PlanetPhysics/Double Groupoid Geometry

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Double Groupoids

The geometry of [[../PiecewiseLinear/|squares]] and their [[../Cod/|compositions]] leads to a common [[../CategoricalGroupRepresentation/|representation]] of a [[../ThinEquivalence/|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' [[../GroupoidHomomorphism2/|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 [[../GroupoidCategory/|category of groupoids]].

Given two groupoids $H,V$ over a set $M$, there is a double groupoid $◻(H,V)$ 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 $h,h^{\prime }\in H,v,v^{\prime }\in V$ 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 [[../TrivialGroupoid/|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 [[../OrthomodularLatticeTheory/|commutative C*--algebra]] $A$ a [[../GeneralizedSuperalgebras/|double algebroid]] (i.e. a set with two [[../Algebroids/|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 [[../PiecewiseLinear/|boundary]]. The two compositions are then bilinear in the obvious sense. Alternatively, we can use the [[../AssociatedGroupoidAlgebraRepresentations/|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 [[../CubicalHigherHomotopyGroupoid/|crossed modules]] of algebroids, developed in (Mosa, 1986, Brown and Mosa, 1986) as well as crossed cubes of (C*) algebras following Ellis (1988).