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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="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <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/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+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='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#as_a_pullback'>As a pullback</a></li> <li><a href='#as_an_equalizer'>As an equalizer</a></li> <li><a href='#as_a_weighted_limit'>As a weighted limit</a></li> <li><a href='#AsARepresentingObject'>As a representing object</a></li> <li><a href='#in_an_category'>In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category</a></li> <li><a href='#other_meanings'>Other meanings</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#property'>Property</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>kernel</em> of a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> is that part of its <a class="existingWikiWord" href="/nlab/show/domain">domain</a> which is sent to <a class="existingWikiWord" href="/nlab/show/zero">zero</a>.</p> <h2 id="definition">Definition</h2> <p>There are various definitions of the notion of kernel, depending on the properties and structures available in the ambient category. We list a few definitions and discuss (in parts) when they are equivalent.</p> <h3 id="as_a_pullback">As a pullback</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> with an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, the <strong>kernel</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f: A \to B</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">ker(f) \to A</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">0 \to B</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ ker(f) &\to& 0 \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>More explicitly, this characterizes the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> as <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> object (unique up to unique <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>) that satisfies the following <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>:</p> <p>for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h : C \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>h</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f\circ h = 0</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>, there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : C \to ker(f)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>p</mi><mo>∘</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">h = p\circ \phi</annotation></semantics></math>.</p> </div> <h3 id="as_an_equalizer">As an equalizer</h3> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphisms</a> (meaning: <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over the <a class="existingWikiWord" href="/nlab/show/category+of+pointed+sets">category of pointed sets</a>), <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <strong>kernel</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f : c \to d</annotation></semantics></math> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and the zero morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex">0_{c,d}</annotation></semantics></math>.</p> </div> <h3 id="as_a_weighted_limit">As a weighted limit</h3> <p>In any category enriched over pointed sets, the kernel of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f:c\to d</annotation></semantics></math> is the universal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">k:a\to c</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f \circ k</annotation></semantics></math> is the basepoint. It is a <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a> in the sense of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>. This applies in particular in any (pre)-<a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>.</p> <p>This is a special case of the construction of <a class="existingWikiWord" href="/nlab/show/generalized+kernels">generalized kernels</a> in enriched categories.</p> <h3 id="AsARepresentingObject">As a representing object</h3> <p>Let <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>. This has all kernels, in particular (it is the archetypical <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>).</p> <p>In every <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched 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>, for every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X\to Y</annotation></semantics></math> 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> there is a <a class="existingWikiWord" href="/nlab/show/subfunctor">subfunctor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">ker f \colon A^{op}\to Ab</annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(-,X)</annotation></semantics></math>, defined on <a class="existingWikiWord" href="/nlab/show/objects">objects</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ker</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> (ker f)(Z) = ker\big(hom(Z,X)\to hom(Z,Y)\big), </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi></mrow><annotation encoding="application/x-tex">ker</annotation></semantics></math> on the right-hand side is the kernel in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>.</p> <p>If the category is in fact <a class="existingWikiWord" href="/nlab/show/preabelian+category">preabelian</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">ker f</annotation></semantics></math> is also representable with <a class="existingWikiWord" href="/nlab/show/representing+object">representing object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">Ker f</annotation></semantics></math>. One has to be careful with the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Coker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">Coker f</annotation></semantics></math>, which does not represent the functor naive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">coker f</annotation></semantics></math> defined as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>coker</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>coker</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">(coker f)(Z) = coker\big(hom(Z,X)\to hom(Z,Y)\big)</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, which is often not representable at all, even in the simple example of the category of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>. Instead, as a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> construction, one should <a class="existingWikiWord" href="/nlab/show/corepresentable+functor">co-represent</a> another functor, namely, the <a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>↦</mo><mi>ker</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Z\mapsto ker\big(hom(Y,Z) \to hom(X,Z)\big)</annotation></semantics></math> (which is a <a class="existingWikiWord" href="/nlab/show/quotient+object">quotent</a> of the corepresentable functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(X,-)</annotation></semantics></math>). In short, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Coker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">Coker f</annotation></semantics></math> is defined by the “double dualization” (passing to <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a>) of the kernel in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Coker</mi><mi>f</mi><mo>=</mo><mo stretchy="false">(</mo><mi>Ker</mi><msup><mi>f</mi> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Coker f = (Ker f^{op})^{op}</annotation></semantics></math>. This is a particular case of the dualization involved in defining any <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> from its corresponding <a class="existingWikiWord" href="/nlab/show/limit">limit</a>.