PlanetPhysics/Hamiltonian Algebroid

Homotopy addition lemma
Let $$f: \boldsymbol{\rho ^\square(X) \to \mathsf D$$ be a morphism of double groupoids with connection. If $$\alpha \in {\boldsymbol{\rho}^\square_2}(X)$$ is thin, then $$f(\alpha)$$ is thin.}

Remarks
The groupoid $${\boldsymbol{\rho}^\square_2}(X)$$ employed here is as defined by the cubically thin homotopy on the set $$R^{\square}_2(X)$$ 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 $$\boldsymbol{\rho}^{\square}_2 (X)$$, 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 $$\mathsf{D}$$.}