# PlanetPhysics/Homotopy Double Groupoid 2

\newcommand{\sqdiagram}{$\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}$ }

===Homotopy double groupoid of a Hausdorff space===


Let $X$ be a Hausdorff space. Also consider the HDA concept of a double groupoid, and how it can be completely specified for a Hausdorff space, $X$ . Thus, in ref.  Brown et al. associated to $X$ a double groupoid, ${\boldsymbol {\rho }}_{2}^{\square }(X)$ , called the 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 objects.

Generally, the geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:

$\displaystyle (1) \D = \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar @<-1ex> [r] _{t^1} \ar @<1ex> [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H \ar[l] \ar @<1ex> [d]^{\,t} \ar @<-1ex> [d]_s \\ V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u]}},$

where $M$ is a set of points', $H,V$ are horizontal' and vertical' groupoids, and $S$ is a set of squares' with two compositions.

The laws for a double groupoid are also defined, more generally, for any topological space $\mathbb {T}$ , and make it also describable as a groupoid internal to the category of groupoids. 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

$pmatrix}"): \begin{pmatrix} & h& \{{center top}}$-0.9ex] v & & v'\{{center top}}[itex]-0.9ex]& h'& \end{pmatrix}$ 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.Thecompositionsaretobeinheritedfromthoseof$H,V$ , that is: \bigbreak $\displaystyle \quadr{h}{v}{v'}{h'} \circ_1\quadr{h'}{w}{w'}{h''} =\quadr{h}{vw}{v'w'}{h''}, \;\quadr{h}{v}{v'}{h'} \circ_2\quadr{k}{v'}{v''}{k'}=\quadr{hk}{v}{v''}{h'k'} ~.$ Alternatively, the data for the above double groupoid $\displaystyle \D$ 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 thin equivalence and the 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 : $matrix}"): \begin{matrix} (\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}_1^{\square} (X) , \partial^{-}_{1} , \partial^{+}_{1} , +_{1} , \varepsilon _{1}) , \boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}^{\square}_1 (X) , \partial^{-}_{2} , \partial^{+}_{2} , +_{2} , \varepsilon _{2})\{{center top}}[itex]3mm] (\boldsymbol{\rho}^{\square}_1 (X) , X , \partial^{-} , \partial^{+} , + , \varepsilon). \end{matrix$ \bigbreak Here ${\boldsymbol {\rho }}_{1}(X)$ denotes the path groupoid of $X$ from ref.  where it was defined as follows. The objects of ${\boldsymbol {\rho }}_{1}(X)$ are the points of $X$ . The morphisms of ${\boldsymbol {\rho }}_{1}^{\square }(X)$ are the equivalence classes of paths in $X$ with respect to the following (thin) equivalence relation $\sim _{T}$ , defined as follows. The data for ${\boldsymbol {\rho }}_{2}^{\square }(X)$ is defined last; furthermore, the symbols specified after the thin square 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 operations ($\partial _{1}^{-}$ , $\varepsilon _{2}$ ... , 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 diagram (0.1), or $\displaystyle \D$ , shown at the top of this page. \end{definition} \begin{definition} Thin Equivalence Let $a,a':x\simeq y$ be paths in $X$ . Then $a$ is 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 . We use $\langle a\rangle :x\simeq y$ to denote the $\sim _{T}$ class of a path $a:x\simeq y$ and call $\langle a\rangle [itex]the{\it {semitrack}}of$ a$. The groupoid structure of ${\boldsymbol {\rho }}_{1}^{\square }(X)$ 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

$matrix}"): \begin{matrix}{r} (a+a') + a'' \simeq a+ (a' +a'') : x \simeq x''' \ ({\it rescale}) \\ a+e_{x'} \simeq a:x \simeq x' ; \ e_{x} + a \simeq a: x \simeq x' \ ({\it dilation}) \\ a+(-a) \simeq e_{x} : x \simeq x \ ({\it cancellation}). \end{matrix}$

The source and target maps of ${\boldsymbol {\rho }}_{1}^{\square }(X)$ are given by $\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,$ if $\langle a\rangle :x\simeq y$ is a semitrack. Identities 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 ${\boldsymbol {\rho }}_{2}^{\square }(X)$ data in Definition 0.1, R. Brown et al. defined as follows a \htmladdnormallink{relation {http://planetphysics.us/encyclopedia/Bijective.html} of cubically thin homotopy} on the set $R_{2}^{\square }(X)$ of squares.

\begin{definition} Cubically Thin Homotopy

Let $u,u'$ be squares in $X$ with common vertices.

1. A {\it cubically thin homotopy} $U:u\equiv _{T}^{\square }u'$ between $u$ and $u'$ is a cube $U\in R_{3}^{\square }(X)$ such that

(i) $U$ is a homotopy between $u$ and $u',$ \begin{center} i.e. $\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$ \end{center} (ii) $U$ is rel. vertices of $I^{2},$ \begin{center} i.e. $\displaystyle \partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2 \partial^{+}_2 (U),\enskip \partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2 \partial^{+}_2 (U)$ are constant,\end{center} (iii) the faces $\partial _{i}^{\alpha }(U)$ are thin for $\alpha =\pm 1,\ i=1,2$ .

1. The square $u$ is {\it cubically} $T$ -{\it equivalent} to

$u',$ denoted $u\equiv _{T}^{\square }u'$ if there is a cubically thin homotopy between $u$ and $u'.$ \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 double category. \end{remark}