PlanetPhysics/Homotopy Double Groupoid 2

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

Let be a Hausdorff space. Also consider the HDA concept of a double groupoid, and how it can be completely specified for a Hausdorff space, . Thus, in ref. [1] Brown et al. associated to a double groupoid, , 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 is a set of `points', are `horizontal' and `vertical' groupoids, and is a set of `squares' with two compositions.

The laws for a double groupoid are also defined, more generally, for any topological space , 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 over a set , there is a double groupoid with as horizontal and vertical edge groupoids, and squares given by quadruples \bigbreak

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for which we assume always that and that the initial and final points of these edges match in as suggested by the notation, that is for example </math>sh=sv, th=sv', \ldots, 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:

where: and 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, , 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})\{{center top}}<math>3mm] (\boldsymbol{\rho}^{\square}_1 (X) , X , \partial^{-} , \partial^{+} , + , \varepsilon). \end{matrix}}

\bigbreak

Here denotes the path groupoid of from ref. [2] where it was defined as follows. The objects of are the points of . The morphisms of are the equivalence classes of paths in with respect to the following (thin) equivalence relation , defined as follows. The data for 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 (, ... , etc) are to be performed relative to the thin square specified for each groupoid, ; 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 pair depicted in the large diagram (0.1), or Failed to parse (unknown function "\D"): {\displaystyle \D} , shown at the top of this page. \end{definition}

\begin{definition} Thin Equivalence

Let be paths in . Then is thinly equivalent to , denoted , if there is a thin relative homotopy between and .

We note that is an equivalence relation, see [1]. We use to denote the class of a path and call a </math>. The groupoid structure of is induced by concatenation, +, of paths. Here one makes use of the fact that if are paths then there are canonical thin relative homotopies

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The source and target maps of are given by Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,} if is a semitrack. Identities and inverses are given by \end{definition}

At the next step, in order to construct the groupoid 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 of squares.

\begin{definition} Cubically Thin Homotopy

Let be squares in with common vertices.


  1. A {\it cubically thin homotopy}

between and is a cube such that

(i) is a homotopy between and

\begin{center} i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',} \end{center} (ii) is rel. vertices of \begin{center} 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,\end{center} (iii) the faces are thin for .

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

denoted if there is a cubically thin homotopy between and


\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}


All Sources[edit]

[2] [1]

References[edit]

  1. 1.0 1.1 1.2 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. 2.0 2.1 K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures , 8 (2000): 209-234.