%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: double groupoid geometry
%%% Primary Category Code: 00.
%%% Filename: DoubleGroupoidGeometry.tex
%%% Version: 4
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\subsection{Double Groupoids}
The geometry of \htmladdnormallink{squares}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} and their \htmladdnormallink{compositions}{http://planetphysics.us/encyclopedia/Cod.html} leads to a common \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of a \emph{\htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/ThinEquivalence.html}} in the following form:
\begin{equation}
\label{squ} \D= \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar @<-1ex> [r]
_{t^1} \ar @<1ex> [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H \ar[l]
\ar @<1ex> [d]^{\,t}
\ar @<-1ex> [d]_s \\
V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u]
}}
\end{equation}
where $M$ is a set of `points', $H,V$ are
`horizontal' and `vertical' \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}, and $S$ is a set of
`squares' with two compositions. The laws for a double groupoid
make it also describable as a groupoid internal to the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}.
Given two groupoids $H,V$ over
a set $M$, there is a double groupoid $\Box(H,V)$ with $H,V$ as
horizontal and vertical edge groupoids, and squares given by
quadruples
\begin{equation}
\begin{pmatrix} & h& \\[-0.9ex] v & & v'\\[-0.9ex]& h'&
\end{pmatrix}
\end{equation}
for which we assume always that $h,h' \in H, \, v,v' \in V$ and
that the initial and final points of these edges match in $M$ as
suggested by the notation, that is for example $sh=sv, th=sv',
\ldots$, etc. The compositions are to be inherited from those of
$H,V$,
that is
\begin{equation}
\quadr{h}{v}{v'}{h'} \circ_1\quadr{h'}{w}{w'}{h''}
=\quadr{h}{vw}{v'w'}{h''}, \;\quadr{h}{v}{v'}{h'}
\circ_2\quadr{k}{v'}{v''}{k'}=\quadr{hk}{v}{v''}{h'k'} ~.
\end{equation}
This construction is right adjoint to the forgetful \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} which
takes the double groupoid as above, to the pair of groupoids $H,V$
over $M$. Now given a general double groupoid as above, we can
define $S\quadr{h}{v}{v'}{h'}$ to be the set of squares with these
as horizontal and vertical edges.
This allows us to construct for at least a \htmladdnormallink{commutative C*--algebra}{http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} $A$ a \htmladdnormallink{double algebroid}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html} (i.e. a set with two \htmladdnormallink{algebroid}{http://planetphysics.us/encyclopedia/Algebroids.html} structures)
\begin{equation}
\label{Rsqu} A\D= \vcenter{\xymatrix @=3pc {AS \ar @<1ex> [r] ^{s^1} \ar @<-1ex> [r]
_{t^1} \ar @<1ex> [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & AH \ar[l]
\ar @<1ex> [d]^{\,t}
\ar @<-1ex> [d]_s \\
AV \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l]
\ar[u] }}
\end{equation}
for which
\begin{equation}
AS\quadr{h}{v}{v'}{h'}
\end{equation}
is the free $A$-module on the set of squares with the given
\htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}. The two compositions are then bilinear in the obvious
sense. Alternatively, we can use the \htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} construction
$\bar{A}\D$ induced by the convolution C*--algebra over $H$ and
$V$. These ideas about algebroids need further development in the light of the
algebra of \htmladdnormallink{crossed modules}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} of algebroids, developed in (Mosa,
1986, Brown and Mosa, 1986) as well as crossed cubes of (C*)
algebras following Ellis (1988).
\end{document}
