PlanetPhysics/Double Groupoid With Connection

Introduction: Geometrically defined double groupoid with connection
In the setting of a geometrically defined double groupoid with connection, as in, (resp. ), there is an appropriate notion of geometrically thin square. It was proven in , (theorem 5.2 (resp. , proposition 4)), that in the cases there specified geometrically and algebraically \htmladdnormallink{thin squares {http://planetphysics.us/encyclopedia/Tree.html} coincide}.

Basic definitions
A map $$ \Phi : |K| \longrightarrow |L| $$ where $$ K $$ and $$ L $$ are (finite) simplicial complexes is PWL ({\it piecewise linear}) if there exist subdivisions of $$ K $$ and $$ L $$ relative to which $$ \Phi$$ is simplicial.

Remarks
We briefly recall here the related concepts involved: A square $$ u:I^{2} \longrightarrow X $$ in a topological space $$ X $$ is thin  if there is a factorisation of $$ u $$, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $$J_{u}$$ is a tree and $$ \Phi_{u} $$ is piecewise linear (PWL, as defined next) on the boundary $$ \partial{I}^{2} $$ of $$ I^{2} $$.

A {\it tree}, is defined here as the underlying space $$ |K| $$ of a finite $$ 1 $$-connected $$ 1 $$-dimensional simplicial complex $$ K $$ boundary $$ \partial{I}^{2} $$ of $$ I^{2} $$.