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<div class="ChapSects"><a href="chap10_mj.html#X7A06103979B92808">10 <span class="Heading">Chain complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7DFB63A97E67C0A1">10.1 <span class="Heading">Introduction</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7AC5660E80079755">10.2 <span class="Heading">Infinite lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8466C7117CEC99D2">10.2-1 IsInfiniteNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F12EE4479A03527">10.2-2 PositiveInfinity</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8759931784478A4D">10.2-3 NegativeInfinity</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X84731DA279E797A5">10.2-4 IsInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X87524BBB7C2F9FCB">10.2-5 IsHalfInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7D21FB1A7D21FB1A"><code>10.2-6 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E01944C83F87EED">10.2-7 MakeHalfInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8458C2698505C12B">10.2-8 StartPosition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X812437C47A208D4A">10.2-9 Direction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7C3728BF7ED7149A">10.2-10 InfListType</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X82AA0979868A6967">10.2-11 RepeatingList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7AF21D62848B270B">10.2-12 ElementFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8728A2107F27D68F">10.2-13 IsStoringValues</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X86888FBD8559B29B">10.2-14 NewValueCallback</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X856E1541834688C6">10.2-15 IsRepeating</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8208162D7F3CC139">10.2-16 InitialValue</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F12274E794510CA">10.2-17 LowestKnownPosition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7B5E57F0859BEE17">10.2-18 HighestKnownValue</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81F7B0AB84D6319B">10.2-19 Shift</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7D8067A7842BC4A4">10.2-20 Cut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X79A67E377B71C540">10.2-21 HalfInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7CE66EBA844DBB57">10.2-22 MakeInfListFromHalfInfLists</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X801DE2517D08C19E">10.2-23 MakeInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7EEB276380E1584F">10.2-24 FunctionInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X78C7FBAE83914790">10.2-25 ConstantInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7D68E0627F556E14">10.2-26 FiniteInfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8134072B7C7309CD">10.2-27 MiddleStart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7EB933657AF3D723">10.2-28 MiddleEnd</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7AC3ECBB7BC93594">10.2-29 MiddlePart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7896D6DD788046E7">10.2-30 PositivePart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81B768838567F98D">10.2-31 NegativePart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8511A0B88003B989">10.2-32 HighestKnownPosition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F12274E794510CA">10.2-33 LowestKnownPosition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A5CB74485184FEE">10.2-34 UpperBound</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X807CC5AD7FDFA4E3">10.2-35 LowerBound</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A311EBE79F31AEA">10.2-36 FinitePartAsList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8028ED6883CFA39C">10.2-37 PositivePartFrom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7863903B7E281CF6">10.2-38 NegativePartFrom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81F7B0AB84D6319B">10.2-39 Shift</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X80153272857B2EDE">10.2-40 Splice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X84FC63C183F0A977">10.2-41 InfConcatenation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F4CF360858E80A6">10.2-42 InfList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X791D879483C59410">10.2-43 IntegersList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7CAF603281B94AC8">10.3 <span class="Heading">Representation of categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F0E546178977963">10.3-1 IsCat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7BB9B02E7E76FF9C">10.3-2 CatOfRightAlgebraModules</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7EC3D95F7C791F7E">10.4 <span class="Heading">Making a complex</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X838C9EBD87B8CB9D">10.4-1 IsQPAComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E0CE8AC87533991">10.4-2 IsZeroComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E51958C86D73D17">10.4-3 Complex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X80F8DD20789BBBC0">10.4-4 ZeroComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7C75868E82A831C8">10.4-5 FiniteComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X80B47D78785F6859">10.4-6 StalkComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A685484784087FE">10.4-7 ShortExactSequence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7F7E0C197FA31B29">10.5 <span class="Heading">Information about a complex</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7BBF131485F3C792">10.5-1 CatOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8239596E87830A51">10.5-2 ObjectOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7B597AC77A3D71F9">10.5-3 DifferentialOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X858D2D1A8674D2EC">10.5-4 DifferentialsOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X85FBFD9C8382C93B">10.5-5 CyclesOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8168D8CC7805FE30">10.5-6 BoundariesOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F8D61447AB57EF3">10.5-7 HomologyOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7DCBCF7682DC87FC">10.5-8 IsFiniteComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A5CB74485184FEE">10.5-9 UpperBound</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X807CC5AD7FDFA4E3">10.5-10 LowerBound</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7C1F81277A5AC8B4">10.5-11 LengthOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X82D615CD796C9C0A">10.5-12 HighestKnownDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X868105FF86FA1B6E">10.5-13 LowestKnownDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X793465497B435197">10.5-14 IsExactSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A6AE1C378E2EEE2">10.5-15 IsExactInDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X87ADD4F685457000">10.5-16 IsShortExactSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X849C525A7F6EBF6D">10.5-17 ForEveryDegree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X8764E5C88284301B">10.6 <span class="Heading">Transforming and combining complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81F7B0AB84D6319B">10.6-1 Shift</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X86B56AC87A433E5B">10.6-2 ShiftUnsigned</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7897A65085171913">10.6-3 YonedaProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X87E1CA507EB6B86E">10.6-4 BrutalTruncationBelow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7CE9481684DD304E">10.6-5 BrutalTruncationAbove</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8547C51781BBB48F">10.6-6 BrutalTruncation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E3E34167B79248A">10.6-7 SyzygyTruncation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7C729A2587E5DA8D">10.6-8 CosyzygyTruncation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X797FC6037F6605E7">10.6-9 SyzygyCosyzygyTruncation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X85F418EB859E7597">10.7 <span class="Heading">Chain maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F76E62D837F0E66">10.7-1 IsChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7BCD94877DF261C4">10.7-2 ChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7C9597F6810F195E">10.7-3 ZeroChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8699B48A783E1900">10.7-4 FiniteChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X805BA1698394DE4C">10.7-5 ComplexAndChainMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7F20B8807ECE404E">10.7-6 MorphismOfChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X792B5B8684A75F72">10.7-7 MorphismsOfChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X78EC487B79B20DDE">10.7-8 ComparisonLifting</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7A95C5E78298E0C6">10.7-9 ComparisonLiftingToProjectiveResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7EF42CF48165D831">10.7-10 MappingCone</a></span>
</div></div>
</div>
<h3>10 <span class="Heading">Chain complexes</span></h3>
<p><a id="X7DFB63A97E67C0A1" name="X7DFB63A97E67C0A1"></a></p>
<h4>10.1 <span class="Heading">Introduction</span></h4>
<p>If <span class="SimpleMath">\(\mathcal{A}\)</span> is an abelian category, then a chain complex of objects of <span class="SimpleMath">\(\mathcal A\)</span> is a sequence</p>
<p class="center">\[
\cdots \longrightarrow C_{i+1} \stackrel{d_{i+1}}{\longrightarrow}
C_i \stackrel{d_i}{\longrightarrow} C_{i-1}
\stackrel{d_{i-1}}{\longrightarrow} \cdots
\]</p>
<p>where <span class="SimpleMath">\(C_i\)</span> is an object of <span class="SimpleMath">\(\mathcal A\)</span> for all <span class="SimpleMath">\(i\)</span>, and <span class="SimpleMath">\(d_i\)</span> is a morphism of <span class="SimpleMath">\(\mathcal A\)</span> for all <span class="SimpleMath">\(i\)</span> such that the composition of two consecutive maps of the complex is zero. The maps are called the differentials of the complex. A complex is called <em>bounded above</em> (resp. below) if there is a bound <span class="SimpleMath">\(b\)</span> such that <span class="SimpleMath">\(C_i = 0\)</span> for all <span class="SimpleMath">\(i>b\)</span> (resp. <span class="SimpleMath">\(i<b\)</span>). A complex is <em>bounded</em> if it is both bounded below and bounded above.</p>
<p>The challenge when representing chain complexes in software is to handle their infinite nature. If a complex is not bounded, or not known to be bounded, how can we represent it in an immutable way? Our solution is to use a category called <code class="code">InfList</code> (for ``infinite list'') to store the differentials of the complex. The properties of the <code class="code">IsInfList</code> category is described in <a href="chap10_mj.html#X7AC5660E80079755"><span class="RefLink">10.2</span></a>. An <code class="code">IsQPAComplex</code> object consists of one <code class="code">IsInfList</code> for the differentials, and it also has an <code class="code">IsCat</code> object as an attribute. The <code class="code">IsCat</code> category is a representation of an abelian category, see <a href="chap10_mj.html#X7CAF603281B94AC8"><span class="RefLink">10.3</span></a>.</p>
<p>To work with bounded complexes one does not need to know much about the <code class="code">IsInfList</code> category. A bounded complex can be created by simply giving a list of the differentials and the degree of the first differential as input (see <code class="func">FiniteComplex</code> (<a href="chap10_mj.html#X7C75868E82A831C8"><span class="RefLink">10.4-5</span></a>)), and to create a stalk complex the stalk object and its degree suffice as input (see <code class="func">StalkComplex</code> (<a href="chap10_mj.html#X80B47D78785F6859"><span class="RefLink">10.4-6</span></a>)). In both cases an <code class="code">IsCat</code> object is also needed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := FiniteComplex(cat, 1, [g,f]);</span>
0 -> 2:(1,0) -> 1:(2,2) -> 0:(1,1) -> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Ms := StalkComplex(cat, M, 3);</span>
0 -> 3:(2,2) -> 0
</pre></div>
<p><a id="X7AC5660E80079755" name="X7AC5660E80079755"></a></p>
<h4>10.2 <span class="Heading">Infinite lists</span></h4>
<p>In this section we give documentation for the <code class="code">IsInfList</code> category. We start by giving a representation of <span class="SimpleMath">\(\pm \infty\)</span>. Then we quickly describe the <code class="code">IsInfList</code> category, before we turn to the underlying structure of the infinite lists -- the half infinite lists (<code class="code">IsHalfInfList</code>). Most of the functionality of the infinite lists come from this category. Finally, we give the constructors for infinite lists, and some methods for manipulating such objects.</p>
<p><a id="X8466C7117CEC99D2" name="X8466C7117CEC99D2"></a></p>
<h5>10.2-1 IsInfiniteNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInfiniteNumber</code></td><td class="tdright">( category )</td></tr></table></div>
<p>A category for infinite numbers.</p>
<p><a id="X7F12EE4479A03527" name="X7F12EE4479A03527"></a></p>
<h5>10.2-2 PositiveInfinity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositiveInfinity</code></td><td class="tdright">( var )</td></tr></table></div>
<p>A global variable representing the number <span class="SimpleMath">\(\infty\)</span>. It is greater than any integer, but it can not be compared to numbers which are not integers. It belongs to the <code class="code">IsInfiniteNumber</code> category.</p>
