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# homotopy addition lemma and corollary

# 0.1 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.*

# 0.1.1 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.

# 0.2 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 $\mathsf{D}$.*

# References

- 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.

## Mathematics Subject Classification

18D05*no label found*55N33

*no label found*55N20

*no label found*55U40

*no label found*

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