1 2-Category of Double Groupoids
This is a topic entry on the 2-category of double groupoids.
1.1 Introduction
Definition 1.1. Let us recall that if X is a topological space, then a double goupoid 𝒟 is
defined by the following categorical diagram of linked groupoids and sets:
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, ∙ 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 X which factors through a tree”
([1]).
1.2 Homotopy double groupoid and homotopy 2-groupoid
The algebraic composition laws, ∙ and ∘, employed above to define a double groupoid 𝒟 allow one
also to define 𝒟 as a groupoid internal to the category of groupoids. Thus, in the particular case of
a Hausdorff space, XH, a double groupoid called the homotopy double groupoid of XH 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 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, 𝒢2– whose objects (or 2-cells) are the above diagrams
𝒟 that define double groupoids, and whose 2-morphisms are functors 𝔽 between double
groupoid 𝒟 diagrams– is called the double groupoid 2-category, or the 2-category of double
groupoids.
Remark 1.1. 𝒢2 is a relatively simple example of a category of diagrams, or a
1-supercategory, §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.
