PlanetPhysics/Homotopy Double Groupoid 2

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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:

Failed to parse (unknown function "\D"): {\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

Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \begin{pmatrix} & h& \<blockquote><math>-0.9ex] v & & v'\<blockquote><math>-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 </math>sh=sv, th=sv', \ldots$,etc.Thecompositionsaretobeinheritedfromthoseof<math>H,V$, that is: \bigbreak

Failed to parse (unknown function "\quadr"): {\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 Failed to parse (unknown function "\D"): {\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.

The data for the homotopy double groupoid, ${\boldsymbol {\rho }}^{\square }(X)$, will be denoted by :

Failed to parse (unknown function "\begin{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})\<blockquote><math>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 Failed to parse (unknown function "\D"): {\displaystyle \D} , shown at the top of this page.

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 $\langle a\rangle <math>the{\it {semitrack}}of$ a </math>. 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

Failed to parse (unknown function "\begin{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 ${\boldsymbol {\rho }}_{1}^{\square }(X)$ are given by Failed to parse (unknown function "\enskip"): {\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 $\varepsilon (x)=\langle e_{x}\rangle \quad \mathrm {resp.} -\langle a\rangle =\langle -a\rangle .$

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.

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 $u',$

i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',}

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

i.e. Failed to parse (unknown function "\enskip"): {\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 $\partial _{i}^{\alpha }(U)$ 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 $u'.$

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