<p><a id="X8759931784478A4D" name="X8759931784478A4D"></a></p>
<h5>10.2-3 NegativeInfinity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NegativeInfinity</code></td><td class="tdright">( var )</td></tr></table></div>
<p>A global variable representing the number <span class="SimpleMath">\(-\infty\)</span>. It is smaller than any integer, but it can not be compared to numbers which are not integers. It belongs to the <code class="code">IsInfiniteNumber</code> category.</p>
<p><a id="X84731DA279E797A5" name="X84731DA279E797A5"></a></p>
<h5>10.2-4 IsInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInfList</code></td><td class="tdright">( category )</td></tr></table></div>
<p>An infinite list is an immutable representation of a list with possibly infinite range of indices. It consists of three parts: The ``middle part'' is finite and covers some range <span class="SimpleMath">\([a,b]\)</span> of indices, the ``positive part'' covers the range <span class="SimpleMath">\([b+1,\infty)\)</span> of indices, and the ``negative part'' covers the range <span class="SimpleMath">\((-\infty,a-1]\)</span> of indices. Note that none of the three parts are mandatory: The middle part may be an empty list, and the positive part may be set to <code class="code">fail</code> to achieve index range ending at <span class="SimpleMath">\(b < \infty\)</span>. Similarly, if the index range has lower bound <span class="SimpleMath">\(a < \infty\)</span>, put the negative part to be <code class="code">fail</code>.</p>
<p>Each of the two infinite parts are described in one of the following ways: (1) A finite list which is repeated indefinitely; (2) A function which takes an index in the list as argument and returns the corresponding list item; (3) A function which takes an item from the list as argument and returns the next item.</p>
<p>The two infinite parts are represented as ``half infinite lists'', see <a href="chap10_mj.html#X87524BBB7C2F9FCB"><span class="RefLink">10.2-5</span></a>. An infinite list can be constructed in the following ways:</p>
<ul>
<li><p>From two half infinite lists and a middle part, <code class="func">MakeInfListFromHalfInfLists</code> (<a href="chap10_mj.html#X7CE66EBA844DBB57"><span class="RefLink">10.2-22</span></a>).</p>
</li>
<li><p>Directly, by giving the same input as when constructing the above, <code class="func">MakeInfList</code> (<a href="chap10_mj.html#X801DE2517D08C19E"><span class="RefLink">10.2-23</span></a>).</p>
</li>
<li><p>If all values of the infinite list are the image of the index under a function <span class="SimpleMath">\(f\)</span>, one can use <code class="func">FunctionInfList</code> (<a href="chap10_mj.html#X7EEB276380E1584F"><span class="RefLink">10.2-24</span></a>).</p>
</li>
<li><p>If all values of the infinite list are the same, one can use <code class="func">ConstantInfList</code> (<a href="chap10_mj.html#X78C7FBAE83914790"><span class="RefLink">10.2-25</span></a>).</p>
</li>
<li><p>If the infinite list has a finite range, one can use <code class="func">FiniteInfList</code> (<a href="chap10_mj.html#X7D68E0627F556E14"><span class="RefLink">10.2-26</span></a>).</p>
</li>
</ul>
<p>In addition, new infinite lists can be constructed from others by shift, splice, concatenation, extracting parts or applying a function to the elements.</p>
<p><a id="X87524BBB7C2F9FCB" name="X87524BBB7C2F9FCB"></a></p>
<h5>10.2-5 IsHalfInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHalfInfList</code></td><td class="tdright">( category )</td></tr></table></div>
<p>A half infinite list is a representation of a list with indices in the range <span class="SimpleMath">\([a,\infty)\)</span> or <span class="SimpleMath">\((-\infty,b]\)</span>. An infinite list is typically made from two half infinite lists, and half infinite lists can be extracted from an infinite list. Hence, the half infinite list stores much of the information about an infinite list. One main difference between an infinite list and a half infinite list is that the half infinite list does not have any finite part, as the ``middle'' part of an infinite list.</p>
<p><a id="X7D21FB1A7D21FB1A" name="X7D21FB1A7D21FB1A"></a></p>
<h5><code>10.2-6 \^</code></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \^</code>( <var class="Arg">list</var>, <var class="Arg">pos</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- either an infinite list or a half infinite list, <var class="Arg">pos</var> -- a valid index for <var class="Arg">list</var>.</p>
<p>Returns: The value at position <var class="Arg">pos</var> of <var class="Arg">list</var>.</p>
<p><a id="X7E01944C83F87EED" name="X7E01944C83F87EED"></a></p>
<h5>10.2-7 MakeHalfInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeHalfInfList</code>( <var class="Arg">start</var>, <var class="Arg">direction</var>, <var class="Arg">typeWithArgs</var>, <var class="Arg">callback</var>, <var class="Arg">repeatifyCallback</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">start</var> -- an integer, <var class="Arg">direction</var> -- either <span class="SimpleMath">\(1\)</span> or <span class="SimpleMath">\(-1\)</span>, <var class="Arg">typeWithArgs</var> -- a list which may have different formats, <var class="Arg">callback</var> -- a function, <var class="Arg">repeatifyCallback</var> -- a function. <br /></p>
<p>Returns: A newly created half infinite list with index range from <var class="Arg">start</var> to <span class="SimpleMath">\(\infty\)</span>, or from <span class="SimpleMath">\(-\infty\)</span> to <var class="Arg">start</var>.</p>
<p>If the range should be <span class="SimpleMath">\([\mathtt{start},\infty)\)</span> then the value of <var class="Arg">direction</var> is <span class="SimpleMath">\(1\)</span>. if the range should be <span class="SimpleMath">\((-\infty,\mathtt{start}]\)</span>, then the value of <var class="Arg">direction</var> is <span class="SimpleMath">\(-1\)</span>.</p>
<p>The argument <var class="Arg">typeWithArgs</var> can take one of the following forms:</p>
<ul>
<li><p><code class="code">[ "repeat", repeatList ]</code></p>
</li>
<li><p><code class="code">[ "next", nextFunction, initialValue ]</code></p>
</li>
<li><p><code class="code">[ "next/repeat", nextFunction, initialValue ]</code></p>
</li>
<li><p><code class="code">[ "pos", posFunction ]</code></p>
</li>
<li><p><code class="code">[ "pos", posFunction, storeValues ]</code></p>
</li>
</ul>
