/table>
<>The <span="SimpleMath"s_1is<em class">\(_*c_1 = b_1*\) sothatitsboundaryis identity. The setofcommutativesquaresin\(\oxG))/> forms sub-double groupoid of< class="">\(BoxG\.


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">a2 := Arrow( Gd8, (6,8), -8, -9 );;</span>
<class">&;<="GAPinput :( Gd8)(,) -,-);/pan
<span class="GAPprompt">gap></span> <span class="GAPinput">d2 := Arrow( Gd8, (5,6,7,8), -9, -8 );; </span>
<span class="GAPprompt">gap><> < =""> :=( DGd8,c1c2d2 );span
[-8]---- 68 ---&; [-java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  |                            |
(5,6(,8        )(,)6)
  V                            V
[-9]----(,67, -----gt; [8java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
< classGAPpromptgap&;<spanclass>=(sq2>
[() : -8 ->}
< classGAPprompt>>/> <spanclassGAPinput IsCommutingSquaresq1IsCommutingSquaresq2;</>
[ false, true ]

</pre></div>

<p><a id="X7F99A468819D7759" name="X7F99A468819D7759"></a></p>

<h5>8.1-4 TransposedSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransposedSquare</code>( <var class="Arg">sq</varampw_1[r]{_1d_2}
<div class="func"><table class="func" width=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Thetranspose of the  <span ="SimpleMath"\(1)<>,aswithmatrix, isobtainedby <spanclass="impleMath>(a_1\ with < ="SimpleMath>(\<span < =SimpleMath\c_1spanwithspan=SimpleMath"\(d_1\)(\)/span>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">~~ \ s_1{_}(delta)
[7 --- 68 ---> -]
  |                         |
(5,7)        (6,8)        (5,6,7,8)
  V                         V
[-8] ---- (5,6)(7,8) ---> [-9]
<span class="GAPprompt">gapspanclass="GAPprompt">gap&;</> < class=GAPinput>eftArrow sq2 ) = RightArrow(  ); <span
false

</-]-- (5,7)(6,8) ----> [-9]

<p><a id="X7D3737FA7E9E3ECA" name="X7D3737FA7E9E3ECA"></a></p>

<h5>8.1-5 HorizontalProduct</h5>

<div class="func"><table class="func" width="100%"><tr,)(57        (7(,)
<p>When <code class="code">RightArrow</code><span                          

<p class="center">\[ 


\vcenter{\xymatrix @=4pc{  
   u_1 \ar[r]^{a_1} \ar[d]_{b_1}  
     & v_1 \ar[r]^{a_2} \ar=">


       & v_2 \ar[d]^{c_2} \ar@{}[dr]|= 
        amp \[r^a_1a_2[d{}  
           ;v_2[^c_2\java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
   w_1 \ar[r]_{d_1} 
     & x_1 \ar[r]_{d_2} 
       & x_2 
         & w_1 \ar[r]_{d_1d_2}
           & x_2 
}} 

\]</

<p>

<p classamp &;   \[]{} ardd_}  
\delta(s_1 (\rightarrow) s_2) ~=~ (d_1d_2 ar[r{}  ar]_b_3
  ~=^{-}d_1-}^{-1}a_1c_1d_2d_2^{}-1}_2c_2)
  ~=~ (\delta s_1)^{d_2} (         &=&;  &amp\ 
\]</p>

<p>(This operation was called <code&;&;  amp;   ar[]_d_3}java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48


<div class="example"

< =GAPpromptgap;/span spanclass>eftArrow ) R( sq1/>
true
< =GAPprompt;/span">sq12=( , )
[-7] ----- (5,7)(6,8) ---->]/>
  |                          |
(6,8)        (5,7)        (5,7)(6java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  
[7] --- ,(, --& [8
<span class="GAPprompt">span=GAPprompts<class"c3=Arrow Gd8 6), -9, -8);;
[  8-; 8java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<spanclass="APprompt>gap> spanc=""(bdy1^d2) * = ;
true

