Quelle  chap8_mj.html   Sprache: HTML

<pspan="GAPprompt">gap;<span>s classGAPinput :=HorizontalIdentities( );</span>

s (rightarrow{1_}=~1(s  

s-]-----( ----gt8





<p> boundary <span="">\(^-}H\<s> isspan class>(^{-1}^{1b =(delta(){}^d{1}\)/>.  arethe products>



<p class="center">\[ 





\vcenter|

   u \ar[58,)()       ,,,)

     & v \ar[r]^{a^{-1}} \ar[d]^{c}

       &; u \[d]^b}

         & v \ar[r]^{a^{-1}} \ar[d]_{c} 

           & u \ar[r]^{a} \ar[d]^{b} 

             & v \ar[d]^{c}  \\ 

   w \[r]_d 

     &

       & w < class class>HorizontalProduct , hid2   ;      </>

         & x \ar

           & w \ar[r]_{d}/></div

             amp

}} 



\]</p>





v class=example">pre>



<pan="GAPprompt"gap/>< class">hinv = orizontalInverse  ); 

[-9] ------         amp

  |                                 |

(5,8,7,6)java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

  V                                 V

[-7] ---- (,)78 ----gt8java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

span=GAPprompt>/pan classGAPinput( hinv ) = [2];/>

true

<span- --6,)--&;[]

true



</pre></div>



<p>Similarly, <span classSimpleMaths)/> has<mvertical</em< classSimpleMaths^1}V\<span product <span="SimpleMath>(\\s(downarrow^-1_{} =1U())/> and class="SimpleMath>(s^-}{}(downarrow) =1D)\a-}^{-1}=\(s)^{1}^c{1\/>.</>



<="center>\[ 





\vcenter{\xymatrix @=4pc{  

   w arr]{}  \ard]{b{-}  

     & x \ar[d]^{c^{-1}} \\ 

   u \ar[r]_{a} 

     & v 

}} 



\]</p>





<div class="example"><pre>



<span class="GAPprompt">gap></span> <span class="GAPinput">vinv := VerticalInverse

[-8] ------(5,6(,8-----gt [7]

  |                                 |

(,67,)(87,)(,67,,8)

  V                                 V

[-8] ----------- (6,8) ----------> [-9]

<span class="GAPprompt">gap></span> <span class=

true\) ^{-}_{} =~~  _L(),\qquad

<spanclass"GAPprompt">gapgt;/span <spanclass="APinput"VerticalProduct( sq24 vinv )) = vid[1]   <span>

true



</pre></div>



<p><a id="X7B8DEF5A7B5AEC46" name="X7B8DEF5A7B5AEC46"></a></p>



<h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<{\ @=pc



<p class="center">\[ 





\vcenter{\xymatrix     & v \ar[r]amp;v\ar[]^{a^1d]{cjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

    \ar[r^{}\ar[d_{}

     & v \ar[d]^{c} \\ 

   w \r[r_{d} 

     & x 

}} 



\]</p>



<p>as an arrow from <span class             & v\[d]^c  \\ 



<>Similarlythe<>vertical groupoid</em  <span="SimpleMath>(\Box_{V(G\)/>on \(\Box(G)\) has the horizontal arrows as objects and pairs of vertical arrows as arrows. The identity arrow at \(a\) is  SimpleMath>\(1Us\<s>,  arrowof < class"SimpleMath"\(s\\<spanis <span="SimpleMath"\(^{-1}_{V}\)</span>.</>



<p> groupoid structures have beenimplemented this.</>



p>a d=" ="X7F319AA17ED13024>a>/>



<h4>8.2 <span class="Heading">Double groupoids with more than one piece</span></h4>



pAswith groupoids double may a union single piece double with object./>



<p><a id="X8589EFAE7935BEA6" name="X8589EFAE7935BEA6"></a></p>



<h5>8.2-1 UnionOfPieces</h5>



<ivclass">width"10">trtd class"tdleft">code "func">82;UnionOfPiecescode(<var"Arg"pieces>)<td<td=tdright>nbspoperationnbsp;)<td<//tr/abledivjava.lang.StringIndexOutOfBoundsException: Index 226 out of bounds for length 226

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 42

<pThe <codeclass=code></code>  theattributecodeclasscode></code> introduced in <a=".htmlX7C7007207FD91BB0"><span class="RefLink">2.5</span></a>, are also used for double groupoids. The pieces are sorted by the least object in their object lists. The <code class="code">ObjectList</code> is the sorted concatenation of the objects in the pieces.</p>



<p>The example shows that, as well as taking the union of two double groupoids, thejava.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4





<div class="example"><pre>



<span class="GAPprompt">gap></span> <span class=/pre<div>

span=GAPpromptgt/> span"GAPinput">DGa4( Ga4);<span>

<span class="GAPprompt">

doublehaving pieces-

1: single piece double groupoid with:

 groupoid = Ga4

    group = a4

  objects = [ -15 .. -11 ]

2: single piece double groupoid with

 groupoid = Gc6

    group = c6

  objects = [ -10 ]



<span class="GAPprompt">gap>w \arr^d  \[d]_{b{-}  

<spanclass="GAPprompt>gap> span class=GAPinput>Pieces DGa4c6);/spanjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86

[ single piece double groupoid\]<<p>

     groupoid = Ga4

        groupdivclassexample>

      objects =<span="GAPprompt"gap&t<s> <spanclass="">vinv= ( sq24)</span>

     groupoid java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 18

       = [ -0 java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25



</pre></div>



<p><a id="X78E24657817C4D30" name="X78E24657817C4D30"></a></p>



<h4>8.3 <span class="Heading">Generators of a double groupoid</span></h4>



<p>Before considering the/></divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12





<l

<li><p>a basic double groupoid with identity group;</p>



</li>

<li><p>a basic double groupoid with a single object.</p>



</li>

</ul>

<p><a id="X80009143808725D5" name="X80009143808725D5"></a></p>



<h5>8.3-1 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0



<iv=""><table=func="10"<><td="tdleft"< class">‣ DoubleGroupoidWithTrivialGroup( >gap>

single double with

 groupoid

    group ( [ ) ] java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

  objects = [ -19 .. -17 ]



<span

81



<>he operation code=""UnionOfPiecescodeand the < class"< classRefLink5<></a> also for groupoids  pieces sorted the object their lists <code="code"ObjectList> is  sorted of objects the.</>



<p><a id="X7D135AFB83995A32" name="X7D135AFB83995A32"></a></p>



<h5>8.3-2 DoubleGroupoidWithSingleObject</h5>



<div ="func>table =func"width0><>td"tdleft"< ="&82; DoubleGroupoidWithSingleObject/>(< =Arg>obj<> )/d>td=tdright>&;operationnbsp)