Homotopy addition lemma

Let be a morphism of double groupoids with connection. If $\alpha \in {\boldsymbol{\rho}^\square_2}(X)$ is thin, then $f(\alpha)$ is thin.

Remarks

Corollary

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 $\boldsymbol{\rho}^{\square} (X)$ ([1], Proposition 5.5). Consequently, if is a morphism of double groupoids with connections, any singular cube in $X$ determines a commutative 3-shell in .

Bibliography

1

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