<p><code class="code">repeatList</code> is a list of values that should be repeated in the half infinite list. <code class="code">nextFunction</code> returns the value at position <span class="SimpleMath">\(i\)</span>, given the value at the previous position as argument. Here <code class="code">initialValue</code> is the value at position <code class="code">start</code>. Similarly, <code class="code">posFunction</code> returns the value at any position <span class="SimpleMath">\(i\)</span>, and it may or may not store the values between the previous computed indices and the newly computed index. The default value of <code class="code">storeValues</code> is <code class="code">true</code> for <code class="code">"next"</code> and <code class="code">"pos"</code>, and <code class="code">false</code> for <code class="code">"repeat"</code>. The type <code class="code">"next/repeat"</code> works exactly like the type <code class="code">"next"</code>, except that when values in the list are computed, the list will try to discover if the values are repeating. If this happens, the function <var class="Arg">repeatifyCallback</var> is called with two arguments: the non-repeating part at the beginning as a normal list (this might be empty) and a new HalfInfList of type <code class="code">"repeat"</code> for the repeating part.</p>
<p>The argument <code class="code">callback</code> is a function that is called whenever a new value of the list is computed. It takes three arguments: The current position, the direction and the type (that is, <code class="code">typeWithArgs[1]</code>). If no callback function is needed, use <code class="code">false</code>.</p>
<p>All the information given to create the list is stored, and can be retrieved later by the operations listed in <a href="chap10_mj.html#X8458C2698505C12B"><span class="RefLink">10.2-8</span></a>--<a href="chap10_mj.html#X7B5E57F0859BEE17"><span class="RefLink">10.2-18</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># make a HalfInfList from 0 to inf which repeats the list [ 2, 4, 6 ]</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">list1 := MakeHalfInfList( 0, 1, [ "repeat", [ 2, 4, 6 ] ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">list1^0;</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">list1^5;</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput"># make a HalfInfList from 0 to inf with x^2 in position x</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := function(x) return x^2; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">list2 := MakeHalfInfList( 0, 1, [ "pos", f, false ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">list2^0;</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">list2^10;</span>
100
<span class="GAPprompt">gap></span> <span class="GAPinput"># make a HalfInfList from 0 to -inf where each new value adds 3</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># to the previous and the value in position 0 is 10</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := function(x) return x+3; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">list3 := MakeHalfInfList( 0, -1, [ "next", g, 7 ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">list3^0;</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">list3^-10;</span>
40
</pre></div>
<p><a id="X8458C2698505C12B" name="X8458C2698505C12B"></a></p>
<h5>10.2-8 StartPosition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StartPosition</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The start position of <var class="Arg">list</var>.</p>
<p><a id="X812437C47A208D4A" name="X812437C47A208D4A"></a></p>
<h5>10.2-9 Direction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Direction</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The direction of <var class="Arg">list</var> (either <span class="SimpleMath">\(1\)</span> or <span class="SimpleMath">\(-1\)</span>).</p>
<p><a id="X7C3728BF7ED7149A" name="X7C3728BF7ED7149A"></a></p>
<h5>10.2-10 InfListType</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfListType</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The type of <var class="Arg">list</var> (either <code class="code">"pos"</code>, <code class="code">"repeat"</code> or <code class="code">"next"</code>).</p>
<p><a id="X82AA0979868A6967" name="X82AA0979868A6967"></a></p>
<h5>10.2-11 RepeatingList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepeatingList</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The repeating list of <var class="Arg">list</var> if <var class="Arg">list</var> is of type <code class="code">"repeat"</code>, and <code class="code">fail</code> otherwise.</p>
<p><a id="X7AF21D62848B270B" name="X7AF21D62848B270B"></a></p>
<h5>10.2-12 ElementFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementFunction</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The element function of <var class="Arg">list</var> if <var class="Arg">list</var> is of type <code class="code">"next"</code> or <code class="code">"pos"</code>, and <code class="code">fail</code> otherwise.</p>
<p><a id="X8728A2107F27D68F" name="X8728A2107F27D68F"></a></p>
<h5>10.2-13 IsStoringValues</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStoringValues</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: <code class="code">true</code> if all elements of the list are stored, <code class="code">false</code> otherwise.</p>
<p><a id="X86888FBD8559B29B" name="X86888FBD8559B29B"></a></p>
<h5>10.2-14 NewValueCallback</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NewValueCallback</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The callback function of the list.</p>
<p><a id="X856E1541834688C6" name="X856E1541834688C6"></a></p>
<h5>10.2-15 IsRepeating</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRepeating</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: <code class="code">true</code> if the type of the list is <code class="code">"repeat"</code>.</p>
<p><a id="X8208162D7F3CC139" name="X8208162D7F3CC139"></a></p>
<h5>10.2-16 InitialValue</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InitialValue</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: If the list is of type <code class="code">"next"</code> then the initial value is returned, otherwise it fails.</p>
<p><a id="X7F12274E794510CA" name="X7F12274E794510CA"></a></p>
<h5>10.2-17 LowestKnownPosition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowestKnownPosition</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The lowest index <span class="SimpleMath">\(i\)</span> such that the value at position <span class="SimpleMath">\(i\)</span> is known without computation (that is, it is either stored, or the list has type <code class="code">"repeat"</code>).</p>
<p><a id="X7B5E57F0859BEE17" name="X7B5E57F0859BEE17"></a></p>
<h5>10.2-18 HighestKnownValue</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HighestKnownValue</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">list</var> -- a half infinite list.<br /></p>
