double groupoid geometry
The geometry of squares
and their compositions
leads to a common representation
of a double groupoid in the following form:
![$\displaystyle \mathsf{D}= \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar...
...> [d]_s \\ V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$](img1.png) |
(0.1) |
where
is a set of `points',
are
`horizontal' and `vertical' groupoids, and
is a set of
`squares' with two compositions. The laws for a double groupoid
make it also describable as a groupoid internal to the category of groupoids.
Given two groupoids
over
a set
, there is a double groupoid
with
as
horizontal and vertical edge groupoids, and squares given by
quadruples
![$\displaystyle \begin{pmatrix}& h& \\ [-0.9ex] v & & v'\\ [-0.9ex]& h'& \end{pmatrix}$](img9.png) |
(0.2) |
for which we assume always that
and
that the initial and final points of these edges match in
as
suggested by the notation, that is for example
, etc. The compositions are to be inherited from those of
,
that is
![$\displaystyle \begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pma...
...}=\begin{pmatrix}& hk& \\ [-1.1ex] v & & v''\\ [-1.1ex]& h'k'& \end{pmatrix} ~.$](img14.png) |
(0.3) |
This construction is right adjoint to the forgetful functor
which
takes the double groupoid as above, to the pair of groupoids
over
. Now given a general double groupoid as above, we can
define
to be the set of squares with these
as horizontal and vertical edges.
This allows us to construct for at least a commutative C*-algebra
a double algebroid
(i.e. a set with two algebroid
structures)
![$\displaystyle A\mathsf{D}= \vcenter{\xymatrix @=3pc {AS \ar @<1ex> [r] ^{s^1} \...
... [d]_s \\ AV \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$](img19.png) |
(0.4) |
for which
![$\displaystyle AS\begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pmatrix}$](img20.png) |
(0.5) |
is the free
-module on the set of squares with the given
boundary. The two compositions are then bilinear in the obvious
sense. Alternatively, we can use the convolution
construction
induced by the convolution C*-algebra over
and
. These ideas about algebroids need further development in the light of the
algebra of crossed modules
of algebroids, developed in (Mosa,
1986, Brown and Mosa, 1986) as well as crossed cubes of (C*)
algebras following Ellis (1988).
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As of this snapshot date, this entry was owned by bci1.