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<p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#ForGeneralSimplicialGroups'>For general simplicial groups</a></li> <li><a href='#ForSimplicialAbelianGroups'>For simplicial abelian groups</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#normalization'>Normalization</a></li> <li><a href='#equivalence_of_categories'>Equivalence of categories</a></li> <li><a href='#homology_and_homotopy_groups'>Homology and homotopy groups</a></li> <li><a href='#HypercrossedComplex'>Hypercrossed complex structure</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Moore complex</em> of a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> – also known in its normalized version as the <strong>complex of normalized chains</strong> – is a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is built from the face maps of the simplicial group.</p> <p>The operation of forming the Moore complex of chains of a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> is one part of the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> that relates <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> (abelian) <a class="existingWikiWord" href="/nlab/show/groups">groups</a> with <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>.</p> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, being in particular a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, may be thought of, in the sense of the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>, as a combinatorial space equipped with a group structure. The Moore complex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> <ul> <li> <p>whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells are the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks with basepoint on their boundary” in this space, with the basepoint sitting on the identity element of the space;</p> </li> <li> <p>the boundary map on which acts literally like a boundary map should: it sends an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disk to its boundary, read as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-disk whose entire boundary is concentrated at the identity point.</p> </li> </ul> <p>This is entirely analogous to how a <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a> is obtained from a <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a>. In fact it is a special case of that, as discussed at <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> in the section on the nonabelian version.</p> <h2 id="definition">Definition</h2> <h3 id="ForGeneralSimplicialGroups">For general simplicial groups</h3> <div class="num_defn" id="NormalizedChainComplexOnGeneralGroup"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, its <em><a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a></em> or <em>Moore complex</em> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mo>∂</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((N G)_\bullet,\partial )</annotation></semantics></math> of (possibly <a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian</a>) groups which</p> <ul> <li> <p>is in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the joint <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mi>ker</mi><mspace width="thinmathspace"></mspace><msubsup><mi>d</mi> <mi>i</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex"> (N G)_n=\bigcap_{i=1}^{n}ker\,d_i^n </annotation></semantics></math></div> <p>of all face maps except the 0-face;</p> </li> <li> <p>with differential given by the remaining 0-face map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub><mo>≔</mo><msubsup><mi>d</mi> <mn>0</mn> <mi>n</mi></msubsup><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_n \coloneqq d_0^n|_{(N G)_n} : (N G)_n \rightarrow (N G)_{n-1} \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> one may equivalently take the joint <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of all but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-face map and take that remaining face map, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>n</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">d_n^n</annotation></semantics></math>, to be the differential.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>We may think of the elements of the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">N G</annotation></semantics></math>, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>, in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> as being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/disks">disks</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> all of whose <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> is captured by a single face:</p> <ul> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>N</mi><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g \in N G_1</annotation></semantics></math> in degree 1 is a 1-disk</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mover><mo>→</mo><mi>g</mi></mover><mo>∂</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 1 \stackrel{g}{\to} \partial g \,, </annotation></semantics></math></div></li> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>N</mi><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h \in N G_2</annotation></semantics></math> is a 2-disk</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mn>1</mn></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><mo>∂</mo><mi>h</mi></mrow></msup></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,, </annotation></semantics></math></div></li> <li> <p>a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mn>1</mn></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><mo>∂</mo><mi>h</mi><mo>=</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,, </annotation></semantics></math></div></li> </ul> <p>etc.