PlanetPhysics/Hamiltonian Algebroid

Homotopy addition lemma

Let Failed to parse (syntax error): {\displaystyle f: \boldsymbol{\rho ^\square(X) \to \mathsf D} be a morphism of double groupoids with connection. If $\alpha \in {{\boldsymbol {\rho }}_{2}^{\square }}(X)$ is thin, then $f(\alpha )$ is thin.}

Remarks

The groupoid ${{\boldsymbol {\rho }}_{2}^{\square }}(X)$ employed here is as defined by the cubically thin homotopy on the set $R_{2}^{\square }(X)$ of squares. Additional explanations of the data, including concepts such as path groupoid and homotopy double groupoid are provided in an attachment.

Corollary

\emph{Let $u:I^{3}\to X$ be a singular cube in a Hausdorff space $X$. Then by restricting $u$ to the faces of $I^{3}$ and taking the corresponding elements in ${\boldsymbol {\rho }}_{2}^{\square }(X)$, we obtain a cube in ${\boldsymbol {\rho }}^{\square }(X)$ which is commutative by the Homotopy addition lemma for ${\boldsymbol {\rho }}^{\square }(X)$ (, proposition 5.5). Consequently, if $f:{\boldsymbol {\rho }}^{\square }(X)\to {\mathsf {D}}$ is a morphism of double groupoids with connections, any singular cube in $X$ determines a [3-shell commutative]{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in ${\mathsf {D}}$.}