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:

Failed to parse (unknown function "\begin{equation}"): {\displaystyle (1) \begin{equation} \label{squ} \D := \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar @<-1ex> [r] _{t^1} \ar @<1ex> [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H \ar[l] \ar @<1ex> [d]^{\,t} \ar @<-1ex> [d]_s \\ V \ar [u] \ar @<1ex> [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

${\boldsymbol {\rho }}_{2}^{\square }(X_{H}):={\mathcal {D}},$

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, Failed to parse (unknown function "\G"): {\displaystyle \G^2} -- whose objects (or $2$-cells) are the above diagrams Failed to parse (unknown function "\D"): {\displaystyle \D} that define double groupoids, and whose $2$-morphisms are functors $\mathbb {F}$ between double groupoid Failed to parse (unknown function "\D"): {\displaystyle \D} diagrams-- is called the double groupoid 2-category, or the 2-category of double groupoids .

Failed to parse (unknown function "\G"): {\displaystyle \G^2} is a relatively simple example of a category of diagrams, or a 1-supercategory, $\S _{1}$.