homotopy double groupoid of a Hausdorff space

Homotopy double groupoid of a Hausdorff space

Let $X$ be a Hausdorff space. Also consider the HDA concept of adouble groupoid (http://planetmath.org/HigherDimensionalAlgebraHDA),and how it can be completely specified for a Hausdorff space, $X$. Thus, inref. [BHKP] Brown et al. associated to $X$ a double groupoid, ${𝝆}_2^{\mathrm{\square }}(X)$, called the homotopy double groupoid of X which is completely defined by the data specified in Definitions 0.1 to 0.3 in this entry and related objects.

Generally, the geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:

$$𝖣=$$(0.1)