<p>Returns: The highest index <span class="SimpleMath">\(i\)</span> such that the value at position <span class="SimpleMath">\(i\)</span> is known without computation (that is, it is either stored, or the list has type <code class="code">"repeat"</code>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># we reuse the IsHalfInfLists from the previous example</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HighestKnownPosition(list1);</span>
+inf
<span class="GAPprompt">gap></span> <span class="GAPinput">HighestKnownPosition(list2);</span>
"none"
<span class="GAPprompt">gap></span> <span class="GAPinput">HighestKnownPosition(list3);</span>
0
</pre></div>
<p><a id="X81F7B0AB84D6319B" name="X81F7B0AB84D6319B"></a></p>
<h5>10.2-19 Shift</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Shift</code>( <var class="Arg">list</var>, <var class="Arg">shift</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- a half infinite list, <var class="Arg">shift</var> -- an integer.<br /></p>
<p>Returns: A new half infinite list which is <var class="Arg">list</var> with all values shifted <var class="Arg">shift</var> positions to the right if <var class="Arg">shift</var> is positive, and to the left if <var class="Arg">shift</var> is negative.</p>
<p><a id="X7D8067A7842BC4A4" name="X7D8067A7842BC4A4"></a></p>
<h5>10.2-20 Cut</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cut</code>( <var class="Arg">list</var>, <var class="Arg">pos</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- a half infinite list, <var class="Arg">pos</var> -- an integer within the range of <var class="Arg">list</var>.<br /></p>
<p>Returns: A new half infinite list which is <var class="Arg">list</var> with some part cut off.</p>
<p>If the direction of <var class="Arg">list</var> is positive, then the new list has range from <code class="code">cut</code> to <span class="SimpleMath">\(\infty\)</span>. If the direction of <var class="Arg">list</var> is negative, then the new list has range from <span class="SimpleMath">\(-\infty\)</span> to <code class="code">cut</code>. The values at position <span class="SimpleMath">\(i\)</span> of the new half infinite list is the same as the value at position <span class="SimpleMath">\(i\)</span> of <var class="Arg">list</var>.</p>
<p><a id="X79A67E377B71C540" name="X79A67E377B71C540"></a></p>
<h5>10.2-21 HalfInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HalfInfList</code>( <var class="Arg">list</var>, <var class="Arg">func</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- a half infinite list, <var class="Arg">func</var> -- a function which takes an element of the list as argument. <br /></p>
<p>Returns: A half infinite list with the same range as <var class="Arg">list</var>, where the value at position <span class="SimpleMath">\(i\)</span> is the image of the value at position <span class="SimpleMath">\(i\)</span> of <var class="Arg">list</var> under <var class="Arg">func</var>.</p>
<p><a id="X7CE66EBA844DBB57" name="X7CE66EBA844DBB57"></a></p>
<h5>10.2-22 MakeInfListFromHalfInfLists</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeInfListFromHalfInfLists</code>( <var class="Arg">basePosition</var>, <var class="Arg">middle</var>, <var class="Arg">positive</var>, <var class="Arg">negative</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">basePosition</var> -- an integer, <var class="Arg">middle</var> -- a list, <var class="Arg">positive</var> -- a half infinite list, <var class="Arg">negative</var> -- a half infinite list. <br /></p>
<p>Returns: An infinite list with <var class="Arg">middle</var> as is middle part, <var class="Arg">positive</var> as its positive part and <var class="Arg">negative</var> as its negative part.</p>
<p>The starting position of <var class="Arg">positive</var> must be <code class="code">basePosition + Length( middle )</code>, and the starting position of <var class="Arg">negative</var> must be <code class="code">basePosition - 1</code>. The returned list has <code class="code">middle[1]</code> in position <var class="Arg">basePosition</var>, <code class="code">middle[2]</code> in position <var class="Arg">basePosition + 1</var> and so on. Note that one probably wants the <var class="Arg">positive</var> half infinite list to have direction <span class="SimpleMath">\(1\)</span>, and the <var class="Arg">negative</var> half infinite list to have direction <span class="SimpleMath">\(-1\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># we want to construct an infinite list with 0 in position</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># 0 to 5, and x^2 in position x where x goes from 6 to inf, </span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># and alternatingly 1 and -1 in position -1 to -inf.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">basePosition := 0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">middle := [0,0,0,0,0,0];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := function(x) return x^2; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">positive := MakeHalfInfList( 6, 1, [ "pos", f, false ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">altList := [ 1, -1 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">negative := MakeHalfInfList( -1, -1, [ "repeat", altList ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">inflist := MakeInfListFromHalfInfLists( basePosition, middle,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> positive, negative );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">inflist^0; inflist^5; inflist^6; inflist^-1; inflist^-4;</span>
0
0
36
1
-1
</pre></div>
<p><a id="X801DE2517D08C19E" name="X801DE2517D08C19E"></a></p>
<h5>10.2-23 MakeInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeInfList</code>( <var class="Arg">basePosition</var>, <var class="Arg">middle</var>, <var class="Arg">positive</var>, <var class="Arg">negative</var>, <var class="Arg">callback</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">basePosition</var> -- an integer, <var class="Arg">middle</var> -- a list, <var class="Arg">positive</var> -- a list describing the positive part, <var class="Arg">negative</var> -- a list describing the negative part. <br /></p>
<p>Returns: An infinite list with <var class="Arg">middle</var> as is middle part, <var class="Arg">positive</var> as its positive part and <var class="Arg">negative</var> as its negative part.</p>
