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 ${\displaystyle X}$  is a topological space, then a double goupoid ${\displaystyle {\mathcal {D}}}$  is defined by the following categorical diagram of linked groupoids and sets:

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

where ${\displaystyle M}$  is a set of points, ${\displaystyle H,V}$  are two groupoids (called, respectively, "horizontal" and "vertical" groupoids) , and ${\displaystyle S}$  is a set of squares with two composition laws, ${\displaystyle \bullet }$  and ${\displaystyle \circ }$ ]] (as first defined and represented in ref. [1] 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 ${\displaystyle X}$  which factors through a tree" ([1]).

Homotopy double groupoid and homotopy 2-groupoid

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

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

where ${\displaystyle \square }$  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. [1]. One notes however a significant distinction between a homotopy 2-groupoid and homotopy double groupoid construction; thus, the construction of the ${\displaystyle 2}$ -cells of the homotopy double groupoid is based upon a suitable cubical approach to the notion of thin ${\displaystyle 3}$ -cube, whereas the construction of the 2-cells of the homotopy ${\displaystyle 2}$ -groupoid can be interpreted by means of a globular notion of thin ${\displaystyle 3}$ -cube. "The homotopy double groupoid of a space, and the related homotopy ${\displaystyle 2}$ -groupoid, are constructed directly from the cubical singular complex and so (they) remain close to geometric intuition in an almost classical way" (viz. [1]).

Defintion of 2-Category of Double Groupoids

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

$\displaystyle \G^2$ is a relatively simple example of a category of diagrams, or a 1-supercategory, ${\displaystyle \S _{1}}$ .

References

1. R. Brown, K.A. Hardie, K.H. Kamps and T. Porter., A homotopy double groupoid of a Hausdorff space , {\it Theory and Applications of Categories} 10 ,(2002): 71-93.
2. R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff. , 17 (1976), 343--362.
3. R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
4. K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures , 8 (2000): 209-234.
5. Al-Agl, F.A., Brown, R. and R. Steiner: 2002, Multiple categories: the equivalence of a globular and cubical approach, Adv. in Math , 170 : 711-118.