PlanetPhysics/Double Groupoid Geometry

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Double Groupoids

The geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:

Failed to parse (unknown function "\D"): {\displaystyle (1) \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] }} }

where $M$ is a set of `points', $H,V$ are `horizontal' and `vertical' groupoids, 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 category of groupoids.

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

Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \begin{pmatrix} & h& \<blockquote><math>-0.9ex] v & & v'\<blockquote><math>-0.9ex]& h'& \end{pmatrix} }

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 </math>sh=sv, th=sv', \ldots$,etc.Thecompositionsaretobeinheritedfromthoseof<math>H,V$, that is

Failed to parse (unknown function "\quadr"): {\displaystyle \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'} ~. }

This construction is right adjoint to the forgetful functor 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 Failed to parse (unknown function "\quadr"): {\displaystyle 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 commutative C*--algebra $A$ a double algebroid (i.e. a set with two algebroid structures)

Failed to parse (unknown function "\D"): {\displaystyle (2) 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] }} }

for which

Failed to parse (unknown function "\quadr"): {\displaystyle AS\quadr{h}{v}{v'}{h'} }

is the free $A$-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 Failed to parse (unknown function "\D"): {\displaystyle \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 crossed modules of algebroids, developed in (Mosa, 1986, Brown and Mosa, 1986) as well as crossed cubes of (C*) algebras following Ellis (1988).