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> the normalized chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(N G)_\bullet</annotation></semantics></math> in def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> is a <a class="existingWikiWord" href="/nlab/show/normal+complex+of+groups">normal complex of groups</a>,</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This means that is easy to take the homology of the complex, even though the groups involved may be non-abelian.</p> </div> <h3 id="ForSimplicialAbelianGroups">For simplicial abelian groups</h3> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a>. Then its <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">(N A)_\bullet \in Ch_\bullet^+</annotation></semantics></math> of def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> is an ordinary <a class="existingWikiWord" href="/nlab/show/connective+chain+complex">connective</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> in the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>.</p> <p>In this abelian cases are two other chain complexes naturally associated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <div class="num_defn" id="AlternatingFaceMapComplex"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> its <strong>alternating face map complex</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(C A)_\bullet</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> which</p> <ul> <li> <p>in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is given by the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math> itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> (C A)_n \coloneqq A_n </annotation></semantics></math></div></li> <li> <p>with <a class="existingWikiWord" href="/nlab/show/differential">differential</a> given by the alternating sum of face maps (using the abelian group structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>)</p> <div class="maruku-equation" id="eq:DifferentialOnChains"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>d</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,. </annotation></semantics></math></div> <p>(see lemma <a class="maruku-ref" href="#AlternatingSumOfFacesInNilpotent"></a>).</p> </li> </ul> </div> <div class="num_lemma" id="AlternatingSumOfFacesInNilpotent"> <h6 id="lemma">Lemma</h6> <p>The differential in def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a> is well-defined in that it indeed squares to 0.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Using the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i \circ d_j = d_{j-1} \circ d_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \lt j</annotation></semantics></math> one finds:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mo>∂</mo> <mi>n</mi></msub><msub><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></msup><msub><mi>d</mi> <mi>k</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="DegenerateElement"> <h6 id="definition_4">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (or in fact any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>), then an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a \in A_{n+1}</annotation></semantics></math> is called <em>degenerate</em> if it is in the <a class="existingWikiWord" href="/nlab/show/image">image</a> of one of the simplicial degeneracy maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">s_i \colon A_n \to A_{n+1}</annotation></semantics></math>. All elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A_0</annotation></semantics></math> are regarded a non-degenerate. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">⟨</mo><msub><mo>∪</mo> <mi>i</mi></msub><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{n+1}</annotation></semantics></math> which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements). Elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">D(A)_{n}</annotation></semantics></math> are often called <em><a class="existingWikiWord" href="/nlab/show/thin+element">thin</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices.</p> </div> <div class="num_defn" id="ComplexModuloDegeneracies"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> its <strong>alternating face maps chain complex modulo degeneracies</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C A)/(D A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> <ul> <li> <p>which in degree 0 equals is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>≔</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">((C A)/D(A))_0 \coloneqq A_0</annotation></semantics></math>;</p> </li> <li> <p>which in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> group obtained by dividing out the group the degenerate elements, def. <a class="maruku-ref" href="#DegenerateElement"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≔</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((C A)/D(A))_{n+1} \coloneqq A_{n+1} / D(A_{n+1}) </annotation></semantics></math></div></li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma <a class="maruku-ref" href="#LeftCosetsDisjoint"></a>).</p> </li> </ul> </div> <div class="num_lemma" id="LeftCosetsDisjoint"> <h6 id="lemma_2">Lemma</h6> <p>Def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a> is indeed well defined in that the alternating face map differential respects the degenerate subcomplex.