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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="additive_and_abelian_categories">Additive and abelian categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></strong></p> <h2 id="context_and_background">Context and background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="categories">Categories</h2> <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> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudo-abelian+category">pseudo-abelian category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>,</p> </li> <li> <p>(AB1) <a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p>(AB2) <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p>(AB5) <a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+exact+category">Quillen exact category</a></p> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a></p> </li> <li> <p>left/right <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></p> </li> </ul> <h2 id="derived_categories">Derived categories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/additive+and+abelian+categories+-+contents">Edit this sidebar</a> </p> </div></div></div> <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='#definition'>Definition</a></li> <li><a href='#via_embedding'>Via embedding</a></li> <li><a href='#via_exact_structure'>Via exact structure</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#quillengabriel_embedding_theorem'>Quillen-Gabriel embedding theorem</a></li> <li><a href='#relation_to_waldhausen_categories_and_algebraic_ktheory'>Relation to Waldhausen categories and algebraic K-theory</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <h2 id="via_embedding">Via embedding</h2> <p>A full <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</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> of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is called <strong>Quillen exact category</strong> if it is closed under <a class="existingWikiWord" href="/nlab/show/extensions">extensions</a> (if in extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>X</mi><mover><mo>→</mo><mi>j</mi></mover><mi>Y</mi><mover><mo>→</mo><mi>p</mi></mover><mi>Z</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\to X\stackrel{j}\to Y\stackrel{p}\to Z\to 0</annotation></semantics></math>, <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> are in <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is in <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>). It is viewed as a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,E)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is the class of all short exact sequences in <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> which are <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>All <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> which appear as <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> in an exact sequence as above are called <strong>inflation</strong>s or <strong>admissible monomorphism</strong>s. All <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> which appear in an exact sequence as above are called <strong>deflation</strong>s or <strong>admissible epimorphism</strong>s.</p> <h2 id="via_exact_structure">Via exact structure</h2> <p>A <strong>Quillen exact category</strong> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,E)</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</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> and a class of sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> called ‘exact’. The following axioms are required for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,E)</annotation></semantics></math>:</p> <p>(QE1) The class of ‘exact’ sequences is closed under <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s and it contains all split extensions. For any ‘exact’ sequence the deflation is the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of inflation and the inflation is the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the deflation.</p> <p>(QE2) The class of deflations is closed under <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> by arbitrary maps. The class of inflations is closed under compositions and <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a> by arbitrary maps.</p> <p>(QE3) If a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>→</mo><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">M\to M'</annotation></semantics></math> having a <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> can factor a deflation <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></mrow><annotation encoding="application/x-tex">N\to M'</annotation></semantics></math> as <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>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">N\to M\to M'</annotation></semantics></math> then it is a deflation. If a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">I\to I'</annotation></semantics></math> having a cokernel can factor an inflation <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\to J</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mi>I</mi><mo>′</mo><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I\to I'\to J</annotation></semantics></math> then it is also an inflation.</p> <h2 id="properties">Properties</h2> <h3 id="quillengabriel_embedding_theorem">Quillen-Gabriel embedding theorem</h3> <p>For every small exact category in the sense of a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,E)</annotation></semantics></math>, there is an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow B</annotation></semantics></math> into an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a class of all sequences which are (short) exact in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <h3 id="relation_to_waldhausen_categories_and_algebraic_ktheory">Relation to Waldhausen categories and algebraic K-theory</h3> <p>Every Quillen exact category can be made into a <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a>. However some information is lost in the process. Moreover, not every Waldhausen category comes from a Quillen exact category. Both Quillen exact categories and Waldhausen categories are devised in order to do <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a>. The K-theory spectrum based on <a class="existingWikiWord" href="/nlab/show/Quillen%27s+Q-construction">Quillen's Q-construction</a> and an exact category agrees with the K-theory spectrum based on the <a class="existingWikiWord" href="/nlab/show/Waldhausen+S-construction">Waldhausen S-construction</a> of the K-theory spectrum from its associated <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a>.</p> <h2 id="references">References</h2> <p>Quillen introduced exact categories in above sense in the article</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, “Higher algebraic K-theory”, in Higher K-theories, pp. 85–147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973.</li> </ul> <p>A nonadditive generalization of exact categories has been introduced by Dyckerhoff and Kapranov and named a <a class="existingWikiWord" href="/nlab/show/proto-exact+category">proto-exact category</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Rosenberg">Alexander Rosenberg</a><a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=3623">introduced</a> one sided generalizations of Quillen exact categories: right ‘exact’ categories involving deflations, and left ‘exact’ categories involving inflations. One of the motivations an alternative definition of higher K-theory of (right exact) categories not involving spectra. In this setup the K-theory is an example of a derived functor in nonabelian homological algebra utilizing roughly the left ‘exact’ structure on the category of essentially small right ‘exact’ categories. It is not known if this K-theory when restricted to the category of essentially small Quillen exact categories agrees with Quillen K-theory. But it has the standard properties of Quillen K-theory (devissage, exactness and so on).</p> <p>The one-sided generalization inspired by ideas introduced by Keller and Vossieck in the build up of the theory of <a class="existingWikiWord" href="/nlab/show/suspended+category">suspended categories</a>.</p> <p>A right ‘exact’ category is a category with an initial object and a Grothendieck pretopology consisting of single maps which are <a class="existingWikiWord" href="/nlab/show/strict+epimorphism">strict epimorphism</a>s. The distinguished class of strict epimorphisms is called a right ‘exact’ structure, or the class of <em>deflations</em>. The construction of derived functors in this generality involves a version of <a class="existingWikiWord" href="/nlab/show/satellite">satellites</a>.</p> <ul> <li>Dmitry Kaledin, Wendy Lowen, <em>Cohomology of exact categories and (non-)additive sheaves</em>, Adv. Math. <strong>272</strong> (2015) 652–698 <a href="https://arxiv.org/abs/1102.5756">arXiv:1102.5756</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 20, 2024 at 15:09:45. 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