0.1 Homotopy addition lemma

Let f : ρ^□(X) →D be a morphism of double groupoids with connection. If α ∈ ρ ₂^□(X) is thin, then f(α) is thin.

0.1.1 Remarks

The groupoid ρ₂^□(X) employed here is as defined by the cubically thin homotopy on the set R₂^□(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 : I³ → X be a singular cube in a Hausdorff space X. Then by restricting u to the faces of I³ and taking the corresponding elements in ρ₂^□(X), we obtain a cube in ρ^□(X) which is commutative by the Homotopy addition lemma for ρ^□(X) ([1], proposition 5.5). Consequently, if f : ρ^□(X) →D is a morphism of double groupoids with connections, any singular cube in X determines a commutative 3-shell in D.

References