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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <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='#with_commuting_differentials'>With commuting differentials</a></li> <li><a href='#AntiCommutingDifferentials'>With anti-commuting differentials</a></li> <li><a href='#EquivalenceOfTheTwoDefinitions'>Equivalence of the two definitions</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#total_complex_of_a_double_complex'>Total complex of a double complex</a></li> <li><a href='#fundamental_lemmas'>Fundamental lemmas</a></li> </ul> <li><a href='#relation_to_homotopy_colimits'>Relation to homotopy colimits</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>double complex</strong> or <strong>bicomplex</strong> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mo>≤</mo></msub><mo>×</mo><msub><mi>ℤ</mi> <mo>≤</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{\leq} \times \mathbb{Z}_{\leq}</annotation></semantics></math> (in some <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</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><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msup><mo>∂</mo> <mi>h</mi></msup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msup><mo>∂</mo> <mi>h</mi></msup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>v</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \vdots && \vdots \\ & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}} \\ \cdots &\to & X_{n,m} &\stackrel{\partial^h}{\to}& X_{n-1,m} & \to & \cdots \\ & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}} \\ \cdots &\to & X_{n,m-1} &\stackrel{\partial^h}{\to}& X_{n-1,m-1} & \to & \cdots \\ & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}} \\ && \vdots && \vdots } </annotation></semantics></math></div> <p>such that each row and each column is a <a class="existingWikiWord" href="/nlab/show/complex">complex</a> (the <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> square to 0: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>v</mi></msup><mo>∘</mo><msup><mo>∂</mo> <mi>v</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial^v \circ \partial^v = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>h</mi></msup><mo>∘</mo><msup><mo>∂</mo> <mi>h</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial^h \circ \partial^h = 0</annotation></semantics></math>) and such that all the squares <a class="existingWikiWord" href="/nlab/show/commuting+diagram"> commute</a>.</p> <p>This means that a <strong>double complex</strong> is a <a class="existingWikiWord" href="/nlab/show/complex">complex</a> in a category of <a class="existingWikiWord" href="/nlab/show/complexes">complexes</a>.</p> <p>Accordingly, a <strong>double chain complex</strong> is a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> in a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msub><mo>=</mo><mrow><mo>[</mo><mi>⋯</mi><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mo>•</mo></mrow></msub><mover><mo>→</mo><mrow><msup><mo>∂</mo> <mi>h</mi></msup></mrow></mover><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>•</mo></mrow></msub><mo>→</mo><mi>⋯</mi><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X_{\bullet, \bullet} = \left[ \cdots \to X_{n,\bullet} \stackrel{\partial^h}{\to} X_{n-1,\bullet} \to \cdots \right] \,. </annotation></semantics></math></div> <p>In the presence of <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>, there is a <em><a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Tot</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Tot X)_\bullet</annotation></semantics></math> associated to a double complex, which 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 the direct sum of all terms of <em>total degree</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></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>Tot</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msub><mo>⊕</mo> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>=</mo><mi>n</mi></mrow></msub><msub><mi>X</mi> <mrow><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Tot X)_n \coloneqq \oplus_{k+l = n} X_{k,l}. </annotation></semantics></math></div> <p>Often it is such total complexes that are of interest.</p> <p>The <a class="existingWikiWord" href="/nlab/show/differential">differential</a> of the total complex is the sum of the horizontal and the vertical differential <em>made anti-commutative by adjusting signs</em>. There is a second convention in which one often sees the double complex defined as above from a complex of complexes, but then with the differentials in every second row (or every second column) multiplied by <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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math>. This is just a different way of sign-bookkeeping, a detailed discussion of this is below. Which convention to use is sometimes influenced by the context, the traditions of the sources in that application of double complexes, and largely a question of taste, that is which one the writer is used to.</p> <p>Double chain complexes often arise from the application of bifunctors – <a class="existingWikiWord" href="/nlab/show/additive+functors">additive functors</a> of two variables – of <a class="existingWikiWord" href="/nlab/show/additive+category">additive categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">C_1, C_2, C_3</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>F</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex"> F : C_1 \times C_2 \to C_3 </annotation></semantics></math></div> <p>to complexes in their two arguments. Combining this with the formation of <a class="existingWikiWord" href="/nlab/show/total+complexes">total complexes</a> then yields bifunctors from categories of complexes to categories of complexes.