<p>The major difference between this construction and the previous is that here the half infinite lists that will make the positive and negative parts are not entered directly as arguments. Instead, one enters ``description lists'', which are of the same format as the argument <var class="Arg">typeWithArgs</var> of <code class="func">MakeHalfInfList</code> (<a href="chap10_mj.html#X7E01944C83F87EED"><span class="RefLink">10.2-7</span></a>). If the positive and/or negative part is specified with type <code class="code">"next/repeat"</code>, then it will initially be of type <code class="code">"next"</code>, but will be replaced by a HalfInfList of type <code class="code">"repeat"</code> if it is discovered that the values are repeating.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># we construct the same infinite list as in the previous example</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">basePosition := 0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">middle := [0,0,0,0,0,0];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := function(x) return x^2; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">altList := [ 1, -1 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inflist2 := MakeInfList( 0, middle, [ "pos", f, false ], [ "repeat", </span>
<span class="GAPprompt">></span> <span class="GAPinput"> altList ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">inflist2^0; inflist2^5; inflist2^6; inflist2^-1; inflist2^-4;</span>
0
0
36
1
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">n := function( x ) return ( x + 1 ) mod 5; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">list := MakeInfList( 0, [ 0 ], [ "next/repeat", n, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "repeat", [ 0 ] ], false );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">list^2;</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRepeating( PositivePart( list ) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">list^11;</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRepeating( PositivePart( list ) );</span>
true
</pre></div>
<p><a id="X7EEB276380E1584F" name="X7EEB276380E1584F"></a></p>
<h5>10.2-24 FunctionInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FunctionInfList</code>( <var class="Arg">func</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">func</var> -- a function that takes an integer as argument.<br /></p>
<p>Returns: An infinite list where the value at position <span class="SimpleMath">\(i\)</span> is the function <var class="Arg">func</var> applied to <span class="SimpleMath">\(i\)</span>.</p>
<p><a id="X78C7FBAE83914790" name="X78C7FBAE83914790"></a></p>
<h5>10.2-25 ConstantInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstantInfList</code>( <var class="Arg">value</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">value</var> -- an object.<br /></p>
<p>Returns: An infinite list which has the object <var class="Arg">value</var> in every position.</p>
<p><a id="X7D68E0627F556E14" name="X7D68E0627F556E14"></a></p>
<h5>10.2-26 FiniteInfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FiniteInfList</code>( <var class="Arg">basePosition</var>, <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">basePosition</var> -- an integer, <var class="Arg">list</var> -- a list of length <span class="SimpleMath">\(n\)</span>.<br /></p>
<p>Returns: An infinite list with <span class="SimpleMath">\(\mathtt{list[1]},\ldots,\mathtt{list[n]}\)</span> in positions <span class="SimpleMath">\(\mathtt{basePosition},\ldots,\)</span> <span class="SimpleMath">\(\mathtt{basePosition + n}\)</span>.</p>
<p>The range of this list is <span class="SimpleMath">\([\mathtt{basePosition}, \mathtt{basePosition + n}]\)</span>.</p>
<p><a id="X8134072B7C7309CD" name="X8134072B7C7309CD"></a></p>
<h5>10.2-27 MiddleStart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MiddleStart</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The starting position of the "middle" part of <var class="Arg">list</var>.</p>
<p><a id="X7EB933657AF3D723" name="X7EB933657AF3D723"></a></p>
<h5>10.2-28 MiddleEnd</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MiddleEnd</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The ending position of the middle part of <var class="Arg">list</var>.</p>
<p><a id="X7AC3ECBB7BC93594" name="X7AC3ECBB7BC93594"></a></p>
<h5>10.2-29 MiddlePart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MiddlePart</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The middle part (as a list) of <var class="Arg">list</var>.</p>
<p><a id="X7896D6DD788046E7" name="X7896D6DD788046E7"></a></p>
<h5>10.2-30 PositivePart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositivePart</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The positive part (as a half infinite list) of <var class="Arg">list</var>.</p>
<p><a id="X81B768838567F98D" name="X81B768838567F98D"></a></p>
<h5>10.2-31 NegativePart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NegativePart</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The negative part (as a halft infinite list) of <var class="Arg">list</var>.</p>
<p><a id="X8511A0B88003B989" name="X8511A0B88003B989"></a></p>
<h5>10.2-32 HighestKnownPosition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HighestKnownPosition</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The highest index <span class="SimpleMath">\(i\)</span> such that the value at position <span class="SimpleMath">\(i\)</span> is known without computation.</p>
<p><a id="X7F12274E794510CA" name="X7F12274E794510CA"></a></p>
<h5>10.2-33 LowestKnownPosition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowestKnownPosition</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The lowest index <span class="SimpleMath">\(i\)</span> such that the value at position <span class="SimpleMath">\(i\)</span> is known without computation.</p>
<p><a id="X7A5CB74485184FEE" name="X7A5CB74485184FEE"></a></p>
<h5>10.2-34 UpperBound</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpperBound</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The highest index in the range of the list.</p>
<p><a id="X807CC5AD7FDFA4E3" name="X807CC5AD7FDFA4E3"></a></p>
<h5>10.2-35 LowerBound</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerBound</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list. <br /></p>
<p>Returns: The lowest index in the range of the list.</p>
<p><a id="X7A311EBE79F31AEA" name="X7A311EBE79F31AEA"></a></p>
<h5>10.2-36 FinitePartAsList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FinitePartAsList</code>( <var class="Arg">list</var>, <var class="Arg">startPos</var>, <var class="Arg">endPos</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list, <var class="Arg">startPos</var> -- an integer, <var class="Arg">endPos</var> -- an integer.<br /></p>
<p>Returns: A list containing the values of <var class="Arg">list</var> in positions <span class="SimpleMath">\(\mathtt{endPos},\ldots,\mathtt{startPos}\)</span>.</p>
<p>Note that both integers in the input must be within the index range of <var class="Arg">list</var>.</p>
<p><a id="X8028ED6883CFA39C" name="X8028ED6883CFA39C"></a></p>
