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# 2-category of double groupoids

# 1 2-Category of Double Groupoids

This is an introduction to the subject of 2-category of double groupoids, including the definition of this new concept. It can also be further generalized to the 2-category of double categories by removing the constrained that all double groupoid homomorphisms be invertible and by replacing the double groupoid with a double category; naturally, double groupoid homomorphisms are then replaced by 2-functors (that need not be invertible) between double categories.

# 1.1 Introduction

###### Definition 1.1.

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:

$\mathcal{D}:=\vbox{\xymatrix@=3pc {S \ar@<1ex> [r]{}^{{s^{1}}} \ar@<-1ex> [r]% {}_{{t^{1}}} \ar@<1ex> [d]^{{\, t_{2}}} \ar@<-1ex> [d]_{{s_{2}}} & H \ar[l] \ar@<1ex> [d]^{{\,t}} \ar@<-1ex> [d]_{s} \\ V \ar[u] \ar@<1ex> [r]{}^{s} \ar@<-1ex> [r]{}_{t} & M \ar[l] \ar[u]}},$ | (1.1) |

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. [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 $X$ which factors through a tree” ([1]).

# 1.2 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 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. [1]. 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. [1]).

# 1.3 Defintion of 2-Category of Double Groupoids

###### Definition 1.2.

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

###### Remark 1.1.

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

# References

- 1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter., A homotopy double groupoid of a Hausdorff space , 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.

## Mathematics Subject Classification

18-00*no label found*18C10

*no label found*

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