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Principal bundles, groupoids and connections

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

Pregroupoids derived from a groupoid

Let G:={G 0G 1}G:=\{G_0\to G_1\} a groupoid, A,BG 0A,B\subset G_0 subsets.

Then the object G(A,B)G(A,B) defined to be the set of arrows in G 1G_1 whose source is in AA and whose target is in BB equipped with the two evident structure maps

s:G(A,B)As:G(A,B)\to A
t:G(A,B)Bt:G(A,B)\to B

is a pregroupoid. If A=BA=B this is a full subgroupoid of GG.

Pregroupoid associated to a principal bundle

Let p:PAp:P\to A be a principal right GG-bundle for a group GG. Then the span ApP!*A\xleftarrow{p}P\xrightarrow{!} * is a pregroupoid.

The category of pregroupoids

There is a category PGrpd\mathbf{PGrpd} with pregroupoids as objects and triples (c 0,c,c 1)(c_0,c,c_1) of morphisms making

A a X b B c 0 c c 1 A a X b B \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:{GrpdPGrpd GG(G 0,G 0)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: PP 1PP^{-1}, P 1PP^{-1}P and E(P)E(P). PP 1PP^{-1} and P 1PP^{-1}P are called edges of the pregroupoid PP, since they appear as edges of some bisimplicial set. E(P)E(P) is called enveloping groupoid for the pregroupoid PP. E(P)E(P) contains the edges of PP as subgroupoids.

The Ehresmann groupoid PP 1PP^{-1}

For a pregroupoid AαPβBA\xleftarrow{\alpha} P\xrightarrow{\beta} B the Ehresmann groupoid PP 1PP^{-1} is defined by PP 0 1:=APP^{-1}_0:=A and

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

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

The ”inverse Ehresmann” groupoid PP 1PP^{-1}

For a pregroupoid AαPβBA\xleftarrow{\alpha} P\xrightarrow{\beta} B the ”inverse Ehresmann” groupoid P 1PP^{-1}P is defined by P 1P 0:=BP^{-1}P_0:=B and

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

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

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

{P 1PG (y 1z:yz)g, yg=z\begin{cases}P^{-1}P\to G\\(y^{-1}z:y\to z)\to g,&yg=z\end{cases}

The enveloping groupoid E(P)E(P)

Let AαPβBA\xleftarrow{\alpha} P\xrightarrow{\beta} B be a pregroupoid.

The enveloping groupoid E(P)E(P) for PP is defined by E(P) 0:=ABE(P)_0:=A\coprod B and E(P) 1:=PP 1P 1PPP 1E(P)_1:=PP^{-1}\coprod P^{-1}P\coprod P\coprod P^{-1}, where P 1P^{-1} denotes the pregroupoid PP with α\alpha and β\beta interchanged. And calculations (see arxiv p.9) show that this is a groupoid.

Properties

  • The edges of PP act on PP principally on the left and the right, respectively and the actions commute with each other.

  • Let p:PAp:P\to A be a principal GG bundle. Then there is a groupoid EE with E 0=A*E_0=A\coprod * and E 1=PE_1=P. In this cases G=E(*,*)G=E(*,*)

  • The functor E():PGrdpGrpdE(-):PGrdp\to Grpd is a faithful left adjoint to the forgetful functor VV, 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)