<h5>10.2-37 PositivePartFrom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositivePartFrom</code>( <var class="Arg">list</var>, <var class="Arg">pos</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list, <var class="Arg">pos</var> -- an integer.<br /></p>
<p>Returns: An infinite list (<em>not</em> a half infinite list) with index range from <code class="code">pos</code> to <code class="code">UpperBound(list)</code>.</p>
<p>The value at position <span class="SimpleMath">\(i\)</span> of the new infinite list is the same as the value at position <span class="SimpleMath">\(i\)</span> of <var class="Arg">list</var>.</p>
<p><a id="X7863903B7E281CF6" name="X7863903B7E281CF6"></a></p>
<h5>10.2-38 NegativePartFrom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NegativePartFrom</code>( <var class="Arg">list</var>, <var class="Arg">pos</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list, <var class="Arg">pos</var> -- an integer.<br /></p>
<p>Returns: An infinite list (<em>not</em> a half infinite list) with index range from <code class="code">LowerBound(list)</code> to <code class="code">pos</code>.</p>
<p>The value at position <span class="SimpleMath">\(i\)</span> of the new infinite list is the same as the value at position <span class="SimpleMath">\(i\)</span> of <var class="Arg">list</var>.</p>
<p><a id="X81F7B0AB84D6319B" name="X81F7B0AB84D6319B"></a></p>
<h5>10.2-39 Shift</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Shift</code>( <var class="Arg">list</var>, <var class="Arg">shift</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list, <var class="Arg">shift</var> -- an integer.<br /></p>
<p>Returns: A new infinite list which is <var class="Arg">list</var> with all values shifted <var class="Arg">shift</var> positions to the right if <var class="Arg">shift</var> is positive, and to the left if <var class="Arg">shift</var> is negative.</p>
<p><a id="X80153272857B2EDE" name="X80153272857B2EDE"></a></p>
<h5>10.2-40 Splice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Splice</code>( <var class="Arg">positiveList</var>, <var class="Arg">negativeList</var>, <var class="Arg">joinPosition</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">positiveList</var> -- an infinite list, <var class="Arg">negativeList</var> -- an infinite list, <var class="Arg">joinPosition</var> -- an integer.<br /></p>
<p>Returns: A new infinite list which is identical to <var class="Arg">positiveList</var> for indices greater than <var class="Arg">joinPosition</var> and identical to <var class="Arg">negativeList</var> for indices smaller than or equal to <var class="Arg">joinPosition</var>.</p>
<p><a id="X84FC63C183F0A977" name="X84FC63C183F0A977"></a></p>
<h5>10.2-41 InfConcatenation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfConcatenation</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">arg</var> -- a number of infinite lists. <br /></p>
<p>Returns: A new infinite list.</p>
<p>If the length of <var class="Arg">arg</var> is greater than or equal to <span class="SimpleMath">\(2\)</span>, then the new infinite list consists of the following parts: It has the positive part of <code class="code">arg[1]</code>, and the middle part is the concatenation of the middle parts of all lists in <var class="Arg">arg</var>, such that <code class="code">MiddleEnd</code> of the new list is the same as <code class="code">MiddleEnd( arg[1] )</code>. The negative part of the new list is the negative part of <code class="code">arg[Length(arg)]</code>, although shiftet so that it starts in the correct position.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># we do an InfConcatenation of three lists.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := function(x) return x; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := function(x) return x+1; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := function(x) return x^2; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">InfList1 := MakeInfList( 0, [ 10 ], [ "pos", f, false ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "repeat", [ 10, 15 ] ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">InfList2 := MakeInfList( 0, [ 20 ], [ "pos", g, false ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "repeat", [ 20, 25 ] ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">InfList3 := MakeInfList( 0, [ 30 ], [ "pos", h, false ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "repeat", [ 30, 35 ] ], false );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">concList := InfConcatenation( InfList1, InfList2, InfList3 );</span>
<object>
<span class="GAPprompt">gap></span> <span class="GAPinput">MiddlePart(concList);</span>
[ 30, 20, 10 ]
</pre></div>
<p>The newly created <code class="code">concList</code> looks as follows around the middle part:</p>
<p class="center">\[
\begin{array}{lrrrrrrrrrrr}
\text{position} & \cdots & 3
& 2 & 1 & 0 & -1 & -2 & -3 & -4 & -5
& \cdots \\
\text{value} & \cdots & 3
& 2 & 1 & 10 & 20 & 30 & 30 & 35 & 30
& \cdots \\
\end{array}
\]</p>
<p><a id="X7F4CF360858E80A6" name="X7F4CF360858E80A6"></a></p>
<h5>10.2-42 InfList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfList</code>( <var class="Arg">list</var>, <var class="Arg">func</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">list</var> -- an infinite list, <var class="Arg">func</var> -- a function which takes an element of the list as argument. <br /></p>
<p>Returns: An infinite list with the same range as <var class="Arg">list</var>, where the value at position <span class="SimpleMath">\(i\)</span> is the image of the value at position <span class="SimpleMath">\(i\)</span> of <var class="Arg">list</var> under <var class="Arg">func</var>.</p>
<p><a id="X791D879483C59410" name="X791D879483C59410"></a></p>
<h5>10.2-43 IntegersList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntegersList</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>An infinite list with range <span class="SimpleMath">\((-\infty,\infty)\)</span> where the value at position <span class="SimpleMath">\(i\)</span> is the number <span class="SimpleMath">\(i\)</span> (that is, a representation of the integers).</p>
<p><a id="X7CAF603281B94AC8" name="X7CAF603281B94AC8"></a></p>
<h4>10.3 <span class="Heading">Representation of categories</span></h4>
<p>A chain complex consists of objects and morphisms from some category. In QPA, this category will usually be the category of right modules over some quotient of a path algebra.</p>
<p><a id="X7F0E546178977963" name="X7F0E546178977963"></a></p>
<h5>10.3-1 IsCat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCat</code></td><td class="tdright">( category )</td></tr></table></div>
<p>The category for categories. A category is a record, storing a number of properties that is specified within each category. Two categories can be compared using <code class="code">=</code>. Currently, the only implemented category is the one of right modules over a (quotient of a) path algebra.</p>