</pre></div>

<p><a id=" |

<h5V                        

<div classspan class=GAPpromptgap;</pan <panclass"bdy3 =B(sq3;span>
<> <code=code></> =">(s_1)\ =">UpArrow/>span =">(s_3>  span/andspan="SimpleMath>(\) <>to \s_1(downarrow \)/> =code=codeVerticalProduct"\(s_1,)\<> :/java.lang.StringIndexOutOfBoundsException: Range [454, 455) out of bounds for length 454

<p class="center">\[ 


\vcenter{\xymatrix @=2pc{  
   u_1 \ar[rr]^{a_1}  \ar[dd]_{b_1}  
     && v_1 \ar[dd]^{c_1} 
         & &  &&  \\ 
     &&  & &  u_1 \ar[rr]^{a_1} \ar[dd_{b_1b_3}  
              &57(,)        (        5876java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
   _ \arrr]_d_1[dd{3   
     && x_1 \ar[dd]^{c_3} 
         &=&
     &&/prediv>
              amp
   w_3
     
}

\]</p>

<ampampv_2ar]{_2 

<p class="center">\[ 
\delta(s_1 (\downarrow) s_3) ~=~ d_3^{-1}(b_1b_3& amp&amp;\ 
  ~=~ (d_3^{-1}b_3^{-&;&&;  amp;& \[rrra_1a_2}ar]_b_1b_3
  = ( )\ )^{}. 
\]</p>

<p>(This operation was called     && x_1arrr{d_2} \ardd|}  


<div class"example"><pre

<span ="APprompt">gap></span> <span"> ( Gd8, 57,
<spanamp;&;  \
<span  \[rrd_3
<span=>&;/><span="sq3:= SquareOfArrows(DGd8 ,b3 c3 d3);
[-7] ---- (5,6,7,8) ---> [-9]
  |                        |
(5,7)       (6,8)       (6,8)
  VV
[-9] ---- (5,8)(6,7) ---> [-8]
<}java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
( 8&; 8java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<{xymatrix
true
<span class="GAPprompt">gap></span> <span class="GAPinput">sq13 := VerticalProduct( sq1, sq3 );&;;-(,) \[dd|(6(,}java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
[7 ---- (,)---gt
  |                           |
(5,7)(6,8)        ()        (5,8,7,6)
  V                           V
[-]----(,(,)--gt-8java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35


</pre&7\java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

<p>Vertical

<p class=


\vcenter=""gap><span class="GAPinput>c4 : ( Gd8, (,,,) -8 -)

     &&span=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 115
         && v_2 \ar[dd]^{c_2} 
             &;\java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
     &V
                  amp;&;ampv_2[dd^c_2c_4\ 
   w_1 \ar[rr]|{d_1} \ar[dd]_{b_3}   
     && x_1 \ar[rr]|{d_2} \ar[dd]|{c_3}  
         &r[dd]^{}
             &=&  &&&  \\ 
     &&;&;&&   \[]{} 
                  &&&
  w_3]d_3
;amprr
         &&),        
}} 

\]</p>

<p>In 

<p class="center">\[ 


\vcenter{\xymatrix @=2pc{  
   -7 \ar[rr]^{(5,7)} \ar[dd]_{(6,8)} 
     && -8 \ar[rrspan class="GAPprompt">gapgt</span<span class=GAPinput
         && -9 \ar6,)        ,67,)        (8,6java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
             ampampamp;;&;\ 
     &&  & -----(, ----&;[9java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
                  amp;&& - \ar[dd_{587,)}\\
   -7 \ar[rr]|{(5,6,7,8)  V                                 V
     ampamp; 9\ar[]|(678}ar]|{,)}
         &&  -8 \ar[
             ;&;&;&\ 
     &&  &&  & &  -9 \ar[rrr]_{(5,7)(6,8)} 
                 &;amp; -\
   -9 \ar[rr]_{(5,8)java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
     &java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
         ;;7
}} 

