PlanetPhysics/Hamiltonian Algebroid

Homotopy addition lemmaEdit

Let Failed to parse (syntax error): {\displaystyle f: \boldsymbol{\rho ^\square(X) \to \mathsf D} be a morphism of double groupoids with connection. If is thin, then is thin.}

RemarksEdit

The groupoid employed here is as defined by the cubically thin homotopy on the set of squares. Additional explanations of the data, including concepts such as path groupoid and homotopy double groupoid are provided in an attachment.

CorollaryEdit

\emph{Let be a singular cube in a Hausdorff space . Then by restricting to the faces of and taking the corresponding elements in , we obtain a cube in which is commutative by the Homotopy addition lemma for ([1], proposition 5.5). Consequently, if is a morphism of double groupoids with connections, any singular cube in determines a [3-shell commutative]{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in .}

All SourcesEdit

[1]

ReferencesEdit

  1. 1.0 1.1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, {\it Theory and Applications of Categories.} 10 ,(2002): 71-93.