homotopy addition lemma and corollary

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.

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

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 $f : \boldsymbol{\rho}^{\square} (X)\to \mathsf{D}$ is a morphism of double groupoids with connections, any singular cube in $X$ determines a commutative 3-shell in $\mathsf{D}$ .

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.

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