\]</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput=tdleft">class">827
products/sources/formale Sprachen/GAP/pkg/groupoids/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 11.8.2025 mit Größe 46 kB image not shown  
/table>
<span="ContSS"< >span"ocss"> </><a hrefhtml"8. What is the double groupoid generated a set squares?
</span>
</div></div>
<div class="ContSect"><span="ocline"<pan"> chap8_mj#X80E0318F78D2AD284spanHeadingw  groupoids>ajava.lang.StringIndexOutOfBoundsException: Index 183 out of bounds for length 183
</span
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X83F6AFC185148621">8.5 <span class="Heading">Double groupoid homomorphisms</span></a>
/>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X87752436787B199D">8.5-1 DoubleGroupoidHomomorphism</a></span>
</div></div>
</div>

<h3>8 <span class="Heading">Double Groupoids</span></h3>

<> em groupoid/m> is a <em>double category</em> in which all the category structures are groupoids. Forthe most type double there  an pre-crossed. In package consider <mbasic groupoids>,which  not pre-crossed. The general will discussed thestrong""></strong.</>

<p>In a double groupoid, as well as objects and arrows, we need a set of <em>squares</em>. A square is bounded by four arrows, two horizontal and two vertical, and there is a <em>horizontal</em> groupoid structure and a <em>vertical</em> groupoid structure on these squares.</p>

<p>Double groupoids can be considered where the vertical arrows come from one groupoid, and the horizontal arrows from another. The double groupoids constructed here are special in that all four arrows come from the same groupoid. We call these <em>edge-symmetric</em> double groupoids.</p>

<p>This addition to the package is very experimental, and will be extended as time permits.</p>

<p><a id="X780DA94780938851" name="X780DA94780938851"></a></p>

<h4>8.1 <span class="Heading">Single piece double groupoids</span></h4>

<p><a id   u_1 \r[]{}  \ar[b_1

<8.- SinglePieceBasicDoubleGroupoid<h5

<iv class=func class ="100"<tr<d="tdleft">code="func">7 </code> < classArg</> )</><td="tdright"(nbspoperation )</td<tr<div
<div class=
<div classWe  the arrowscode="code>()/> code class""LeftArrows(s)/>  <code class>(s)<code<p
<div="func"<table class=func="10"<><td="tdleft"< class>#27 </code(var="">dgpd> )/td class> category&bsp></r>/able>
<p>Let classSimpleMath(\<span a connected groupoid object < classSimpleMath(Omega)/span>double < classSimpleMath(Box<span <span="SimpleMath>G)/>i constructedbythe codeclass"">inglePieceBasicDoubleGroupoid()/>

<p>The global function <code class="code">DoubleGroupoid</code> may be used instead of this operation, and will work with various other input parameters.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">DGd8 := SinglePieceBasicDoubleGroupoid( Gd8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DGd8!.groupoid;</span>
Gd8
<span class="GAPprompt">gap></span> <span class="GAPinput">DGd8!.objects;</span>
[ -9 - 7]
<span class="GAPprompt">gap></
<span="GAPprompt">/> <span="GAPinput> IsDoubleGroupoid ) (DGd8 );
[ true376,3176]

></div

p>a ="X823A3A7481B90EB7"name="

<h5-2SquareOfArrows<h5>

<div="func"" width=10"<>tdtdleft=func2;SquareOfArrows>< =Arg/> var">up )/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightArrow</code>( <var7 ---- (,)---&; [-java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
div="">< classfunc="%>>< class=tdleft>&27DownArrow class">sq

Quelle  chap8_mj.html   Sprache: HTML

 
 products/sources/formale Sprachen/GAP/pkg/groupoids/doc/chap8_mj.html


<xml version=1"=UTF-8">

<DOCTYPE html PUBLIC/W3C XHTML//EN
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<tml xmlns="ttp/www../99xhtml :lang="en
<head>
<script type
  srccdnn/@M.sconfigjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta
<link> a  groupoid well as and,we a is by  arrows horizontaland twovertical, and there a <></em  structure a <>vertical/emgroupoidstructure onthese squares/pjava.lang.StringIndexOutOfBoundsException: Index 278 out of bounds for length 278
<script="manualjs"type="/>/script>
<script type=
</head>addition to the is  experimental,andwill beX780DA94780938851name=">/pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<body class="chap8"  onload="jscontent()">


