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= ="-"java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for lengt <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/java.lang.StringIndexOutOfBoundsException: Index 29 out of bo <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (XMod) - Chapter 1,0,0][-,-1,,0,0][0,, <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap10" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</[0,,0,][-,,,0,[,01,0]j <p id="mathjaxlink001/2]], <p><a id="X80B3A81B7E5CA3A9" name="X80B3A81B7E5CA3A9"></a></p> <div class="ChapSects"><a href="chap10.html#X80B3A81B7E5CA3A9">10 <span class="Heading">Cr <div class="ContSect"><span class="tocline"><span class="nocss"> </span:[100,010,00,]]java. </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap10.html#X78F89CAB7A281B8F"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap10.html#X86CD034F82F5F029"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap10.html#X7B76F2BF82E075FF"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap10.html#X858EB4F97D04D012"> <span class="ContSS"><br /><span class="nocss"> [0,-,][100,001] </div></div> </div> <h3>10 <span class="Heading">Crossed modules of groupoids</span></h3> <p>The material documented in this chapter is experimental, and is likely to be changed in due cou <p><a id="X847F4ED77F50528C" name="X847F4ED77F50528C"></a></p> <h4>10.1 <[0,10],0,0,1]], <p>A typical example of a crossed module <span class="SimpleMath">calX</span> over a groupoid has f <p><a id="X78F89CAB7A281B8F" name="X78F89CAB7A281B8F"></a></p> <h5>10.1-1 PreXModWithObjectsByBoundaryAndAction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code[[1,1,0],[-,,0 <p>This is the groupoid generalisation of the operation <code class="code">PreXModByBoundaryA <p><a id="X86CD034F82F5F029" name="X86CD034F82F5F029"></a></p> <h5>10.1-2, <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>At present the experimental operation <code rec(1:=recgenerators:[[0,,1,0,-,1,,,][0,, <p>In the example the crossed module <code class="code">DX4</code> has discrete source, while the c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( (1,2,3,4), (3,4) );; </sp <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s4, "s4" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">a4 := Subgroup( s4, [ (1,2,3), (2,3,4 <pan class="GAPprompt">gap></span> <span class="GAPinput">SetName( a4, "a4" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">X4 := XModByNormalSubgroup( s4, a4 ) <span class="GAPprompt">gap></span> <span class="GAPinput">CX4 := SinglePiecePreXModWithObje single piece precrossed module with objects source groupoid: single piece groupoid: < a4, [ -6, -5, -4 ] > and range groupoid: single piece groupoid: < s4, [ -6, -5, -4 ] > <span class="GAPprompt">gap></span> <span class=[,0,/2,1], <span class="GAPprompt">gap></span> <span class="GAPinput">Ca4 := Source( CX4 );; SetName( Ca4, " <span class="GAPprompt">gap></span> <span class="GAPinput">Cs4 := Range( CX4 );; SetName( Cs4, "C </pre></div> <p><a id="X7B76F2BF82E075FF" name="X7B76F2BF82E075FF"></a></p> <h5>10.1-3 IsXModWithObjects</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# div classfunc><table class="func" width"100"<tr><td class=tdleft"<code func">‣ IsPreXModWithObjects</code>( <var class="Arg">pxmod</var> )</td><td class="tdr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>The precrossed module <code class="code">DX4</code> belongs to the category <code class="code"> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Set( KnownPropertiesOfObject( CX4 [ "CanEasilyCompareElements", "CanEasilySortElements", "IsAssociative", "IsDuplicateFree", "IsGeneratorsOfSemigroup", "IsPreXModWithObjects", "", "IsXModWithObjects </pre></div> <p><a id="X858EB4F97D04D012" name="X858EB4F97D04D012"></a></p> <h5>10.1-4 IsPermPreXModWithObjects</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div[[0,-1,0][-0,0][,0,]], <p>To test these properties we test the precrossed modules from which they were constructed.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsPermPreXModWithObjects( CX4 );</ true <span class="GAPprompt">gap></span> <span class="GAPinput">IsPcPreXModWithObjects( CX4 ); </sp false <span class="GAPprompt">gap></span> <span class="GAPinput">IsFpPreXModWithObjects( CX4 );</sp false </pre></div> <p><a id="X797B1CD07C3682EE" name="X797B1CD07C3682EE"></a></java.lang.StringIndexOutOfBou <h5>10.1-5 Root2dGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>The attributes of a precrossed module with objects include the standard <code class="code">So <p>Note that <code class="code">XModAction</code> is now a groupoid homomorphism[0,-,,0],[-,,0 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Root2dGroup( CX4 ); </span> [a4->s4] <span class="GAPprompt">gap></span> <span class="GAPinput">actC := XModAction( CX4 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">Size( Range( actC ) ); </span> 20736 <span class="GAPprompt">gap></span> <span class="GAPinput">r1 := Arrow( Cs4, (1,2,3,4), -4, -5 ); <span class=asis:=[[1,,0],[0,,java.lang.StringIndexOutOfBoundsException: Index 33 [groupoid homomorphism : Ca4 -> Ca4 [ [ [(1,2,3) : -6 -> -6], [(2,3,4) : -6 -> -6], [() : -6 -> -5], [() : -6 -> -4] ], [ (23,) : -4 -> -4],] [((11,3,3,) -4 -&t;-], ( : --> -6], [() : -4 -> -5] ] ] : 0 -> 0] <span class="GAPprompt">gap></span> <span class="GAPinput">s1 := Arrow( Ca4, (1,2,4), -5, -5 );;</ <span class="GAPprompt">gap></span> <span class="GAPinput">## calculate s1^r1 </span> <span class="GAPprompt">gap></span> <span class="GAPinput">ims1 := ImageElmXModAction( CX4, s1 [(1,2,3) : -6 -> -6] </pre></div> <p>There is 0-,][1,0,00,], <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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