PlanetPhysics/2 Category of Double Groupoids

2-Category of Double Groupoids
This is a topic entry on the 2-category of double groupoids.

Introduction
Let us recall that if $$X$$ is a topological space, then a double goupoid $$\mathcal{D}$$ is defined by the following categorical diagram of linked groupoids and sets:

$$ (1) \begin{equation} \label{squ} \D := \vcenter{\xymatrix @=3pc {S \ar @ [r] ^{s^1} \ar @<-1ex> [r] _{t^1} \ar @ [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H \ar[l] \ar @ [d]^{\,t} \ar @<-1ex> [d]_s \\ V \ar [u] \ar @ [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u]}}, \end{equation}, $$

where $$M$$ is a set of points, $$H,V$$ are two groupoids (called, respectively, "horizontal" and "vertical" groupoids) , and $$S$$ is a set of squares with two composition laws, $$\bullet$$ and $$\circ$$]] (as first defined and represented in ref. by Brown et al.). A simplified notion of a thin square is that of "a continuous map from the unit square of the real plane into $$X$$ which factors through a tree".

Homotopy double groupoid and homotopy 2-groupoid
The algebraic composition laws, $$\bullet$$ and $$\circ$$, employed above to define a double groupoid $$\mathcal{D}$$ allow one also to define $$\mathcal{D}$$ as a groupoid internal to the category of groupoids. Thus, in the particular case of a Hausdorff space, $$X_H$$, a double groupoid called the homotopy Thin Equivalence double groupoid of $$X_H$$ can be denoted as follows

$$\boldsymbol{\rho}^{\square}_2 (X_H) := \mathcal{D} ,$$

where $$\square$$ is in this case a thin square. Thus, the construction of a homotopy double groupoid is based upon the geometric notion of thin square that extends the notion of thin relative homotopy as discussed in ref. . One notes however a significant distinction between a homotopy 2-groupoid and homotopy double groupoid construction; thus, the construction of the $$2$$-cells of the homotopy double groupoid is based upon a suitable cubical approach to the notion of thin $$3$$-cube, whereas the construction of the 2-cells of the homotopy $$2$$-groupoid can be interpreted by means of a globular notion of thin $$3$$-cube. "The homotopy double groupoid of a space, and the related homotopy $$2$$-groupoid, are constructed directly from the cubical singular complex and so (they) remain close to geometric intuition in an almost classical way" (viz. ).

Defintion of 2-Category of Double Groupoids
The 2-category, $$\G^2$$-- whose objects (or $$2$$-cells) are the above diagrams $$\D$$ that define double groupoids, and whose $$2$$-morphisms are functors $$\mathbb{F}$$ between double groupoid $$\D$$ diagrams-- is called the double groupoid 2-category, or the 2-category of double groupoids.

$$\G^2$$ is a relatively simple example of a category of diagrams, or a 1-supercategory, $$\S_1$$.