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<p id="mathjaxlink" class="pcenter"><a href="chap8.html">[MathJax off]</a></p>
<p><a id="X83B7E8A287C9284A" name="X83B7E8A287C9284A"></a></p>
<div class="ChapSects"><a href="chap8_mj.html#X83B7E8A287C9284A">8 <span class="Heading">Double Groupoids</span></a>
<div ="ContSect"<spanclass"ocline"<span class"nocss> 8.1 Single piece double groupoids<
</span>
<div class"ContSSBlock>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7F5D722C83CB9139">8.1-1 SinglePieceBasicDoubleGroupoid</a></span>
< classContSS >span="nocss>&;nbspa =chap8_mj.tmlX823A3A7481B90EB7">8.- SquareOfArrows<a</span
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X83DF8EEE7BA72AA3">8.1-3 IsCommutingSquare</a></span>
<span class="ContSS"><br /><span <p>Let <span class="SimpleMath">\(G>be groupoid objectset <span ="SimpleMath"\Omegagroupoidclass">\\Box(G)\)/> on SimpleMathG)spanconstructed operation lass>()code/java.lang.StringIndexOutOfBoundsException: Index 330 out of bounds for length 330
<span class="ContSS"><br
spanContSS"8.- VerticalProduct/span>
<span class="ContSS"><
span=""><br/> classnocss&; span ="chap8_mjhtml#">8.- <pan="eading>Horizontal and vertical groupoids in < class="">\(BoxG\/>
<span>
</div><java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
<div class="ContSect"><span class="tocline"><span[ -,-,7]
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href= classGAPpromptgap&;<span class"SetName( DGd8,"" );
<div/iv
<div class="ContSect"><span class="tocline"><span class[true ]
</span>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class=
<span>8.-class"< "" width"0"<>tdclass""< class"">̻SquareOfArrowslt Argrt  =Arg"dn/> )( operation 
td class"">(&;operation )/iv
<div class="func"span="">gap;<span class> := d1*^-1 *a1;</spanjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
<div="func"< classfunc ="10%>>tdclass"">code class="&87 <code< class">sq/> )( attributenbsp;		(nbsp; )/td>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BoundaryOfSquare</code
<div class="func"><table class="func" width="100%"><(,):- -&;-]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:red'>span="">gap;<span class"LeftArrow( sq1 )
<p>Let <span class="[56(,):- &; -9]

<p class="center">\[ 


\vcenter{\xymatrix @=4pc{  
   u_1 \ar[r]^{a_1}span=GAPprompt>/> <span="GAPinput>BoundaryOfSquare )
     ; v_1[d]^c_1
  w_1 ar{} 
     &
}} 

\]</>

<p>We

<p>We think of the square <span class="SimpleMath">\(java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 0

<p>The>8.-IsCommutingSquare


<div="example"<>

<span="GAPprompt"gap;<> <span="">[ ( DGd8 (*8)4;<>
[ 331776, java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<span="GAPprompt"gap/>< =GAPinput := Arrow(, (5,7,- 8);<span
<span="GAPprompt"gap;<"">b1 Arrow(, (68,-,-);/>
<span class="GAPprompt">gap class>>span classGAPinput := ( Gd8,,,) - 8);<span
<pan=GAPprompt>span class"d1: Arrow (,,,8,-, ;
<span class="GAPprompt">[8 ----(,)----&t -]
[(6,) - -; -
<span class="GAPprompt">gap></span> <span class6(,8                57(8java.lang.StringIndexOutOfBoundsException: Range [38, 39) out of bounds for length 38
[-]---- 5, ---&; [-8
  |                          |
    (,)(,)(,)
  V                          V
[-7] ----- (5,6,7,8) ----> [-9]
<span class="GAPprompt"< classGAPpromptgapgt;span>span="">[ IsCommutingSquaresq1(sq2<spanjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
true
<span classp< id=X7F99A468819D7759="


[(5>8.1-4h5
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[6)  7- -]
<span class="GAPprompt">gap></span> <spanclass> class"width=10>sq&;property )d>/>
/>
java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 26
<span class="GAPprompt">gap ="example"pre
[(,78):- > 9]
<span class="GAPprompt">gap></span> <span class="GAPinput">BoundaryOfSquare( sq1- ----(, --; 7
[(6,8)(7)       (,)(567)
<span class="GAPprompt">gap></span                         
DGd8
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDoubleGroupoidElement( sq1 );</span>
true