<p><a id="X7BB9B02E7E76FF9C" name="X7BB9B02E7E76FF9C"></a></p>
<h5>10.3-2 CatOfRightAlgebraModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CatOfRightAlgebraModules</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">A</var> -- a (quotient of a) path algebra.<br /></p>
<p>Returns: The category mod <span class="SimpleMath">\(A\)</span>.</p>
<p>mod <span class="SimpleMath">\(A\)</span> has several properties, which can be accessed using the <code class="code">.</code> mark. Some of the properties store functions. All properties are demonstrated in the following example.</p>
<ul>
<li><p><code class="code">zeroObj</code> -- returns the zero module of mod <span class="SimpleMath">\(A\)</span>.</p>
</li>
<li><p><code class="code">isZeroObj</code> -- returns true if the given module is zero.</p>
</li>
<li><p><code class="code">zeroMap</code> -- returns the ZeroMapping function.</p>
</li>
<li><p><code class="code">isZeroMapping</code> -- returns the IsZero test.</p>
</li>
<li><p><code class="code">composeMaps</code> -- returns the composition of the two given maps.</p>
</li>
<li><p><code class="code">ker</code> -- returns the Kernel function.</p>
</li>
<li><p><code class="code">im</code> -- returns the Image function.</p>
</li>
<li><p><code class="code">isExact</code> -- returns true if two consecutive maps are exact.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">alg;</span>
<algebra-with-one over Rationals, with 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput"># L, M, and N are alg-modules</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># f: L --> M and g: M --> N are non-zero morphisms</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cat := CatOfRightAlgebraModules(alg);</span>
<cat: right modules over algebra>
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.zeroObj;</span>
<right-module over <algebra-with-one over Rationals, with 7 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.isZeroObj(M);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.zeroMap(M,N);</span>
<mapping: <3-dimensional right-module over AlgebraWithOne( Rationals,
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*v4], [(1)*a], [(1)*b], [(1)*c] ])> ->
<1-dimensional right-module over AlgebraWithOne( Rationals,
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*v4], [(1)*a], [(1)*b], [(1)*c] ] )> >
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.composeMaps(g,f);</span>
<mapping: <1-dimensional right-module over AlgebraWithOne( Rationals,
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*v4], [(1)*a], [(1)*b], [(1)*c]]
-> <1-dimensional right-module over AlgebraWithOne( Rationals,
[ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*v4], [(1)*a], [(1)*b], [(1)*c] ] )> >
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.ker(g);</span>
<2-dimensional right-module over <algebra-with-one over Rationals,
with 7 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">cat.isExact(g,f);</span>
false
</pre></div>
<p><a id="X7EC3D95F7C791F7E" name="X7EC3D95F7C791F7E"></a></p>
<h4>10.4 <span class="Heading">Making a complex</span></h4>
<p>The most general constructor for complexes is the function <code class="func">Complex</code> (<a href="chap10_mj.html#X7E51958C86D73D17"><span class="RefLink">10.4-3</span></a>). In addition to this, there are constructors for common special cases:</p>
<ul>
<li><p><code class="func">ZeroComplex</code> (<a href="chap10_mj.html#X80F8DD20789BBBC0"><span class="RefLink">10.4-4</span></a>)</p>
</li>
<li><p><code class="func">StalkComplex</code> (<a href="chap10_mj.html#X80B47D78785F6859"><span class="RefLink">10.4-6</span></a>)</p>
</li>
<li><p><code class="func">FiniteComplex</code> (<a href="chap10_mj.html#X7C75868E82A831C8"><span class="RefLink">10.4-5</span></a>)</p>
</li>
<li><p><code class="func">ShortExactSequence</code> (<a href="chap10_mj.html#X7A685484784087FE"><span class="RefLink">10.4-7</span></a>)</p>
</li>
</ul>
<p><a id="X838C9EBD87B8CB9D" name="X838C9EBD87B8CB9D"></a></p>
<h5>10.4-1 IsQPAComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQPAComplex</code></td><td class="tdright">( category )</td></tr></table></div>
<p>The category for chain complexes.</p>
<p><a id="X7E0CE8AC87533991" name="X7E0CE8AC87533991"></a></p>
<h5>10.4-2 IsZeroComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZeroComplex</code></td><td class="tdright">( category )</td></tr></table></div>
<p>Category for zero complexes, subcategory of <code class="func">IsQPAComplex</code> (<a href="chap10_mj.html#X838C9EBD87B8CB9D"><span class="RefLink">10.4-1</span></a>).</p>
<p><a id="X7E51958C86D73D17" name="X7E51958C86D73D17"></a></p>
<h5>10.4-3 Complex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Complex</code>( <var class="Arg">cat</var>, <var class="Arg">baseDegree</var>, <var class="Arg">middle</var>, <var class="Arg">positive</var>, <var class="Arg">negative</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A newly created chain complex</p>
<p>The first argument, <var class="Arg">cat</var> is an <code class="func">IsCat</code> (<a href="chap10_mj.html#X7F0E546178977963"><span class="RefLink">10.3-1</span></a>) object describing the category to create a chain complex over.</p>
<p>The rest of the arguments describe the differentials of the complex. These are divided into three parts: one finite ("middle") and two infinite ("positive" and "negative"). The positive part contains all differentials in degrees higher than those in the middle part, and the negative part contains all differentials in degrees lower than those in the middle part. (The middle part may be placed anywhere, so the positive part can -- despite its name -- contain some differentials of negative degree. Conversely, the negative part can contain some differentials of positive degree.)</p>
<p>The argument <var class="Arg">middle</var> is a list containing the differentials for the middle part. The argument <var class="Arg">baseDegree</var> gives the degree of the first differential in this list. The second differential is placed in degree <span class="SimpleMath">\(\textit{baseDegree}+1\)</span>, and so on. Thus, the middle part consists of the degrees</p>
<p class="center">\[
<var class="Arg">baseDegree</var>,\quad
<var class="Arg">baseDegree</var> + 1,\quad
\ldots\quad
<var class="Arg">baseDegree</var> + \text{Length}(<var class="Arg">middle</var>).
\]</p>
<p>Each of the arguments <var class="Arg">positive</var> and <var class="Arg">negative</var> can be one of the following:</p>
<ul>
<li><p>The string <code class="code">"zero"</code>, meaning that the part contains only zero objects and zero morphisms.</p> | | |