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Using the mixed <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> we find that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">s_j(a) \in A_n</annotation></semantics></math> a degenerate element, its boundary is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>s</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><mi>a</mi><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mi>j</mi></msub><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mi>j</mi></msub><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned} </annotation></semantics></math></div> <p>which is again a combination of elements in the image of the degeneracy maps.</p> </div> <h2 id="properties">Properties</h2> <h3 id="normalization">Normalization</h3> <div class="num_prop" id="NormalizedIntoModuloDegeneraciesIsIsomorpism"> <h6 id="proposition_2">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the evident composite of natural morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>CA</mi><mover><mo>→</mo><mi>p</mi></mover><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N A \stackrel{i}{\hookrightarrow} CA \stackrel{p}{\to} (C A)/(D A) </annotation></semantics></math></div> <p>from the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>, into the alternating face map complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, (inclusion followed by projection to the quotient) is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of chain complexes.</p> </div> <p>(<a href="#GoerssJardine">Goerss-Jardine, theorem III 2.1</a>, see also <a href="#SchwedeShipley03">Schwede-Shipley 03, Section 2.1</a>).</p> <div class="num_cor" id="SplittingOffDegenerateCells"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a>, there is a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splitting</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>N</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A) </annotation></semantics></math></div> <p>of the alternating face map complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a> as a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>, where the first direct summand is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> of def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> and the second is the degenerate cells from def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By prop. <a class="maruku-ref" href="#NormalizedIntoModuloDegeneraciesIsIsomorpism"></a> there is an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> to the diagonal composite in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,. </annotation></semantics></math></div> <p>This hence exhibits a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splitting</a> of the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> given by the quotient by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">D A</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>D</mi><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↙</mo> <mrow><msub><mpadded width="0"><mo>≃</mo></mpadded> <mi>iso</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,. </annotation></semantics></math></div></div> <p> <div class='num_theorem' id='EMTheorem'> <h6>Theorem</h6> <p><strong>(Eilenberg-MacLane)</strong> <br /> Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>N</mi><mi>A</mi><mo>↪</mo><mi>C</mi><mi>A</mi></mrow><annotation encoding="application/x-tex"> i \colon N A \hookrightarrow C A </annotation></semantics></math></div> <p>of the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> into the full alternating face map complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a>, is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> and in fact a natural chain <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, i.e. the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_\bullet(X)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/null+homotopy">null-homotopic</a> (a <a class="existingWikiWord" href="/nlab/show/contractible+chain+complex">contractible chain complex</a>).</p> </div> </p> <p>(<a href="#GoerssJardine">Goerss-Jardine, theorem III 2.4</a>)</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Following the proof of (<a href="#GoerssJardine">Goerss-Jardine, theorem III 2.1</a>) we look for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">j \lt n</annotation></semantics></math> at the groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mo>≔</mo><msubsup><mo>∩</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>j</mi></msubsup><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> N_n(A)_j \coloneqq \cap_{i=0}^j ker (d_i) \subset A_n </annotation></semantics></math></div> <p>and similarly at</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mo>=</mo><mo stretchy="false">{</mo><msub><mi>s</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>≤</mo><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>A</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> D_n(A)_j = \{s_{i}\}_{i \leq j}(A_{n-1}) \subset A_n \,, </annotation></semantics></math></div> <p>the subgroup generated by the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> degeneracies.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j= n-1</annotation></semantics></math> these coincide with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_n(A)</annotation></semantics></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_n(A)</annotation></semantics></math>, respectively. We show by induction on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> that the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mo>↪</mo><msub><mi>A</mi> <mi>n</mi></msub><mover><mo>→</mo><mrow></mrow></mover><msub><mi>A</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> N_n(A)_j \hookrightarrow A_n \stackrel{}{\to} A_n/D_n(A)_j </annotation></semantics></math></div> <p>is an isomorphism of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">j \lt n</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j = n-1</annotation></semantics></math> this is then the desired result.