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Principal bundles, groupoids and connections

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

Pregroupoids derived from a groupoid

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

$$s:G(A,B)\to A$$s:G(A,B)\to A

$$t:G(A,B)\to B$$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 $A\stackrel{p}{\leftarrow }P\stackrel{!}{\to }*$ is a pregroupoid.

The category of pregroupoids

There is a category $\mathrm{PGrpd}$ with pregroupoids as objects and triples $(c_0,c,c_1)$ of morphisms making

$$\begin{array}{ccccc}A& \stackrel{a}{\leftarrow }& {}^X& \stackrel{b}{\to }& B\\ {}_{c_0}\downarrow & & {}_c\downarrow & & {}_{c_1}\downarrow \\ A^{\prime }& \stackrel{a^{\prime }}{\leftarrow }& X^{\prime }& \stackrel{b^{\prime }}{\to }& B^{\prime }\end{array}$$\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

$$V:\{\begin{array}{l}\mathrm{Grpd}\to \mathrm{PGrpd}\\ G\to G(G_0,G_0)\end{array}$$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 $E(P)$. ${\mathrm{PP}}^{-1}$ 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.

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

For a pregroupoid $A\stackrel{\alpha }{\leftarrow }P\stackrel{\beta }{\to }B$ the Ehresmann groupoid ${\mathrm{PP}}^{-1}$ is defined by ${\mathrm{PP}}_0^{-1}:=A$ and

$${\mathrm{PP}}^{-1}(a,a^{\prime }):={\mathrm{xy}}^{-1}:=\{(x,y)\mid \alpha (x)=a,\alpha (y)=a^{\prime }\}/\sim$$PP^{-1}(a,a^\prime):=xy^{-1}:=\{(x,y)|\alpha(x)=a,\,\alpha(y)=a^\prime\}/\sim

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

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

For a pregroupoid $A\stackrel{\alpha }{\leftarrow }P\stackrel{\beta }{\to }B$ the ”inverse Ehresmann” groupoid $P^{-1}P$ is defined by $P^{-1}{P}_0:=B$ and

$$P^{-1}P(b,b^{\prime }):=y^{-1}z:=\{(y,z)\mid \beta (y)=b,\alpha (z)=b^{\prime }\}/\sim$$P^{-1}P(b,b^\prime):=y^{-1}z:=\{(y,z)|\beta(y)=b,\,\alpha(z)=b^\prime\}/\sim

where $(y,z)\sim (x,u)$ 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

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

The enveloping groupoid $E(P)$

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

The enveloping groupoid $E(P)$ for $P$ is defined by $E(P{)}_0:=A\coprod B$ and $E(P{)}_1:={\mathrm{PP}}^{-1}\coprod P^{-1}P\coprod P\coprod P^{-1}$, 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 $G=E(*,*)$

• The functor $E(-):\mathrm{PGrdp}\to \mathrm{Grpd}$ 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),