# PlanetPhysics/2 Category of Double Groupoids

## 2-Category of Double GroupoidsEdit

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

### IntroductionEdit

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:

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 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-groupoidEdit

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* Thin Equivalence 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 GroupoidsEdit

The 2-category, **Failed to parse (unknown function "\G"): {\displaystyle \G^2}**
-- whose objects (or -cells) are the above diagrams **Failed to parse (unknown function "\D"): {\displaystyle \D}**
that define double groupoids, and whose -morphisms are functors 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, .

## All SourcesEdit

^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}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. - ↑
R. Brown and C.B. Spencer: Double groupoids and crossed modules,
*Cahiers Top. Géom.Diff.*,**17**(1976), 343--362. - ↑ R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
- ↑
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff
*Applied Categorical Structures*,**8**(2000): 209-234. - ↑
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.