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.

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

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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