2-Category of Double Groupoids

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

Introduction

Definition 1.1   Let us recall that if is a topological space, then a double goupoid is defined by the following categorical diagram of linked groupoids and sets:

(1.1)

where is a set of points, are two groupoids (called, respectively, “horizontal” and “vertical” groupoids) , and is a set of squares with two composition laws, and (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 which factors through a tree” ([1]).

Homotopy double groupoid and homotopy 2-groupoid

The algebraic composition laws, and , employed above to define a double groupoid allow one also to define as a groupoid internal to the category of groupoids. Thus, in the particular case of a Hausdorff space, , a double groupoid called the homotopy double groupoid of can be denoted as follows

where 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 -cells of the homotopy double groupoid is based upon a suitable cubical approach to the notion of thin -cube, whereas the construction of the 2-cells of the homotopy -groupoid can be interpreted by means of a globular notion of thin -cube. “The homotopy double groupoid of a space, and the related homotopy -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

Definition 1.2   The 2-category, - whose objects (or -cells) are the above diagrams that define double groupoids, and whose -morphisms are functors between double groupoid diagrams- is called the double groupoid 2-category, or the 2-category of double groupoids.

Remark 1.1   is a relatively simple example of a category of diagrams, or a 1-supercategory, .

Bibliography

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.