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<p><a id="X78D1062D78BE08C1" name="X78D1062D78BE08C1"></a></p>
<div class="ChapSects"><a href="chap8.html#X78D1062D78BE08C1">8 <span class="Heading"> Functors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7CFDEEC07F15CF82">8.1 <span class="Heading"> </span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X81BA486D7E532469">8.1-1 ExtendScalars</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X788F3B5E7810E309">8.1-2 HomToIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7E0216028756963B">8.1-3 HomToIntegersModP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X81FED0E9858E413A">8.1-4 HomToIntegralModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7F5BAB35811AB0D1">8.1-5 TensorWithIntegralModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7CF7B8A3842D498B">8.1-6 HomToGModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7D686D5D78FEF5C9">8.1-7 InduceScalars</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8456E06D7E76707B">8.1-8 LowerCentralSeriesLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X83BA99787CBE2B7D">8.1-9 TensorWithIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X829DD3868410FE2E">8.1-10 FilteredTensorWithIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7A0B33D085067A38">8.1-11 TensorWithTwistedIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8122D25786C83565">8.1-12 TensorWithIntegersModP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X873096CB823BFD1B">8.1-13 TensorWithTwistedIntegersModP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X809BA8A87F61EEDA">8.1-14 TensorWithRationals</a></span>
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<h3>8 <span class="Heading"> Functors</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>8.1 <span class="Heading"> </span></h4>
<p><a id="X81BA486D7E532469" name="X81BA486D7E532469"></a></p>
<h5>8.1-1 ExtendScalars</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtendScalars</code>( <var class="Arg">R</var>, <var class="Arg">G</var>, <var class="Arg">EltsG</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZH</span>-resolution <span class="SimpleMath">R</span>, a group <span class="SimpleMath">G</span> containing <span class="SimpleMath">H</span> as a subgroup, and a list <span class="SimpleMath">EltsG</span> of elements of <span class="SimpleMath">G</span>. It returns the free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">(R ⊗_ZH ZG)</span>. The returned resolution <span class="SimpleMath">S</span> has S!.elts:=EltsG. This is a resolution of the <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">(Z ⊗_ZH ZG)</span>. (Here <span class="SimpleMath">⊗_ZH</span> means tensor over <span class="SimpleMath">ZH</span>.)</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X788F3B5E7810E309" name="X788F3B5E7810E309"></a></p>
<h5>8.1-2 HomToIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">X=R</span>, or an equivariant chain map <span class="SimpleMath">X = (F:R ⟶ S)</span>. It returns the cochain complex or cochain map obtained by applying <span class="SimpleMath">HomZG( _ , Z)</span> where <span class="SimpleMath">Z</span> is the trivial module of integers (characteristic 0).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../tutorial/chap8.html">3</a></span> , <span class="URL"><a href="../tutorial/chap10.html">4</a></span> , <span class="URL"><a href="../tutorial/chap13.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>
<p><a id="X7E0216028756963B" name="X7E0216028756963B"></a></p>
<h5>8.1-3 HomToIntegersModP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegersModP</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and returns the cochain complex obtained by applying <span class="SimpleMath">HomZG( _ , Z_p)</span> where <span class="SimpleMath">Z_p</span> is the trivial module of integers mod <span class="SimpleMath">p</span>. (At present this functor does not handle equivariant chain maps.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">4</a></span> </p>
<p><a id="X81FED0E9858E413A" name="X81FED0E9858E413A"></a></p>
<h5>8.1-4 HomToIntegralModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegralModule</code>( <var class="Arg">R</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and a group homomorphism <span class="SimpleMath">f:G ⟶ GL_n(Z)</span> to the group of <span class="SimpleMath">n×n</span> invertible integer matrices. Here <span class="SimpleMath">Z</span> must have characteristic 0. It returns the cochain complex obtained by applying <span class="SimpleMath">HomZG( _ , A)</span> where <span class="SimpleMath">A</span> is the <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">Z^n</span> with <span class="SimpleMath">G</span> action via <span class="SimpleMath">f</span>. (At present this function does not handle equivariant chain maps.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../tutorial/chap14.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">4</a></span> </p>
<p><a id="X7F5BAB35811AB0D1" name="X7F5BAB35811AB0D1"></a></p>
<h5>8.1-5 TensorWithIntegralModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegralModule</code>( <var class="Arg">R</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and a group homomorphism <span class="SimpleMath">f:G ⟶ GL_n(Z)</span> to the group of <span class="SimpleMath">n×n</span> invertible integer matrices. Here <span class="SimpleMath">Z</span> must have characteristic 0. It returns the chain complex obtained by tensoring over <span class="SimpleMath">ZG</span> with the <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">A=Z^n</span> with <span class="SimpleMath">G</span> action via <span class="SimpleMath">f</span>. (At present this function does not handle equivariant chain maps.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> </p>
<p><a id="X7CF7B8A3842D498B" name="X7CF7B8A3842D498B"></a></p>
<h5>8.1-6 HomToGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToGModule</code>( <var class="Arg">R</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and an abelian G-outer group A. It returns the G-cocomplex obtained by applying <span class="SimpleMath">HomZG( _ , A)</span>. (At present this function does not handle equivariant chain maps.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">4</a></span> </p>
<p><a id="X7D686D5D78FEF5C9" name="X7D686D5D78FEF5C9"></a></p>
<h5>8.1-7 InduceScalars</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InduceScalars</code>( <var class="Arg">R</var>, <var class="Arg">hom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZQ</span>-resolution <span class="SimpleMath">R</span> and a surjective group homomorphism <span class="SimpleMath">hom:G→ Q</span>. It returns the unduced non-free <span class="SimpleMath">ZG</span>-resolution.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8456E06D7E76707B" name="X8456E06D7E76707B"></a></p>
