# PlanetPhysics/Homotopy Double Groupoid 2

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### Homotopy double groupoid of a Hausdorff space

Let ${\displaystyle X}$ be a Hausdorff space. Also consider the HDA concept of a double groupoid, and how it can be completely specified for a Hausdorff space, ${\displaystyle X}$. Thus, in ref. [1] Brown et al. associated to ${\displaystyle X}$ a double groupoid, ${\displaystyle {\boldsymbol {\rho }}_{2}^{\square }(X)}$ , 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 ${\displaystyle M}$ is a set of points', ${\displaystyle H,V}$ are horizontal' and vertical' groupoids, and ${\displaystyle S}$ is a set of squares' with two compositions.

The laws for a double groupoid are also defined, more generally, for any topological space ${\displaystyle \mathbb {T} }$, 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 ${\displaystyle H,V}$ over a set ${\displaystyle M}$, there is a double groupoid ${\displaystyle \Box (H,V)}$ with ${\displaystyle H,V}$ as horizontal and vertical edge groupoids, and squares given by quadruples \bigbreak

$pmatrix}"): {\displaystyle \begin{pmatrix} & h& \
$-0.9ex] v & & v'\ [itex]-0.9ex]& h'& \end{pmatrix}$ for which we assume always that ${\displaystyle h,h'\in H,\,v,v'\in V}$ and that the initial and final points of these edges match in ${\displaystyle M}$ as suggested by the notation, that is for example$sh=sv, th=sv', \ldots${\displaystyle ,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: ${\displaystyle (D_{2},D_{1},\partial _{1}^{-},\partial _{1}^{+},+_{1},\varepsilon _{1}),(D_{2},D_{1},\partial _{2}^{-},\partial _{2}^{+},+_{2},\varepsilon _{2}),(D_{1},D_{0},\partial _{1}^{-},\partial _{1}^{+},+,\varepsilon ),}$ where: ${\displaystyle D_{0}=M~,~D_{1}=V=H~,~D_{2}=S,}$ ${\displaystyle s^{1}=\partial _{2}^{-}~,~t^{1}=\partial _{2}^{+}~,~s_{2}=s=\partial _{1}^{-}}$ and ${\displaystyle t_{2}=t=\partial _{1}^{+}.}$ 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. The data for the homotopy double groupoid, ${\displaystyle {\boldsymbol {\rho }}^{\square }(X)}$, will be denoted by : $matrix}"): {\displaystyle \begin{matrix} (\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}_1^{\square} (X) , \partial^{-}_{1} , \partial^{+}_{1} , +_{1} , \varepsilon _{1}) , \boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}^{\square}_1 (X) , \partial^{-}_{2} , \partial^{+}_{2} , +_{2} , \varepsilon _{2})\ [itex]3mm] (\boldsymbol{\rho}^{\square}_1 (X) , X , \partial^{-} , \partial^{+} , + , \varepsilon). \end{matrix}$ \bigbreak Here ${\displaystyle {\boldsymbol {\rho }}_{1}(X)}$ denotes the path groupoid of ${\displaystyle X}$ from ref. [2] where it was defined as follows. The objects of ${\displaystyle {\boldsymbol {\rho }}_{1}(X)}$ are the points of ${\displaystyle X}$. The morphisms of ${\displaystyle {\boldsymbol {\rho }}_{1}^{\square }(X)}$ are the equivalence classes of paths in ${\displaystyle X}$ with respect to the following (thin) equivalence relation ${\displaystyle \sim _{T}}$, defined as follows. The data for ${\displaystyle {\boldsymbol {\rho }}_{2}^{\square }(X)}$ 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 (${\displaystyle \partial _{1}^{-}}$, ${\displaystyle \varepsilon _{2}}$... , etc) are to be performed relative to the thin square specified for each groupoid, ${\displaystyle \rho _{1}~or~\rho _{2}}$; 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 ${\displaystyle (H,V)}$ pair depicted in the large diagram (0.1), or $\displaystyle \D$ , shown at the top of this page. Thin Equivalence  Let ${\displaystyle a,a':x\simeq y}$ be paths in ${\displaystyle X}$. Then ${\displaystyle a}$ is thinly equivalent to ${\displaystyle a'}$, denoted ${\displaystyle a\sim _{T}a'}$, if there is a thin relative homotopy between ${\displaystyle a}$ and ${\displaystyle a'}$. We note that ${\displaystyle \sim _{T}}$ is an equivalence relation, see [1]. We use ${\displaystyle \langle a\rangle :x\simeq y}$ to denote the ${\displaystyle \sim _{T}}$ class of a path ${\displaystyle a:x\simeq y}$ and call ${\displaystyle \langle a\rangle [itex]the{\it {semitrack}}of}$ a$. The groupoid structure of ${\displaystyle {\boldsymbol {\rho }}_{1}^{\square }(X)}$ is induced by concatenation, +, of paths. Here one makes use of the fact that if ${\displaystyle a:x\simeq x',\ a':x'\simeq x'',\ a'':x''\simeq x'''}$ are paths then there are canonical thin relative homotopies

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

The source and target maps of ${\displaystyle {\boldsymbol {\rho }}_{1}^{\square }(X)}$ are given by $\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,$ if ${\displaystyle \langle a\rangle :x\simeq y}$ is a semitrack. Identities and inverses are given by ${\displaystyle \varepsilon (x)=\langle e_{x}\rangle \quad \mathrm {resp.} -\langle a\rangle =\langle -a\rangle .}$

At the next step, in order to construct the groupoid ${\displaystyle {\boldsymbol {\rho }}_{2}^{\square }(X)}$ 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 ${\displaystyle R_{2}^{\square }(X)}$ of squares.

Cubically Thin Homotopy
`

Let ${\displaystyle u,u'}$ be squares in ${\displaystyle X}$ with common vertices.

1. A {\it cubically thin homotopy} ${\displaystyle U:u\equiv _{T}^{\square }u'}$

between ${\displaystyle u}$ and ${\displaystyle u'}$ is a cube ${\displaystyle U\in R_{3}^{\square }(X)}$ such that

(i) ${\displaystyle U}$ is a homotopy between ${\displaystyle u}$ and ${\displaystyle u',}$

i.e. $\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$

(ii) ${\displaystyle U}$ is rel. vertices of ${\displaystyle I^{2},}$

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,

(iii) the faces ${\displaystyle \partial _{i}^{\alpha }(U)}$ are thin for ${\displaystyle \alpha =\pm 1,\ i=1,2}$.

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

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

By removing from the above double groupoid construction the condition that all morphisms must be invertible one obtains the prototype of a double category.

## References

1. 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.
2. K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures , 8 (2000): 209-234.