# PlanetPhysics/Hamiltonian Algebroid

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

#### Remarks

The groupoid ${\displaystyle {{\boldsymbol {\rho }}_{2}^{\square }}(X)}$ employed here is as defined by the cubically thin homotopy on the set ${\displaystyle R_{2}^{\square }(X)}$ 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 ${\displaystyle u:I^{3}\to X}$ be a singular cube in a Hausdorff space ${\displaystyle X}$. Then by restricting ${\displaystyle u}$ to the faces of ${\displaystyle I^{3}}$ and taking the corresponding elements in ${\displaystyle {\boldsymbol {\rho }}_{2}^{\square }(X)}$, we obtain a cube in ${\displaystyle {\boldsymbol {\rho }}^{\square }(X)}$ which is commutative by the Homotopy addition lemma for ${\displaystyle {\boldsymbol {\rho }}^{\square }(X)}$ ([1], proposition 5.5). Consequently, if ${\displaystyle f:{\boldsymbol {\rho }}^{\square }(X)\to {\mathsf {D}}}$ is a morphism of double groupoids with connections, any singular cube in ${\displaystyle X}$ determines a [3-shell commutative]{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in ${\displaystyle {\mathsf {D}}}$.}

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## References

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.