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>Ch</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>Ch</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Ch</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde F : Ch(C_1) \times Ch(C_2) \to Ch(C_3) \,. </annotation></semantics></math></div> <p>The most important examples of this are induced by the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> and the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> functor together with their <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a> and <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>.</p> <p>Notice that under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> and with sufficient <a class="existingWikiWord" href="/nlab/show/resolutions">resolutions</a>, such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde F</annotation></semantics></math> can be understood as the internal hom or tensor products, etc., between <a class="existingWikiWord" href="/nlab/show/infinity-groupoid">higher groupoids</a>.</p> <p>Although we suggest (and prefer) the ‘complex of complexes’ definition, as above, rather than the equivalent <em>anticommutiing diagram</em> one, we give both and will discuss how to go between them in some detail.</p> <h2 id="definition">Definition</h2> <h3 id="with_commuting_differentials">With commuting differentials</h3> <p>A double complex, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, is a commutative diagram in an additive category in which the objects are bi-indexed by the integers,</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><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ X_{p,q} \mid p,q\in \mathbb{Z} \}</annotation></semantics></math></div> <p>and with two classes of ‘<a class="existingWikiWord" href="/nlab/show/differentials">differentials</a>’ or ‘boundary morphisms’:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>:</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_X^v: X_{p,q}\to X_{p,q-1}</annotation></semantics></math> called the ‘vertical boundary morphisms’ or ‘vertical differentials’, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_X^v d_X^v = 0</annotation></semantics></math>;</li> </ul> <p>and</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><mo>:</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">d_X^h: X_{p,q}\to X_{p-1,q}</annotation></semantics></math> called the ‘horizontal boundary morphisms’ or ‘horizontal differentials’, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_X^h d_X^h = 0</annotation></semantics></math>;</li> </ul> <p>such that all <a class="existingWikiWord" href="/nlab/show/commuting+diagram">squares commute</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><mo>∘</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>=</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>∘</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex">d_X^h \circ d_X^v = d_X^v \circ d_X^h</annotation></semantics></math>.</li> </ul> <p>(To state the obvious, this means <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>−</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_X^h d_X^v - d_X^v d_X^h=0</annotation></semantics></math>, in contrast to the formula in the second version <a href="#AntiCommutingDifferentials">below</a>.)</p> <p>This gives a commutative diagram:</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><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \vdots && \vdots \\ & & \downarrow^{d_X^v} && \downarrow^{d_X^v} \\ \cdots &\to & X_{n,m} &\stackrel{d_X^h}{\to}& X_{n,m-1} & \to & \cdots \\ & & \downarrow^{d_X^v} && \downarrow^{d_X^v} \\ \cdots &\to & X_{n-1,m} &\stackrel{d_X^h}{\to}& X_{n-1,m-1} & \to & \cdots \\ & & \downarrow^{d_X^v} && \downarrow^{d_X^v} \\ && \vdots && \vdots } </annotation></semantics></math></div> <h3 id="AntiCommutingDifferentials">With anti-commuting differentials</h3> <p>A double complex, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, is an anticommutative diagram in an additive category in which the objects are bi-indexed by the integers,</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><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ X_{p,q} \mid p,q\in \mathbb{Z} \}</annotation></semantics></math></div> <p>and with two classes of ‘differentials’ or ‘boundary morphisms’:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>:</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_X^v: X_{p,q}\to X_{p,q-1}</annotation></semantics></math> called the ‘vertical boundary morphisms’ or ‘vertical differentials’, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_X^v d_X^v = 0</annotation></semantics></math>;</li> </ul> <p>and</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup><mo>:</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\bar{d}_X^h: X_{p,q}\to X_{p-1,q}</annotation></semantics></math> called the ‘horizontal boundary morphisms’ or ‘horizontal differentials’, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\bar{d}_X^h \bar{d}_X^h = 0</annotation></semantics></math>;</li> </ul> <p>in addition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>+</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\bar{d}_X^h d_X^v + d_X^v \bar{d}_X^h = 0</annotation></semantics></math>.