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![[parent]](https://images.physicslibrary.org/images/uparrow.png)
double groupoid geometry
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(Definition)
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The geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:
![$\displaystyle \mathsf{D}= \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar... ... [d]_s \ V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$](http://images.physicslibrary.org/cache/objects/830/l2h/img1.png "$\displaystyle \mathsf{D}= \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar... ... [d]_s \ V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$") |
(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
![$\displaystyle \begin{pmatrix}& h& \\ [-0.9ex] v & & v'\\ [-0.9ex]& h'& \end{pmatrix}$](http://images.physicslibrary.org/cache/objects/830/l2h/img9.png "$\displaystyle \begin{pmatrix}& h& \\ [-0.9ex] v & & v'\\ [-0.9ex]& h'& \end{pmatrix}$") |
(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
![$\displaystyle \begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pma... ...}=\begin{pmatrix}& hk& \\ [-1.1ex] v & & v''\\ [-1.1ex]& h'k'& \end{pmatrix} ~.$](http://images.physicslibrary.org/cache/objects/830/l2h/img14.png "$\displaystyle \begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pma... ...}=\begin{pmatrix}& hk& \\ [-1.1ex] v & & v''\\ [-1.1ex]& h'k'& \end{pmatrix} ~.$") |
(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
![$S\begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pmatrix}$](http://images.physicslibrary.org/cache/objects/830/l2h/img17.png "$S\begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pmatrix}$") 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)
![$\displaystyle A\mathsf{D}= \vcenter{\xymatrix @=3pc {AS \ar @<1ex> [r] ^{s^1} \... ...[d]_s \ AV \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$](http://images.physicslibrary.org/cache/objects/830/l2h/img19.png "$\displaystyle A\mathsf{D}= \vcenter{\xymatrix @=3pc {AS \ar @<1ex> [r] ^{s^1} \... ...[d]_s \ AV \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }}$") |
(0.4) |
for which
![$\displaystyle AS\begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pmatrix}$](http://images.physicslibrary.org/cache/objects/830/l2h/img20.png "$\displaystyle AS\begin{pmatrix}& h& \\ [-1.1ex] v & & v'\\ [-1.1ex]& h'& \end{pmatrix}$") |
(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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"double groupoid geometry" is owned by bci1.
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This object's parent.
Cross-references: crossed modules, convolution, boundary, algebroid, double algebroid, commutative C*--algebra, functor, category of groupoids, groupoids, double groupoid, representation, compositions, squares
This is version 4 of double groupoid geometry, born on 2010-01-28, modified 2010-02-13.
Object id is 830, canonical name is DoubleGroupoidGeometry.
Accessed 400 times total.
Classification:
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Pending Errata and Addenda
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