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