</p> <p>(…)</p> </div> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then the projection <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (C A) \longrightarrow (C A)/(D A) </annotation></semantics></math></div> <p>from its alternating face maps complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a>, to the alternating face map complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Consider the pre-composition of the map with the inclusion of the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } </annotation></semantics></math></div> <p>By theorem <a class="maruku-ref" href="#EMTheorem"></a> the vertical map is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> and by prop. <a class="maruku-ref" href="#NormalizedIntoModuloDegeneraciesIsIsomorpism"></a> the composite diagonal map is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence in particular also a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>. Since quasi-isomorphisms satisfy the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property, it follows that also the map in question is a quasi-isomorphism.</p> </div> <h3 id="equivalence_of_categories">Equivalence of categories</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> restricts on <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+groups">simplicial abelian groups</a> to an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>sAb</mi><mover><mo>⟶</mo><mo>≃</mo></mover><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N \colon sAb \stackrel{\simeq}{\longrightarrow} Ch_\bullet^+(A) </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/sAb">sAb</a> and the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> in non-negative degree.</p> </div> <p>This is the statement of the <em><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></em>. See there for details.</p> <h3 id="homology_and_homotopy_groups">Homology and homotopy groups</h3> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> underlying any <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> (as described there) is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> \pi_n(G) \;\;\; n \in \mathbb{N} </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-th <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Notice that due to the group structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in this case also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(G)</annotation></semantics></math> is indeed canonically a group, not just a set.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a simplicial abelian group there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_n(A,0) \simeq H_n(N A) \simeq H_n(A) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a>s and the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups of the unnormalized and of the normalized chain complexes.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The first isomorphism follows with the <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a>. The second directly from the <a href="#EMTheorem">Eilenberg-MacLane theorem</a> above.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAb</mi></mrow><annotation encoding="application/x-tex">sAb</annotation></semantics></math> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">Ch_\bullet^+</annotation></semantics></math> are naturally <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> given by those morphisms that induce <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s on all <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a> and on all <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups, respectively. So the above statement says that the Moore complex functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> respects these weak equivalences.</p> <p>In fact, it induces an equivalence of categories also on the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>. And even better, it induces a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> with respect to the standard <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures that refine the structures of categories of weak equivalences. All this is discussed at <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>.</p> </div> <h3 id="HypercrossedComplex">Hypercrossed complex structure</h3> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The Moore complex of a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> is naturally a <a class="existingWikiWord" href="/nlab/show/hypercrossed+complex">hypercrossed complex</a>.</p> </div> <p>This has been established in (<a href="#Carrasco-Cegarra">Carrasco-Cegarra</a>). In fact, the analysis of the Moore complex and what is necessary to rebuild the simplicial group from its Moore complex is the origin of the abstract motion of hypercrossed complex, so our stated proposition is almost a tautology!</p> <p>Typically one has pairings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mi>p</mi></msub><mo>×</mo><mi>N</mi><msub><mi>G</mi> <mi>q</mi></msub><mo>→</mo><mi>N</mi><msub><mi>G</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N G_p \times N G_q \to N G_{p+q}</annotation></semantics></math>. These use the <a class="existingWikiWord" href="/nlab/show/Conduch%C3%A9+decomposition+theorem">Conduché decomposition theorem</a>, see the discussion at <a class="existingWikiWord" href="/nlab/show/hypercrossed+complex">hypercrossed complex</a>.