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%%% Primary Title: homotopy double groupoid of a Hausdorff space
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\begin{document}
\subsection{Homotopy double groupoid of a Hausdorff space}
Let $X$ be a Hausdorff space. Also consider the \htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of a
\htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebraHDA.html},
and how it can be completely specified for a Hausdorff space, $X$. Thus, in
ref. \cite{BHKP} Brown et al. associated to $X$ a double groupoid, $\boldsymbol{\rho}^{\square}_2 (X)$
, called the {\em homotopy double groupoid of X} which is completely defined by the data specified in Definitions 0.1 to 0.3 in this entry and related \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}.
Generally, the geometry of \htmladdnormallink{squares}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} and their \htmladdnormallink{compositions}{http://planetphysics.us/encyclopedia/Cod.html} leads to a common \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of a \emph{double groupoid} in the following form:
\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 `horizontal' and `vertical' \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}, and $S$ is a set of
`squares' with two compositions.
The laws for a double groupoid are also defined, more generally, for any \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space $\mathbb{T}$, and make it also describable as a groupoid internal to the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}. Further details of this general definition are provided next.
Given two groupoids $H,V$ over a set $M$, there is a double groupoid $\Box(H,V)$ with $H,V$ as
horizontal and vertical edge groupoids, and squares given by
quadruples
\bigbreak
\begin{equation}
\begin{pmatrix} & h& \\[-0.9ex] v & & v'\\[-0.9ex]& h'&
\end{pmatrix}
\end{equation}
for which we assume always that $h,h' \in H, \, v,v' \in V$ and
that the initial and final points of these edges match in $M$ as
suggested by the notation, that is for example $sh=sv, th=sv',
\ldots$, etc. The compositions are to be inherited from those of
$H,V$,
that is:
\bigbreak
\begin{equation}
\quadr{h}{v}{v'}{h'} \circ_1\quadr{h'}{w}{w'}{h''}
=\quadr{h}{vw}{v'w'}{h''}, \;\quadr{h}{v}{v'}{h'}
\circ_2\quadr{k}{v'}{v''}{k'}=\quadr{hk}{v}{v''}{h'k'} ~.
\end{equation}
Alternatively, the data for the above double groupoid $\D$ can be specified as a triple of groupoid structures:
$$(D_2,D_1, \partial^{-}_{1}, \partial^{+}_{1}, +_1,\varepsilon_1), (D_2,D_1,\partial^{-}_{2}, \partial^{+}_{2}, +_2, \varepsilon_2), (D_1, D_0, \partial^{-}_{1}, \partial^{+}_{1}, + , \varepsilon),$$
where:
$$D_0 = M ~,~ D_1= V = H ~,~ D_2 = S,$$
$$s^1 = \partial^{-}_{2}~,~ t^1 = \partial^{+}_{2}~,~ s_2 = s= \partial^{-}_{1}$$ and
$$t_2 = t = \partial^{+}_{1}.$$
Then, as a first step, consider this data for the homotopy double groupoid specified in the following definition; in order to specify completely such data one also needs to define the related concepts of \emph{thin equivalence} and the \emph{relation of cubically thin homotopy}, as provided in the two definitions following the homotopy double groupoid data specified above and in the (main) Definition 0.1.
\begin{definition}
The data for the homotopy double groupoid, $\boldsymbol{\rho}^{\square} (X) $,
will be denoted by :
\[
\begin{array}{c}
(\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}_1^{\square} (X) ,
\partial^{-}_{1} , \partial^{+}_{1} , +_{1} , \varepsilon _{1}) ,
\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}^{\square}_1 (X) ,
\partial^{-}_{2} , \partial^{+}_{2} , +_{2} , \varepsilon _{2})\\[3mm]
(\boldsymbol{\rho}^{\square}_1 (X) , X , \partial^{-} , \partial^{+} , + , \varepsilon).
