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name="query" value="Search" 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"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homological_algebra'>Homological algebra</h4> <div class='hide'> <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' title='schreiber'>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+sheaf'>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/exact+sequence'>short exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/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/projective+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+%28infinity%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-infinity-category'>A-∞-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%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+limit'>homotopy colimit</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/BV-BRST+formalism'>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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#AbelianCategoryStructure'>Abelian category</a></li><li><a href='#closed_monoidal_structure'>Closed monoidal structure</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/additive+category'>additive category</a>.</p> <p>Recall the notion of <em><a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complex</a></em>, of <em><a class='existingWikiWord' href='/nlab/show/chain+map'>chain map</a></em> between chain complexes and of <em><a class='existingWikiWord' href='/nlab/show/chain+homotopy'>chain homotopy</a></em> between chain maps in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> <p>Call a <a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complex</a> <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>C_\bullet</annotation></semantics></math></p> <ul> <li> <p><strong>bounded below</strong> if there is <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>≤</mo><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>C_{n \leq k} = 0</annotation></semantics></math>;</p> </li> <li> <p><strong>bounded above</strong> if there is <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>C_{n \geq k} = 0</annotation></semantics></math>;</p> </li> <li> <p><strong>bounded</strong> if it is bounded below and bounded above. We have</p> </li> </ul> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/category'>category</a> whose <a class='existingWikiWord' href='/nlab/show/object'>objects</a> are <a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complexes</a> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are <a class='existingWikiWord' href='/nlab/show/chain+map'>chain maps</a> between these.</p> <p>This is the <strong>category of chain complexes</strong> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> </div> <p>Several variants of this category are of relevance.</p> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>b</mi></mrow></msubsup><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet^{+,-,b}(\mathcal{A}) \hookrightarrow Ch_\bullet(\mathcal{A})</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> on the chain complexes which are, respectively, bounded above, bounded below or bounded.</p> </div> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\mathcal{A})</annotation></semantics></math> for the category obtained from <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> by identifying <a class='existingWikiWord' href='/nlab/show/chain+homotopy'>homotopic</a> chain maps.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>≔</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>chain</mi><mo>−</mo><mi>homotopy</mi><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'> K(\mathcal{A})(C_\bullet, D_\bullet) \coloneqq Ch_\bullet(C_\bullet, D_\bullet)/chain-homotopy \,. </annotation></semantics></math></div> <p>Accordingly <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>K</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>b</mi></mrow></msup><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K^{+,-,b}(\mathcal{A}) \hookrightarrow K(\mathcal{A})</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> on the chain complexes bounded above, bounded below or bounded, respectively.</p> </div> <p>This is sometimes called the <em><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+chain+complexes'>homotopy category of chain complexes</a></em>. But see the warning on terminology there, as this term is also appropriate for the category in the following remark.</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is moreover an <a class='existingWikiWord' href='/nlab/show/abelian+category'>abelian category</a>, then there is also the <strong><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a></strong> <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(\mathcal{A})</annotation></semantics></math>, obtained from <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\mathcal{A})</annotation></semantics></math> by <a class='existingWikiWord' href='/nlab/show/localization'>universally inverting</a> all <a class='existingWikiWord' href='/nlab/show/quasi-isomorphism'>quasi-isomorphisms</a>. See at <em><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a></em> for more on this.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='AbelianCategoryStructure'>Abelian category</h3> <div class='num_theorem' id='IsAbelian'> <h6 id='theorem'>Theorem</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/abelian+category'>abelian category</a> also the category of chain complexes <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> is again an abelian category.</p> </div> <p>We discuss the ingredients that go into this statement.</p> <p>(…)</p> <div class='num_prop' id='KernelsOfChainComplexes'> <h6 id='proposition'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>D</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>f : C_\bullet \to D_\bullet</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/chain+map'>chain map</a>,</p> <ul> <li> <p>the complex <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ker</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>ker(f)</annotation></semantics></math> of degreewise <a class='existingWikiWord' href='/nlab/show/kernel'>kernels</a> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is the kernel of <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math>;</p> </li> <li> <p>the complex <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>coker</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>coker(f)</annotation></semantics></math> of degreewise <a class='existingWikiWord' href='/nlab/show/cokernel'>cokernels</a> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is the cokernel of <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math>.</p> </li> </ul> </div> <div class='num_remark' id='ShortExactSequencesDegreewise'> <h6 id='remark_2'>Remark</h6> <p>A sequence of chain complexes <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/exact+sequence'>short exact sequence</a> in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> precisely if each component <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>0 \to A_n \to B_n \to C_n \to 0</annotation></semantics></math> is a short exact sequence in <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> </div> <p>(…)</p> <h3 id='closed_monoidal_structure'>Closed monoidal structure</h3> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>Equipped with the standard <a class='existingWikiWord' href='/nlab/show/tensor+product+of+chain+complexes'>tensor product of chain complexes</a> <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math> the category of chain complexes is a <a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>R</mi><mi>Mod</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ch_\bullet(R Mod), \otimes)</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/unit+object'>unit object</a> is the chain complex concentrated in degree 0 on the tensor unit <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>R Mod</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>In fact <math class='maruku-mathml' display='inline' id='mathml_df133d1f4ec51d7a7cc0206786ba5e4209e33ce9_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ch_\bullet, \otimes)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed monoidal category</a>, the <a class='existingWikiWord' href='/nlab/show/internal+hom'>internal hom</a> is the standard <a class='existingWikiWord' href='/nlab/show/internal+hom+of+chain+complexes'>internal hom of chain complexes</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+chain+complexes'>homotopy category of chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+chain+complexes'>model structure on chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a>, <a class='existingWikiWord' href='/nlab/show/derived+functor'>derived functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+chain+complexes'>(∞,1)-category of chain complexes</a></p> </li> </ul> <h2 id='references'>References</h2> <p>A basic introduction is in chapter 1 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Charles+Weibel'>Charles Weibel</a>, <em><a class='existingWikiWord' href='/nlab/show/An+Introduction+to+Homological+Algebra'>An Introduction to Homological Algebra</a></em></li> </ul> <p><div class='property'> category: <a class='category_link' href='/nlab/list/category'>category</a></div></p> <p> </p> <p> </p> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on June 20, 2014 at 14:03:43 by <a href="/nlab/author/Tobias+Fritz" style="color: #005c19">Tobias Fritz</a> See the <a href="/nlab/history/category+of+chain+complexes" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/4146/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/category+of+chain+complexes/17" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (6 more)</span><span class="backintime"><a href="/nlab/revision/category+of+chain+complexes/15" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (15 more)</span><a href="/nlab/show/category+of+chain+complexes" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/category+of+chain+complexes/16" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/category+of+chain+complexes" accesskey="S" class="navlink" id="history" rel="nofollow">History (21 revisions)</a><a href="/nlab/rollback/category+of+chain+complexes?rev=16" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/category+of+chain+complexes/16/cite" style="color: black">Cite</a> <a href="/nlab/source/category+of+chain+complexes/16" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>