</p> <p>These Moore complexes are easily understood in low dimensions:</p> <ul> <li> <p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a simplicial group with Moore complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">N G</annotation></semantics></math>, which satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mi>k</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N G_k = 1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k\gt 1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_1,G_0,d_1,d_0)</annotation></semantics></math> has the structure of a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>. The <a class="existingWikiWord" href="/nlab/show/interchange+law">interchange law</a> is satisfied since the corresponding equation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> is always the image of an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N G_2</annotation></semantics></math>, and here that must be trivial. If one thinks of the 2-group as being specified by a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,P,\delta, a)</annotation></semantics></math>, then in terms of the original simplicial group, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>=</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">N G_0 = G_0 = P</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mn>1</mn></msub><mo>≅</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">N G_1 \cong C</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>=</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex"> \partial = \delta</annotation></semantics></math> and the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> translates to an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">N G_0</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N G_1</annotation></semantics></math> using conjugation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_0(p)</annotation></semantics></math>, i.e., for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p\in G_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>N</mi><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">c\in N G_1</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi>c</mi><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>p</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">a(p)(c) = s_0(p)c s_0(p)^{-1}.</annotation></semantics></math></div></li> <li> <p>Suppose next that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>G</mi> <mi>k</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N G_k = 1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k \gt 2</annotation></semantics></math>, then the Moore complex is a <a class="existingWikiWord" href="/nlab/show/2-crossed+module">2-crossed module</a>.</p> </li> </ul> <h2 id="Examples">Examples</h2> <p> <div class='num_remark' id='ChainsOnThe1Simplex'> <h6>Example</h6> <p><strong>(chain on the 1-simplex)</strong> <br /> Consider the 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[\Delta[1]]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> which in each degree is the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> on the simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>.</p> <p>This simplicial abelian group starts out as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mover><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mo>⟶</mo></mover><msup><mi>ℤ</mi> <mn>4</mn></msup><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><msup><mi>ℤ</mi> <mn>3</mn></msup><mover><munder><mo>⟶</mo><mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></mover></mover><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right) </annotation></semantics></math></div> <p>(where we are indicating only the face maps for notational simplicity).</p> <p>Here the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>=</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of two copies of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, is the group of 0-chains generated from the two endpoints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, i.e. the abelian group of formal linear combinations of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>≃</mo><mrow><mo>{</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,. </annotation></semantics></math></div> <p>The second <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>3</mn></msup><mo>≃</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}</annotation></semantics></math> is the abelian group generated from the three (!) 1-simplicies in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, namely the non-degenerate edge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0\to 1)</annotation></semantics></math> and the two degenerate cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0 \to 0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1 \to 1)</annotation></semantics></math>, hence the abelian group of formal linear combinations of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>3</mn></msup><mo>≃</mo><mrow><mo>{</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>ℤ</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,. </annotation></semantics></math></div> <p>The two face maps act on the basis 1-cells as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo>→</mo><mi>j</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \partial_1 \colon (i \to j) \mapsto (i) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo>→</mo><mi>j</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_0 \colon (i \to j) \mapsto (j) \,. </annotation></semantics></math></div> <p>Now of course most of the (infinitely!) many simplices inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math> are degenerate. In fact the only non-degenerate simplices are the two 0-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1)</annotation></semantics></math> and the 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0 \to 1)</annotation></semantics></math>. Hence