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\section{Introduction}
\begin{definition}
Let us recall that if $X$ is a topological space, then a \emph{double goupoid} $\D$
is defined by the following categorical diagram of linked groupoids and sets:
\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 $M$ is a set of points, $H,V$ are two groupoids (called, respectively, ``horizontal'' and ``vertical'' groupoids)
, and $S$ is a set of \PMlinkname{squares with two composition laws, $\bullet$ and $\circ$}{ThinSquare} (as first defined and represented in ref. \cite{BHKP} 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'' (\cite{BHKP}).
\end{definition}
\subsubsection{Homotopy double groupoid and homotopy 2-groupoid}
The algebraic composition laws, $\bullet$ and $\circ$, employed above to define a double groupoid $\D$ allow one also to define $\D$ as a groupoid internal to the \PMlinkname{category of groupoids}{GroupoidCategory}. Thus, in the particular case of a Hausdorff space, $X_H$, a double groupoid called the \emph{homotopy double groupoid of $X_H$} can be denoted as follows
$$\boldsymbol{\rho}^{\square}_2 (X_H) := \D ,$$
where $\square$ is in this case a \PMlinkname{thin square}{ThinSquare}. 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. \cite{BHKP}. 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. \cite{BHKP}).
\section{2-category of double groupoids}
\begin{definition}
The \PMlinkname{2-category}{2Category}, $\G^2$-- whose objects (or $2$-cells) are the above diagrams $\D$ that define double groupoids, and whose $2$-morphisms are functors $\mathbb{F}$ between double groupoid $\D$ diagrams-- is called the \emph{double groupoid 2-category}, or the \emph{2-category of double groupoids}.
\end{definition}
\begin{remark}
$\G^2$ is a relatively simple example of a category of diagrams, or a 1-supercategory, $\S_1$.
\end{remark}
\begin{thebibliography}{9}
\bibitem{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter.,
\PMlinkexternal{A homotopy double groupoid of a Hausdorff space}{http://www.tac.mta.ca/tac/volumes/10/2/10-02.pdf} ,
{\it Theory and Applications of Categories} \textbf{10},(2002): 71-93.
\bibitem{BS1}
R. Brown and C.B. Spencer: Double groupoids and crossed modules, \emph{Cahiers Top. G\'eom.Diff.},
\textbf{17} (1976), 343--362.
\bibitem{BMos}
R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
\bibitem{HKK}
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff
\emph{Applied Categorical Structures}, \textbf{8} (2000): 209-234.
\bibitem{Agl-Br-St2k2}
Al-Agl, F.A., Brown, R. and R. Steiner: 2002, Multiple categories: the equivalence of a globular and cubical approach, \emph{Adv. in Math}, \textbf{170}: 711-118.
\end{thebibliography}
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