</p> <p>This gives an anticommutative diagram:</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><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd><msub><mo>⇙</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>X</mi> <mi>h</mi></msubsup></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \vdots && \vdots \\ & & \downarrow^{d_X^v} && \downarrow^{d_X^v} \\ \cdots &\to & X_{n,m} &\stackrel{\bar{d}_X^h}{\to}& X_{n,m-1} & \to & \cdots \\ & & \downarrow^{d_X^v} & \swArr_{-1} & \downarrow^{d_X^v} \\ \cdots &\to & X_{n-1,m} &\stackrel{\bar{d}_X^h}{\to}& X_{n-1,m-1} & \to & \cdots \\ & & \downarrow^{d_X^v} && \downarrow^{d_X^v} \\ && \vdots && \vdots } </annotation></semantics></math></div> <h3 id="EquivalenceOfTheTwoDefinitions">Equivalence of the two definitions</h3> <p>Which definition is ‘better’? ‘Commuting squares’, i.e., the first version, is convenient if you want to define a double complex as a chain complex in the category of chain complexes. On the other hand, ‘anticommuting squares’ and version 2 is sometimes convenient for defining the total complex (for computing total homology). Does it matter which you use? The following says they are just two views of the same situation.</p> <p>One makes a double complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with commutative squares into a double complex with anticommutative squares by using the same vertical differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mi>v</mi></msup></mrow><annotation encoding="application/x-tex">d^v</annotation></semantics></math> but taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>d</mi><mo stretchy="false">¯</mo></mover> <mi>h</mi></msup><mo>:</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\bar{d}^h : X_{p,q} \to X_{p,q-1}</annotation></semantics></math> to be <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>p</mi></msup><msup><mi>d</mi> <mi>h</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^p d^h</annotation></semantics></math>. The same trick can, of course, be used to make a double complex with anticommutative squares into a double complex with commutative squares.</p> <h2 id="properties">Properties</h2> <h3 id="total_complex_of_a_double_complex">Total complex of a double complex</h3> <p>The <strong><a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a></strong> of a double complex (under the convention that squares commute) is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>tot</mi> <mo>⊕</mo> <mi>k</mi></msubsup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>=</mo><mi>k</mi></mrow></munder><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> tot_{\oplus}^k = \bigoplus_{m+n=k} X_{n,m} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mrow><msub><mi>tot</mi> <mo>⊕</mo></msub></mrow> <mi>k</mi></msubsup><msub><mo stretchy="false">|</mo> <mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow></msub><mo>=</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex"> d^k_{tot_\oplus}|_{X_{n,m}} = d^v_X + (-1)^\bullet d_X^h </annotation></semantics></math></div> <p>Similarly, one can define the product total complex as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>tot</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>k</mi></msubsup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>=</mo><mi>k</mi></mrow></munder><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> tot_{\prod}^k = \prod_{m+n=k} X_{n,m} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mrow><msub><mi>tot</mi> <mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></msub></mrow> <mi>k</mi></msubsup><msub><mo stretchy="false">|</mo> <mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow></msub><mo>=</mo><msubsup><mi>d</mi> <mi>X</mi> <mi>v</mi></msubsup><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><msubsup><mi>d</mi> <mi>X</mi> <mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex"> d^k_{tot_\prod}|_{X_{n,m}} = d^v_X + (-1)^\bullet d_X^h </annotation></semantics></math></div> <p>Note that these two coincide when the set of non-zero objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n,m}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n + m = k</annotation></semantics></math> is finite, for example, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a first quadrant double complex.</p> <h3 id="fundamental_lemmas">Fundamental lemmas</h3> <p>There is series of basic <a class="existingWikiWord" href="/nlab/show/lemmas">lemmas</a> in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> which determine the horizontal/vertical <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> of a double complex in some row or column from exactness information in other columns. The most fundamental of these is maybe the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/salamander+lemma">salamander lemma</a></li> </ul> <p>from which a series of others follow:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a> (see also <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> and <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+homology">long exact sequence in homology</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-lemma">5-lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a> .</p> </li> </ul> <h2 id="relation_to_homotopy_colimits">Relation to homotopy colimits</h2> <p>The total complex of the double complex induced by a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> is a model for the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of that map, see at <em><a href="mapping+cone#ConeViaDoubleComplex">mapping cone – via double complexes</a> for more.</em></p> </body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2021 at 06:24:56. 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