the alternating face maps complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[\Delta[1]]</annotation></semantics></math> is simply this:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mover><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,. </annotation></semantics></math></div> <p>Notice that alternatively we could consider the topological 1-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^1 = [0,1]</annotation></semantics></math> and its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(\Delta^1)</annotation></semantics></math> in place of the smaller <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, then the free simplicial abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}(Sing(\Delta^1))</annotation></semantics></math> of that. The corresponding alternating face map chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\mathbb{Z}(Sing(\Delta^1)))</annotation></semantics></math> is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.</p> </div> </p> <p> <div class='num_remark' id='NormalizedChainsOnEZTwo'> <h6>Example</h6> <p><strong>(normalized chain complex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E\mathbb{Z}_2</annotation></semantics></math>)</strong> <br /> We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>≡</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \,\equiv\, \mathbb{Z}/2\mathbb{Z}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a>, whose <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/set">set</a> we denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0, 1\}</annotation></semantics></math>.</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> (see <a href="simplicial+classifying+space#eq:SimplicialClassifyingSpaceAsNerve">here</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≡</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><munderover><mo>⇉</mo><mrow><msub><mi>pr</mi> <mn>2</mn></msub></mrow><mrow><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munderover><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> W \mathbb{Z}_2 \;=\; N\big( \mathbf{E} \mathbb{Z}_2 \big) \;\equiv\; N\big( \mathbb{Z}_2 \times \mathbb{Z}_2 \underoverset {pr_2} {(\text{-})\cdot(-)} {\rightrightarrows} \mathbb{Z}_2 \big) \,. </annotation></semantics></math></div> <p>Forming degree-wise <a class="existingWikiWord" href="/nlab/show/linear+spans">linear spans</a> gives a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>Ab</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \;\; \in \;\; Ab^{\Delta^{op}} \,. </annotation></semantics></math></div> <p>with <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> denoted</p> <div class="maruku-equation" id="eq:NormalizedChainsOnWZTwo"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_\bullet \big( \mathbb{Z} [ W \mathbb{Z}_2 ] \big) \;\; \in \;\; Ch_{\geq 0} \,. </annotation></semantics></math></div> <p>Now, since the only non-<a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a> morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{E}\mathbb{Z}_2</annotation></semantics></math> are the two morphsism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \to 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mn>0</mn><mo>,</mo></mrow><annotation encoding="application/x-tex">1 \to 0,</annotation></semantics></math> the non-degenerate <a class="existingWikiWord" href="/nlab/show/n-simplex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-simplices</a> of <a class="maruku-eqref" href="#eq:NormalizedChainsOnWZTwo">(3)</a> are alternating sequences on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1\}</annotation></semantics></math>. By the alternating property, these are fully determined by their first element (in particular), whence there are exactly two non-degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. On these, the differential is given as follows, using <a class="maruku-eqref" href="#eq:DifferentialOnChains">(1)</a> and Prop. <a class="maruku-ref" href="#NormalizedIntoModuloDegeneraciesIsIsomorpism"></a>, according to which only the 0th and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/face+maps">face maps</a> contribute in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the latter with sign <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^n</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:NormalizedChainsOnWZTwo"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="3em" minsize="3em">[</mo><mi>⋯</mi><mo>→</mo><mover><mo>→</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mover><msub><mi>ℤ</mi> <mn>0101</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>1010</mn></msub><mover><mo>→</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mover><msub><mi>ℤ</mi> <mn>010</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>101</mn></msub><mover><mo>→</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mover><msub><mi>ℤ</mi> <mn>01</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>10</mn></msub><mover><mo>→</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mover><msub><mi>ℤ</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>1</mn></msub><mo maxsize="3em" minsize="3em">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_\bullet \mathbb{Z} \big[ W \mathbb{Z}_2 \big] \;\simeq\; \Bigg[ \cdots \to \xrightarrow{ \left[ \array{ +1 & +1 \\ +1 & +1 } \right] } \mathbb{Z}_{0101} \oplus \mathbb{Z}_{1010} \xrightarrow{ \left[ \array{ -1 & +1 \\ +1 & -1 } \right] } \mathbb{Z}_{010} \oplus \mathbb{Z}_{101} \xrightarrow{ \left[ \array{ +1 & +1 \\ +1 & +1 } \right] } \mathbb{Z}_{01} \oplus \mathbb{Z}_{10} \xrightarrow{ \left[ \array{ -1 & +1 \\ +1 & -1 } \right] } \mathbb{Z}_0 \oplus \mathbb{Z}_1 \Bigg] \,. </annotation></semantics></math></div> <p>One may think