PlanetPhysics/Double Groupoid With Connection

Introduction: Geometrically defined double groupoid with connection

In the setting of a [[../PiecewiseLinear/|geometrically defined double groupoid with connection]], as in , (resp. ), there is an appropriate notion of geometrically thin [[../PiecewiseLinear/|square]]. It was proven in , ([[../Formula/|theorem]] 5.2 (resp. , [[../Predicate/|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 $\mathrm{\Phi }:|K|⟶|L|$ where $K$ and $L$ are (finite) [[../PiecewiseLinear/|simplicial complexes]] is PWL ({\it [[../PiecewiseLinear/|piecewise linear]]}) if there exist subdivisions of $K$ and $L$ relative to which $\mathrm{\Phi }$ is [[../PiecewiseLinear/|simplicial]].

Remarks

We briefly recall here the related [[../PreciseIdea/|concepts]] involved: A square $u:I^2⟶X$ in a [[../CoIntersections/|topological]] space $X$ is thin if there is a factorisation of $u$, $u:I^2\stackrel{{\mathrm{\Phi }}_u}{⟶}{J}_u\stackrel{p_u}{⟶}X,$ where $J_u$ is a [[../Tree/|tree]] and ${\mathrm{\Phi }}_u$ is piecewise linear (PWL, as defined next) on the [[../PiecewiseLinear/|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 $I^2$.