

For a pregroupoid $A\stackrel{\alpha }{←}P\stackrel{\beta }{\to }B$ the Ehresmann groupoid ${\mathrm{PP}}^{-1}$ is defined by ${\mathrm{PP}}_{0}^{-1}:=A$ and

${\mathrm{PP}}^{-1}\left(a,{a}^{\prime }\right):={\mathrm{xy}}^{-1}:=\left\{\left(x,y\right)\mid \alpha \left(x\right)=a,\phantom{\rule{thinmathspace}{0ex}}\alpha \left(y\right)={a}^{\prime }\right\}/\sim$PP^{-1}(a,a^\prime):=xy^{-1}:=\{(x,y)|\alpha(x)=a,\,\alpha(y)=a^\prime\}/\sim

where $\left(x,y\right)\sim \left(u,z\right)$ iff $u={\mathrm{xy}}^{-1}z$

### The ”inverse Ehresmann” groupoid ${\mathrm{PP}}^{-1}$

For a pregroupoid $A\stackrel{\alpha }{←}P\stackrel{\beta }{\to }B$ the ”inverse Ehresmann” groupoid ${P}^{-1}P$ is defined by ${P}^{-1}{P}_{0}:=B$ and

${P}^{-1}P\left(b,{b}^{\prime }\right):={y}^{-1}z:=\left\{\left(y,z\right)\mid \beta \left(y\right)=b,\phantom{\rule{thinmathspace}{0ex}}\alpha \left(z\right)={b}^{\prime }\right\}/\sim$P^{-1}P(b,b^\prime):=y^{-1}z:=\{(y,z)|\beta(y)=b,\,\alpha(z)=b^\prime\}/\sim

where $\left(y,z\right)\sim \left(x,u\right)$ iff $u={\mathrm{xy}}^{-1}z$

If $P$ is a principal $G$-bundle, the groupoid ${P}^{-1}P$ is canonically isomorphic to the one object groupoid $G$ by

$\left\{\begin{array}{l}{P}^{-1}P\to G\\ \left({y}^{-1}z:y\to z\right)\to g,& \mathrm{yg}=z\end{array}$\begin{cases}P^{-1}P\to G\\(y^{-1}z:y\to z)\to g,&yg=z\end{cases}

## The enveloping groupoid $E\left(P\right)$

Let $A\stackrel{\alpha }{←}P\stackrel{\beta }{\to }B$ be a pregroupoid.

The enveloping groupoid $E\left(P\right)$ for $P$ is defined by $E\left(P{\right)}_{0}:=A\coprod B$ and $E\left(P{\right)}_{1}:={\mathrm{PP}}^{-1}\coprod {P}^{-1}P\coprod P\coprod {P}^{-1}$, where ${P}^{-1}$ denotes the pregroupoid $P$ with $\alpha$ and $\beta$ interchanged. And calculations (see arxiv p.9) show that this is a groupoid.

## Properties

• 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 $G=E\left(*,*\right)$

• The functor $E\left(-\right):\mathrm{PGrdp}\to \mathrm{Grpd}$ is a faithful left adjoint to the forgetful functor $V$, the unit for the adjunction is injective.

## References

Anders Kock:

• Principal bundles, groupoids and connections (arxiv),
Created on October 17, 2011 13:53:18 by Stephan (79.227.189.118)