of the two generators in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> as corresponding to two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/hemispheres">hemispheres</a> with common <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> an <a class="existingWikiWord" href="/nlab/show/equator">equator</a> similarly formed by two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/hemispheres">hemispheres</a>, and so on. Thereby this chain complex is seen to be isomorphically that for the <a class="existingWikiWord" href="/nlab/show/cellular+homology">cellular homology</a> of the standard <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> on the <a href="sphere#InfiniteDimensionalSphere">infinite-dimensional sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math> (cf. also at <em><a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a></em> the section <em><a href="real+projective+space#RelationToClassifyingSpace">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-classifying space</a></em>).</p> <p>Indeed, one sees immediately that <a class="maruku-eqref" href="#eq:NormalizedChainsOnWZTwo">(3)</a> has vanishing <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> in all <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degrees and homology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\simeq \mathbb{Z}</annotation></semantics></math> in degree 0, whence the canonical map to the normalized chain complex on the point is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><munder><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>qiso</mi><mspace width="thickmathspace"></mspace></mrow></munder><msub><mi>N</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">[</mo><mo>*</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_\bullet\,\mathbb{Z}[W \mathbb{Z}_2] \xrightarrow[\;qiso\;]{} N_\bullet\,\mathbb{Z}[\ast] \,. </annotation></semantics></math></div> <p>This reflects the <a class="existingWikiWord" href="/nlab/show/contractible+homotopy+type">contractible homotopy type</a> of both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E \mathbb{Z}_2</annotation></semantics></math> (the total space of the <a href="simplicial+principal+bundle#UniversalSimplicialBundle">universal principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-bundle</a>) and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_remark' id='NormalizedChainDGAlgebraOnEZTwo'> <h6>Example</h6> <p><strong>(normalized chain dg-algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E\mathbb{Z}_2</annotation></semantics></math>)</strong></p> <p>On the normalized chain complex of Ex. <a class="maruku-ref" href="#NormalizedChainsOnEZTwo"></a>, the <a class="existingWikiWord" href="/nlab/show/group">group</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">W \mathbb{Z}_2</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/simplicial+ring">simplicial ring</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a>, the “simplicial <a class="existingWikiWord" href="/nlab/show/group+ring">group ring</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>Ring</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \;\; \in \;\; Ring^{\Delta^{op}} \,. </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msup><mi>Ab</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><msub><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">N_\bullet \,\colon\, Ab^{\Delta^{op}} \to Ch_{\geq 0}</annotation></semantics></math> is (see <a href="monoidal+Dold-Kan+correspondence#ChainsIsLaxMonoidal">here</a>) a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> via the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a>, this induces on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">N_\bullet\big( \mathbb{Z}[W \mathbb{Z}_2] \big)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>.</p> <p>To write this out, denote the two generators in each degree by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>ℤ</mi> <mrow><mn>0</mn><mi>⋯</mi></mrow></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mrow><mn>1</mn><mi>⋯</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>N</mi> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (0), (1) \;\in\; \mathbb{Z}_{0\cdots} \oplus \mathbb{Z}_{1\cdots} \;=\; N_n\big( \mathbb{Z}[W \mathbb{Z}_2] \big) \,. </annotation></semantics></math></div> <p>The non-degenerate cells of the tensor product simplicial group are similarly labeled (via <a href="product+of+simplices#NonDegenerateSimplicesInProductOfSimplices">this Prop.</a>) by</p> <ol> <li> <p>a pair of first elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1\}</annotation></semantics></math></p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p, q)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/shuffle">shuffle</a>,</p> </li> </ol> <p>hence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>sh</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">}</mo> <mn>2</mn></msup><mo>×</mo><msub><mo>⊔</mo> <mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>n</mi></mrow></msub><mi>Sh</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>N</mi> <mi>n</mi></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>N</mi> <mi>n</mi></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo>⊗</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( (g, g'), sh \big) \;\in\; \mathbb{Z} \big[ \{0,1\}^2 \times \sqcup_{p+q = n} Sh(p,q) \big] \;\simeq\; N_{n} \Big( \mathbb{Z}\big[ W \mathbb{Z}_2 \times W \mathbb{Z}_2 \big] \Big) \;\simeq\; N_{n} \Big( \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \otimes \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \Big) \,. </annotation></semantics></math></div> <p>One immediately finds that on these generators the induced product map is just the group operation, independent of the shuffle:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mover></mtd> <mtd><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>sh</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \otimes \mathbb{Z}[W \mathbb{Z}_2] \big) & \xrightarrow{ N_\bullet \big( \mathbb{Z}[(\text{-})\cdot(\text{-})] \big) } & N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \big) \\ \big( g, g', sh \big) &\mapsto& (g \cdot g') \mathrlap{\,.