</pre></div>

<p><a id="X83DF8EEE7BA72AA3" name="java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 5

<h5>8.1-3 <aid"="></a>/>

<ivclass=func< ="width"0%<>tdtdleft class""> IsCommutingSquare<code(<ar lasssq> )<td classtdright"(nbsppropertynbsp)/>

<span class="GAPprompt">gap></span> <span class="GAPinput">d4 := Arrow( Gd8, (5,6)(7,8), -8, -7 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sq4 := SquareOfArrows( DGd8, d2, c3> is singleidentity  the < =code>HorizontalProductcodeand < =<code but arefor , em>left/em>,  > /m, emup<em and <emdown</>      squares  isequal  squarespan=""\s\<span  theother  are left  span=1L\)span the span="">(1_Rs\</>forclass>p
[-9] ------- (5,6,7,w\[java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 16
  |                             
(6,8)        (5,6,7,8) class="example"><pre
  V                             
[]---(),)---&;-
<span --- )--&;[8
true
<span class="V
true

<
[7 ---576[]
 |
(5,7)        (5,8)(
                                
prejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<span class="GAPprompt">gap></span> <span class="GAPinput">sq1234 := VerticalProduct( sq12, sq34 );</span>
[-7] --------- (5,7)(
  |                                  
(5,u\r[^a  \[_1  
  V&; v ard^1 
[-[&; w \[r]^{}\[d]

<span[r]_{}
[- ; v 
  |                                 |
(5,8,7,6)        (5,6,7,8)        (5,8,7,6)
  V                                 V
[8 ----- 56(,)-----; -]

<span class="GAPprompt">gap>}
<]<p>
true

</pre></div>

<p><a id="X79E6E4E97A23C257" name

<h51- HorizontalIdentities/>

<div class classGAPprompt>/> <spanclass="GAPinput">vid :=="GAPinput"vidVerticalIdentities(sq24 );<span>
<ivfuncclass" width"0%"

<span class="GAPprompt">gap></span> <span class="GAPinput">hinv := HorizontalInverse( sq24 ); </span>
[-9] ----------- (6,8) ----------> [-8]
  |                                 |
(5,8,7,6)        (5,6,7,> operation class""UnionOfPiecescode>and attribute < =""Pieces,introduced sectiona href"hap2_mjhtml#java.lang.StringIndexOutOfBoundsException: Range [172, 171) out of bounds for length 414
  V                                 V
[-7] -------- (5,6)(7,8) -------> [-8]
<span class="GAPprompt">gap></span> <span class="GAPinput">HorizontalProduct( hinv, sq24 ) = hid[2];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HorizontalProduct( sq24, hinv ) = hid[1];      </span>
true

<><div

<p>Similarly, <span class="SimpleMath"> class="GAPprompt">gap&;<span<span class=>:= SinglePieceBasicDoubleGroupoid ); </span

<p class="center groupoid having 2 :java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34


\vcenter{\xymatrixjava.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 14
   \[]{}ar{^1}
     & x \ar[ class">( <>
   u \ar[r]_{a} 
     & v 
}} 

\]/


< =""><pre

< class>g;/pan classGAPinput :VerticalInverse ;<span
[-8] -------- (5,6)(7,8) - = Gc6
  |                                 |
(5,6,7,8)        (5,8,7,6)        (5,6,7,8)
  V                                 V
[-8] ---------- bjects1 ]]
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<span class="GAPprompt">gap></span> <span class="GAPinput">VerticalProduct( sq24, vinv ) = vid[1];   </java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 0
true

<prediv>

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

<h5>8.1-8 <span <>

<p>Now <span class="SimpleMath">\(\Box(G)\)<<p>Now <span class="SimpleMath">\(java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0

<p class="center">\[ 


\vcenter{\xymatrix @=4pc{  
   u \ar   
     < classfunc class"" width0%>tr class>code="func"></var<td>td ="tdright"(nbsp td><table>
   w \ar[r]_{d} 
     ampjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
}} 