\end{array}\]
\bigbreak
Here $\boldsymbol{\rho}_1 (X)$ denotes the \emph{path groupoid} of $X$
from ref. \cite{HKK} where it was defined as follows. The objects of
$\boldsymbol{\rho}_1 (X) $ are the points of $ X $. The \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of
$\boldsymbol{\rho}^\square_1 (X) $ are the equivalence classes of paths in $ X$ with respect to the following (thin) \htmladdnormallink{equivalence relation}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $ \sim_{T} $, defined as follows. The data for $\boldsymbol{\rho}^{\square}_2 (X)$ is defined last; furthermore, the symbols specified after the \htmladdnormallink{thin square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} symbol specify both the sides (or the groupoid `dimensions') of the square which are involved (i.e., 1 and 2, respectively), and also the order in which the shown \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} ($\partial^{-}_{1}$, $\varepsilon _{2}$... , etc) are to be performed relative to the thin square specified for each groupoid, $\rho_1 ~ or~ \rho_2$; moreover, all such symbols are explicitly and precisely defined in the related entries of the concepts involved in this definition. These two groupoids can also be pictorially represented as the $(H,V)$ pair depicted in the large \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} (0.1), or $\D$, shown at the top of this page.
\end{definition}
\begin{definition} \emph{Thin Equivalence}
Let $ a,a' : x \simeq y $ be paths in $ X $. Then
$ a$ is \emph{ thinly equivalent} to $ a' $, denoted $ a \sim_{T} a' $, if
there is a thin relative homotopy between $ a $ and $ a' $.
We note that $ \sim_{T} $ is an equivalence relation, see
\cite{BHKP}. We use $ \langle a \rangle : x \simeq y $ to denote
the $ \sim_{T} $ class of a path $ a: x \simeq y $ and call $
\langle a \rangle $ the {\it semitrack} of $ a $. The groupoid
structure of $ \boldsymbol{\rho}^\square_1 (X) $ is induced by concatenation,
+, of paths. Here one makes use of the fact that if $ a: x \simeq
x', \ a' : x' \simeq x'', \ a'' : x'' \simeq x''' $ are paths then
there are canonical thin relative homotopies
\[
\begin{array}{r}
(a+a') + a'' \simeq a+ (a' +a'') : x \simeq x''' \ ({\it rescale}) \\
a+e_{x'} \simeq a:x \simeq x' ; \ e_{x} + a \simeq a: x \simeq x' \
({\it dilation}) \\
a+(-a) \simeq e_{x} : x \simeq x \ ({\it cancellation}).
\end{array}
\]
The source and \htmladdnormallink{target maps}{http://planetphysics.us/encyclopedia/SmallCategory.html} of $\boldsymbol{\rho}^\square_1 (X)$ are given by
$$\partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1}
\langle a\rangle =y,$$
if $\langle a\rangle :x\simeq y$ is a semitrack. \htmladdnormallink{Identities}{http://planetphysics.us/encyclopedia/Cod.html} and inverses
are given by
$$\varepsilon (x)=\langle e_x\rangle \quad \mathrm{ resp.} -\langle a\rangle
=\langle -a \rangle.$$
\end{definition}
At the next step, in order to construct the groupoid $\boldsymbol{\rho}^{\square}_2 (X)$ data in Definition 0.1, R. Brown et al. defined as follows a
\emph{\htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{cubically thin homotopy}{http://planetphysics.us/encyclopedia/CubicallyThinHomotopy2.html}} on the set $R^{\square}_2(X)$ of squares.
\begin{definition} \textbf{Cubically Thin Homotopy}
Let $u,u'$ be squares in $X$ with common vertices.
\begin{enumerate}
\item A {\it cubically thin homotopy} $U:u\equiv^{\square}_T u'$
between $u$ and $u'$ is a cube $U\in R^{\square}_3(X)$ such that
(i) $U$ is a homotopy between $u$ and $u',$
\begin{center}
i.e. $\partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$\end{center}
(ii) $U$ is rel. vertices of $I^2,$
\begin{center}
i.e. $\partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2
\partial^{+}_2 (U),\enskip
\partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2
\partial^{+}_2 (U)$ are
constant,\end{center}
(iii) the faces $ \partial^{\alpha}_{i} (U) $ are thin for $ \alpha =
\pm 1, \ i = 1,2 $.
\item The square $u$ is {\it cubically} $T$-{\it equivalent} to
$u',$ denoted $u\equiv^{\square}_T u'$ if there is a cubically
thin homotopy between $u$ and $u'.$
\end{enumerate}
\end{definition}
\begin{remark}
By removing from the above double groupoid construction the condition that all morphisms must be invertible one obtains the prototype of a
\emph{\htmladdnormallink{double category}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}}.
\end{remark}
\begin{thebibliography}{9}
\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{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter.,
\htmladdnormallink{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.
\end{thebibliography}
\end{document}