} } </annotation></semantics></math></div> <p>Composed with the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a> this gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>N</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⊗</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>N</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo></mrow></mphantom></mover></mtd> <mtd><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mover></mtd> <mtd><msub><mi>N</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>W</mi><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>g</mi><msub><mo stretchy="false">)</mo> <mi>p</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><msub><mo stretchy="false">)</mo> <mi>q</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mstyle displaystyle="true"><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mpadded width="0" lspace="-50%width"><mrow><mi>sh</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mpadded></munder><mspace width="thickmathspace"></mspace><mi>sgn</mi><mo stretchy="false">(</mo><mi>sh</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>sh</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mstyle></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle displaystyle="true"><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mpadded width="0" lspace="-50%width"><mrow><mi>sh</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mpadded></munder><mspace width="thickmathspace"></mspace><mi>sgn</mi><mo stretchy="false">(</mo><mi>sh</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>g</mi><mo>′</mo><msub><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \big( N_\bullet \, \mathbb{Z}[W \mathbb{Z}_2] \big) \otimes \big( N_\bullet \, \mathbb{Z}[W \mathbb{Z}_2] \big) &\xrightarrow{\phantom{----}}& N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \otimes \mathbb{Z}[W \mathbb{Z}_2] \big) & \xrightarrow{ N_\bullet \big( \mathbb{Z}[(\text{-})\cdot(\text{-})] \big) } & N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \big) \\ (g)_p \otimes (g')_q &\mapsto& \displaystyle{ \underset{ \mathclap{ sh \in Sh(p,q) } }{\sum} } \; sgn(sh) \, \big( g,g', sh \big) &\mapsto& \Big( \displaystyle{ \underset{ \mathclap{ sh \in Sh(p,q) } }{\sum} } \; sgn(sh) \Big) \, (g \cdot g')_{p + q} \mathrlap{\,.} } </annotation></semantics></math></div> <p></p> </div> </p> <h2 id="references">References</h2> <p>Original sources:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Homotopie des complexes monoïdaux, I.</em> Séminaire Henri Cartan <strong>7</strong> 2 (1954-1955), Exposé <strong>18</strong> (<a href="http://www.numdam.org/item?id=SHC_1954-1955__7_2_A8_0">numdam:SHC_1954-1955__7_2_A8_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Semi-simplicial complexes and Postnikov systems</em>, Symposium international de topologia algebraica, Mexico (1958) p. 243</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Semi-simplicial Complexes, seminar notes</em>, Princeton University (1956)</p> </li> </ul> <p>There is also a never published</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+C.+Moore">John C. Moore</a>, <em>Algebraic homotopy theory</em>. Princeton 1956. Mimeographed notes [<a href="https://dmitripavlov.org/scans/moore-algebraic-homotopy-theory.pdf">pdf</a>]</li> </ul> <p>A proof by Cartan is in</p> <ul> <li>Cartan, <em>Quelques questions de topologies</em> seminar, 1956-57</li> </ul> <p>A standard textbook reference for the abelian version is</p> <ul> <li id="GoerssJardine"><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, chapter III.2 of <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em></li> </ul> <p>Notice that these authors write “normalized chain complex” for the complex that elsewhere in the literature would be called just “Moore complex”, whereas what Goerss–Jardine call “Moore complex” is sometime maybe just called “alternating sum complex”.</p> <p>See also</p> <ul> <li id="SchwedeShipley03"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, Section 2.1 of: <em>Equivalences of monoidal model categories</em> , Algebr. Geom. Topol. 3 (2003), 287–334 (<a href="http://arxiv.org/abs/math.AT/0209342">arXiv:math.AT/0209342</a>, <a href="https://projecteuclid.org/euclid.agt/1513882376">euclid:euclid.agt/1513882376</a>)</li> </ul> <p>A discussion with an emphasis of the generalization to non-abelian simplicial groups is in section 1.3.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em><a class="existingWikiWord" href="/nlab/show/Crossed+Menagerie">Crossed Menagerie</a></em></li> </ul> <p>The discusson of the <a class="existingWikiWord" href="/nlab/show/hypercrossed+complex">hypercrossed complex</a> structure on the Moore complex of a general simplicial group is in</p> <ul> <li id="Carrasco-Cegarra"><a class="existingWikiWord" href="/nlab/show/Pilar+Carrasco">P. Carrasco</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+Cegarra">A. M. Cegarra</a>, <em>Group-theoretic Algebraic Models for Homotopy Types</em> , J. Pure Appl. Alg., 75, (1991), 195–235 <a href="https://doi.org/10.1016/0022-4049(91)90133-M">https://doi.org/10.1016/0022-4049(91)90133-M</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 7, 2023 at 15:48:46. 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