<h5>8.1-8 LowerCentralSeriesLieAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerCentralSeriesLieAlgebra</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerCentralSeriesLieAlgebra</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pcp group <span class="SimpleMath">G</span>. If each quotient <span class="SimpleMath">G_c/G_c+1</span> of the lower central series is free abelian or p-elementary abelian (for fixed prime p) then a Lie algebra <span class="SimpleMath">L(G)</span> is returned. The abelian group underlying <span class="SimpleMath">L(G)</span> is the direct sum of the quotients <span class="SimpleMath">G_c/G_c+1</span> . The Lie bracket on <span class="SimpleMath">L(G)</span> is induced by the commutator in <span class="SimpleMath">G</span>. (Here <span class="SimpleMath">G_1=G</span>, <span class="SimpleMath">G_c+1=[G_c,G]</span> .)</p>
<p>The function can also be applied to a group homomorphism <span class="SimpleMath">f: G ⟶ G'</span> . In this case the induced homomorphism of Lie algebras <span class="SimpleMath">L(f):L(G) ⟶ L(G')</span> is returned.</p>
<p>If the quotients of the lower central series are not all free or p-elementary abelian then the function returns fail.</p>
<p>This function was written by Pablo Fernandez Ascariz</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLie.html">3</a></span> </p>
<p><a id="X83BA99787CBE2B7D" name="X83BA99787CBE2B7D"></a></p>
<h5>8.1-9 TensorWithIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">X=R</span>, or an equivariant chain map <span class="SimpleMath">X = (F:R ⟶ S)</span>. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers (characteristic 0).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap3.html">2</a></span> , <span class="URL"><a href="../tutorial/chap6.html">3</a></span> , <span class="URL"><a href="../tutorial/chap7.html">4</a></span> , <span class="URL"><a href="../tutorial/chap10.html">5</a></span> , <span class="URL"><a href="../tutorial/chap11.html">6</a></span> , <span class="URL"><a href="../tutorial/chap13.html">7</a></span> , <span class="URL"><a href="../tutorial/chap14.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAspherical.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">12</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">13</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCocycles.html">14</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">15</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">16</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">17</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">18</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">19</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoxeter.html">20</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">21</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDavisComplex.html">22</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDefinitions.html">23</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">24</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">25</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">26</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutFunctorial.html">27</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGraphsOfGroups.html">28</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">29</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">30</a></span> </p>
<p><a id="X829DD3868410FE2E" name="X829DD3868410FE2E"></a></p>
<h5>8.1-10 FilteredTensorWithIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FilteredTensorWithIntegers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> for which "filteredDimension" lies in NamesOfComponents(R). (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) It returns the filtered chain complex obtained by tensoring with the trivial module of integers (characteristic 0).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap10.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">2</a></span> </p>
<p><a id="X7A0B33D085067A38" name="X7A0B33D085067A38"></a></p>
<h5>8.1-11 TensorWithTwistedIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithTwistedIntegers</code>( <var class="Arg">X</var>, <var class="Arg">rho</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">X=R</span>, or an equivariant chain map <span class="SimpleMath">X = (F:R ⟶ S)</span>. It also inputs a function <span class="SimpleMath">rho: G→ Z</span> where the action of <span class="SimpleMath">g ∈ G</span> on <span class="SimpleMath">Z</span> is such that <span class="SimpleMath">g.1 = rho(g)</span>. It returns the chain complex or chain map obtained by tensoring with the (twisted) module of integers (characteristic 0).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap3.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">4</a></span> </p>
<p><a id="X8122D25786C83565" name="X8122D25786C83565"></a></p>
<h5>8.1-12 TensorWithIntegersModP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">X</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">X=R</span>, or a characteristics 0 chain complex, or an equivariant chain map <span class="SimpleMath">X = (F:R ⟶ S)</span>, or a chain map between characteristic 0 chain complexes, together with a prime <span class="SimpleMath">p</span>. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo <span class="SimpleMath">p</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDefinitions.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>
<p><a id="X873096CB823BFD1B" name="X873096CB823BFD1B"></a></p>
<h5>8.1-13 TensorWithTwistedIntegersModP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithTwistedIntegersModP</code>( <var class="Arg">X</var>, <var class="Arg">p</var>, <var class="Arg">rho</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">X=R</span>, or an equivariant chain map <span class="SimpleMath">X = (F:R ⟶ S)</span>, and a prime <span class="SimpleMath">p</span>. It also inputs a function <span class="SimpleMath">rho: G→ Z</span> where the action of <span class="SimpleMath">g ∈ G</span> on <span class="SimpleMath">Z</span> is such that <span class="SimpleMath">g.1 = rho(g)</span>. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo <span class="SimpleMath">p</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">1</a></span> </p>
<p><a id="X809BA8A87F61EEDA" name="X809BA8A87F61EEDA"></a></p>
<h5>8.1-14 TensorWithRationals</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithRationals</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and returns the chain complex obtained by tensoring with the trivial module of rational numbers.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap10.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">3</a></span> </p>
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