\]</java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

<<amp;v ar^)}\ 

<p>Similarly the <em>vertical groupoid</     amp

<p>These groupoid structures have not]pjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>8.2 <span class piece groupoid:

<p<group =Group(  )

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

<h5>8.2-1 UnionOfPieces</h5>

<div

<>he operation< classcode></> and  attributecode="code>Pieces/>, introduced in section >2./spana> are used double.The are by least in object. The class></code the concatenation the in pieces/>

<p>The example shows


<div class=divclass"" ="10%"codeclass=func>#27; DoubleGroupoidWithSingleObject"gp/> < class""obj"(nbsp&;<>

<span class="GAPprompt">gap></span> <span class="GAPinput">DGc6 := SinglePieceBasicDoubleGroupoid( Gc6 );; </span>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanclass=GAPprompt>/pan><span ="GAPinput"DGc6s4DoubleGroupoidD, DGa4</>
double
1single double with
 groupoid
    group = a4
  objects = [ -15 .. -11 ]
2: single piece double groupoid with:
 groupoid = Gc6
    group = c6
  objects = [ -10 ]

<span class="GAPprompt">gap>groupoid =  piece: lt( [ (,,,4],  gt;
<span class="GAPprompt">gap><span> span=GAPinputPieces  )span
[ single piece
      = Ga4
        group = a4
      objects6
     groupoid = Gc6
        grouppre/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  doubleby of?spanh5

</

<a="X78E24657817C4D30>p

p class

<p>Before consideringar]{}ar_{  


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

</li>
<><p>b double with object</>

</li>
</ul>
<>< ="X80009143808725D5" nameX80009143808725D5>a<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5ampv

<div class=&; java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
<p>When <span class="}

<p class="center">\[ 


vcenter pc
   u \ar[r]^{()}  \ar[d]_{()}  
     & v \ar[d]^{()} \\ 
   w \ar[r]_{()} 
     & x
}} 

\]</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">DGtriv := DoubleGroupoidWithTrivialGroup( [-19..-17] 
single double :
 groupoid = single piece
    
  objects =

<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 39
81

</pre></div>

<p><a id="X7D135AFB83995A32" name="X7D135AFB83995A32"></[w_1,w_2 \[r]_{d1] 

<h5>8.3-2 DoubleGroupoidWithSingleObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleGroupoidWithSingleObject</code>( <var class="Arg">gp</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Given a group <span class="SimpleMath">\(G\)</span> we can form the corresponding groupoid with a single object, and from that a double groupoid on that object. The number of squares is <span class="SimpleMath">\(|<div="


<div class="example"><pre>

<span class=" piecedouble groupoid with:
  with
 groupoidobjects=java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 53
    group = Group( [ (1,2,3,4) ] ) classGAPprompt>>><span=GAPinput"emb2 Embedding(Gd8c6, 2;
  objects = [ 0 ]

<span class="GAPprompt(,3: 9-0 &;-, -1 ]]
256

</pre></div>

<2  ,-0 -;  8-0]

<h5>8.3-3 <span class="Heading">What is the double groupoid generated by a set of squares?</spanspan="GAPprompt"gap/span<classGAPinput:Arrow,(1,1 0);/>

<p>This is a very experimental section. Let us 

<pclass>java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21


\vcenter{\xymatrix @=4pc{  
    \[]a  \[d]{}java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
      & v \ar[d]^{1} 
        &; v []^1}\ar]1java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
          & -0]]--(,)---gt8-10]
            & u \ar
              & u \ar
    u \ar>  ="
      &id="7367 =X87752436787B199D"<<java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
        ampr]1
          & u
java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 30
              & u
}} 

\]</p>

<p>The

<p><a id="X80E0318F78D2AD28" name="X80E0318F78D2AD28"></a></p>

<h4>8.4 <span class="Heading">Starting with two groupoids</span></h4>

<p>In the literature on double groupoids the construction often starts with two groupoids <span class="SimpleMath">\(G_1, G_2\)</span>, and squares <span="GAPprompt>><=>endDGd8 :( DGd8DGd8,md8;/>

<=\


\vcenter{
   [u_1,u_2] \ar,,78 -gt(587)
     & [v_1,v_2] \ar[d]^{[1,c]} \\ 
[,w_2]_[,]}
     & [x_1 :  )(,)(,)java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
}} 