</p> <h3 id="in_an_category">In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category</h3> <p>The kernel of a morphism in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categorical <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> as in the pullback definition above: the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>.</p> <p>See also <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a>.</p> <h3 id="other_meanings">Other meanings</h3> <p>In some fields, the term ‘kernel’ refers to an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> that category theorists would see as a <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a>. This is especially important in fields such as <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> theory where both notions exist but are not equivalent (while in <a class="existingWikiWord" href="/nlab/show/group">group</a> theory they are equivalent).</p> <p>In <a class="existingWikiWord" href="/nlab/show/ring">ring</a> theory, even when one assumes that rings have units preserved by ring homomorphisms, the traditional notion of kernel (an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>) exists in the category of non-unital rings (and is not itself a unital ring in general). A purely category-theoretic theory of unital rings can be recovered either by using the kernel pair instead or (to fit better the usual language) moving to a category of <a class="existingWikiWord" href="/nlab/show/modules">modules</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>, this may be handled in the framework of <a class="existingWikiWord" href="/nlab/show/Malcev+variety">Mal'cev varieties</a>.</p> <p>Kashiwara-Schapira, following the terminology of EGA, uses kernel as a synonym of equalizer (and co-kernel of co-equalizer).</p> <h2 id="Properties">Properties</h2> <div class="num_prop"> <h6 id="property">Property</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a category with <a class="existingWikiWord" href="/nlab/show/pullback">pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>.</p> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the kernel of a kernel is 0.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the <a href="http://nlab.mathforge.org/nlab/show/pullback#Pasting">pasting law for pullbacks</a> we have that the total square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ker</mi><mi>ker</mi><mi>f</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ker</mi><mi>f</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>d</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ ker ker f &\to& ker f &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& c &\stackrel{f}{\to}& d } </annotation></semantics></math></div> <p>is a pullback. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">0 \to c</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> and the pullback of a monomorphism along itself is the domain of the monomorphis, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mi>ker</mi><mi>f</mi><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ker ker f \simeq 0</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This statement crucially fails to be true in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>. There, the kernel of a kernel is the based <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>. For this reason where one has <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequences</a> in 1-category theory, there are instead long <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequences</a> in higher category theory.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>In a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/pullback">pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/pushout">pushouts</a> and a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>, kernel and <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> form a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functors</a> on the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>coker</mi><mo>⊣</mo><mi>ker</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>ker</mi></munder><mover><mo>←</mo><mi>coker</mi></mover></mover><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (coker \dashv ker) : Arr(C) \stackrel{\overset{coker}{\leftarrow}}{\underset{ker}{\to}} Arr(C) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We check the hom-isomorphism of a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functors</a>. An element in the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Arr</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>ker</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr_C(g,ker f)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>c</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ c &\to& ker(f) &\to& 0 \\ {}^{\mathllap{g}}\downarrow && \downarrow && \downarrow \\ d &\to& a &\stackrel{f}{\to}& b } \,. </annotation></semantics></math></div> <p>By the universal property of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, this is the same as a 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><mi>c</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ c &\to& &\to& 0 \\ {}^{\mathllap{g}}\downarrow && && \downarrow \\ d &\to& a &\stackrel{f}{\to}& b } \,. </annotation></semantics></math></div> <p>By the dual reasoning, an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Arr</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>coker</mi><mi>g</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr_C(coker g, f)</annotation></semantics></math> is a 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><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>coker</mi><mi>g</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ c &\stackrel{g}{\to}& d &\to& a \\ \downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ 0 &\to& coker g &\to& b } \,. </annotation></semantics></math></div> <p>By the universal property of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> this is equivalently a 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><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ c &\stackrel{g}{\to}& d &\to& a \\ \downarrow && && \downarrow^{\mathrlap{f}} \\ 0 &\to& &\to& b } \,. </annotation></semantics></math></div></div> <p>(This also follows from the general theory of <a class="existingWikiWord" href="/nlab/show/generalized+kernels">generalized kernels</a>.)</p> <h2 id="examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of abelian groups, the kernel of a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(0)</annotation></semantics></math> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> that are sent to the zero-element of <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> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, this is true in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>: the kernel of a morphism of modules is the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+kernel">generalized kernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rank-nullity+theorem">rank-nullity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber">fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 11, 2023 at 20:53:24. 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