\]</p>


<div class="example"><pre>

<s class>><spanspan="GAPinput>
single piece groupoid: < Group( [ (1,2,3,4), (1,3), (5,6,7)(8,9]-- (5,7) ------> [-8]
[ [ -9, -10 ], [ -8, -10 ], [ -7, -10 ] ] >
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName(68        ,)       56)7,)
<span class=7 - 5,,8 -&; [-9]
single groupoid:
 groupoid = Gd8c6
    group =
  objects =,)       7        5876)

< class=""gap<><classGAPinputemb1(1;/java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
<span class
< classGAPprompt>span < class"GAPinput" :Arrow, 57,-, <>
<span
[(div< =>oto: /spanhref"Topa> ="chap1_mj"1 < href=chap2_mj=chap3_mjhtml"3/a>  < =html"4>< href".html<  < href.html6> href."7/> href"."8/>< ="."9/> < href=.html"0<  ahref"Bib/a>a"htmlInd <div
<span class="GAPprompt">gap></span> <span class="GAPinput">d5 := Arrow( Gd8, (6,8), -9, -8 );;</span>
span="GAPprompt">></span> span="GAPinput>=(emb1,d5 );/>
[(2,4) : [ -9, -10 ] -> [ -8, -10 ]]
<spanGAPpromptgap</span =APinput:Arrow1,23,0,10)</pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">b6bodyjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
[(5,6,7) : [ -9, -10 ] -> [ -9, -10 ]]
<span class="GAPprompt">gap></span> <span class="GAPinput">c6 := Arrow( Gd8c6, (8,9), [-7,-10], [-8,-10] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sq := SquareOfArrows( DGd8c6, a6, b6, c6, d6 );</span>
[[ -9, -10 ]] ----- (1,3) ----> [[ -7, -10 ]]
  |                                        |
(5,6,7)        (1,3)(2,4)(5,7,6)(8,9)        (8,9)
  V                                        V
[[ -9, -10 ]] ----- (2,4) ----> [[ -8, -10 ]]

</pre></div>

<p><a id="X83F6AFC185148621" name="X83F6AFC185148621"></a></p>

<h4>8.5 <span class="Heading">Double groupoid homomorphisms</span></h4>

<p><a id="X87752436787B199D" name="X87752436787B199D"></a></p>

<h5>8.5-1 DoubleGroupoidHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleGroupoidHomomorphism</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDoubleGroupoidHomomorphism</code>( <var class="Arg">mwohom</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A homomorphism of double groupoids is determined by a homomorphism <code class="code">mor</code> between the underlying groupoids since <code class="code">mor</code> determines the images of the four arrows in every square.</p>

<p>In the example we take the endomorphism <code class="code">md8</code> of <code class="code">Gd8</code>, constructed in section <a href="chap5_mj.html#X83DDA9528396451A"><span class="RefLink">5.2-1</span></a>, to produce an endomorphism of <code class="code">DGd8</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ad8 := GroupHomomorphismByImages( d8, d8,</span>
<span class="GAPprompt">></span> <span class="GAPinput">              [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (6,8) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">md8 := GroupoidHomomorphism( Gd8, Gd8, ad8, </span>
<span class="GAPprompt">></span> <span class="GAPinput">              [-7,-9,-8], [(),(5,7),(6,8)] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">endDGd8 := DoubleGroupoidHomomorphism( DGd8, DGd8, md8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( endDGd8 );            </span>
double groupoid homomorphism: [ DGd8 ] -> [ DGd8 ]
with underlying groupoid homomorphism:
homomorphism to single piece groupoid: Gd8 -> Gd8
root group homomorphism:
(5,6,7,8) -> (5,8,7,6)
(5,7) -> (6,8)
object map: [ -9, -8, -7 ] -> [ -7, -9, -8 ]
ray images: [ (), (5,7), (6,8) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDoubleGroupoidHomomorphism( endDGd8 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">sq1;</span>
[-7] ------- (5,7) ------> [-8]
  |                          |
(6,8)        (6,8)        (5,6)(7,8)
  V                          V
[-7] ----- (5,6,7,8) ----> [-9]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( endDGd8, sq1 );</span>
[-8] ------- (5,7) ------> [-9]
  |                         |
(5,7)        (5,7)        (5,8,7,6)
  V                         V
[-8] ---- (5,8)(6,7) ---> [-7]

</pre></div>


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¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.21Bemerkung:  ¤

*Bot Zugriff

vartd=tdrightnbsp;<>tr/>
< classfunc class"width"0%>tr class"< class=func">#827 HorizontalInverses>< =">sq)/>		><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VerticalInverses</code>( <var class="Arg">sq|
<p>There

< class=center\ 


\vcenter{\xymatrix @=4pc{  
   u \ar[r]^{1}  \ar[d]_{b}  
     &  V
       & v \ar[r]^{1}  \[- ---(,)7)--gt7java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
         &; v \[d]^c \
   w \ar[r]_{1} 
     &; w \rr]{}
       & class"gap&t(sq24,vid[])=sq24;/pan>
         amp
}} 

\]</p>


< class"

< classGAPprompt><span <pan="">hid HorizontalIdentities(sq24;span>
<span\) s^-}{H ~~  1L),  \qquad
[8]-----( ----&; [-8
  |                          
(5,8,The of classSimpleMaths^-}{})/pan <span="SimpleMath"\dca-1 =(deltas)-1){^-}<spanHere  two:</p>
  V                          V
[-8] --------

[-9] --------- () --------> [-9]
  |                          
(5,76         (5876
  V                          V
[-7] ------       ampar{java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<span class="GAPprompt">gap ar{d}
true
span="GAPprompt">gap></span> <span="GAPinput"(sq24[2)=sq24span
true

<pre>

<p>Similarly&; x

<p 


\vcenter
    class=example">pre>
     &
       < class">>><pre>

<span class="GAPprompt">gap></span> <span class ---- 56(,)---&; [-]
<span class="GAPprompt">gap&< class"">gap;</> <span="">HorizontalProduct, sq24hid<span
[-]-- (, -gt [9
  |                   
()         ()         ()
  V                   V
[="">\(\<span a <> inverse> span="">\(^-}{})/> whose with class">(\/> isan p or :( \) s{1_V =1Us\<span <span"\s{1_V ()s _(s))/> The boundary span =SimpleMath"\^{1bdc  (delta-){^{1})<spanp

[-8] ---- (5,6)(7,8) ---> [-7]
  |                        |
()            < class"\
  V                        V
[-8] ---- (5,6)(7,8) --->\[^d  [_^1}java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
<span classjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
true
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true

</pre ---- ,6)(78 --&;-]

<p>Confusingly, <span class="SimpleMath">\(s\)</5,,8        5,,,6        (,7,java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43

<p class="center">\[
s (rightarrows{1{}~=  1L(s  qquad  
s^{-1}_{H =&<><=>(,   [;/java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
\]<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 62

<p>The boundary of <span class="SimpleMath">\(s^{-1}_{H}\)</span> is <span class="SimpleMath">\(dc^{-1}a^{-1}b = (\delta(s)^{-1})^{d^{-1}}\)</span>. Here are the two products:</p>

<p class="center">\[ 


\vcenter\xymatrix=pc{  
   u \ar[r]^{a}  \ar[djava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     & arr^a^-}} \ar[d^}
       & u \ar[d]^{b}
         & v \ar[r]^{a^{-1}} \ar[d]u\ar[r]{}  ard]b  
           & u \ar\r]{java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
             amp ar^} 
   w \ar[r]_{d} 
     & x \ar[r]Similarly  em /> structure class"\Box_V})\1_())/panand the inverse of span=SimpleMath>s)/> is class>s/p>
       & w 
         & x \ar[r]_{d^{-1}}
           & w \ar[pThesestructures not  in packagep
             &<i=X7F319AA17ED13024nameX7F319AA17ED13024">/a>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
}} 

\]</> with, groupoids comprise ofpiece groupoids disjoint sets<pjava.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123


<div class="d class="func " width="%<><tdclass=">#27 nionOfPieces( class=Arg> />< class""(&;&;





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