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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> additive functor </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4343/#Item_11" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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"> <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="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> <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='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_enriched_functors'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enriched functors</a></li> <li><a href='#CharacterizationByExactness'>Characterization of right exact additive functors</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</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: \mathcal{A} \to \mathcal{B}</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/additive+category">additive categories</a> is itself called <strong>additive</strong> if it preserves finite <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a>.</p> <p>That is,</p> <ol> <li> <p><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> maps a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> to a zero object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F(0) \simeq 0 \in \mathcal{B}</annotation></semantics></math>;</p> </li> <li> <p>given any two <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">x, y \in \mathcal{A}</annotation></semantics></math>, there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>⊕</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x \oplus y) \cong F(x) \oplus F(y)</annotation></semantics></math>, and this respects the inclusion and projection maps of the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>:</p> </li> </ol> <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>x</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>x</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>y</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi><mo>⊕</mo><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>x</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="1em"></mspace><mspace width="1em"></mspace><mover><mo>↦</mo><mi>F</mi></mover><mspace width="1em"></mspace><mspace width="1em"></mspace><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>⊕</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { x & & & & y \\ & {}_{\mathllap{i_x}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) } </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In practice, functors between additive categories are generally assumed to be additive.</p> </div> <div class="num_remark" id="SufficientConditions"> <h6 id="remark_2">Remark</h6> <p>Each of the following conditions is sufficient for guaranteeing that a functor <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">\mathcal{A} \to \mathcal{B}</annotation></semantics></math> preserves biproducts (where <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">\mathcal{A}</annotation></semantics></math> and <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">\mathcal{B}</annotation></semantics></math> are categories with a zero object):</p> <ul> <li>The functor preserves finite products (for instance, because it’s a right adjoint) and any product 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">\mathcal{B}</annotation></semantics></math> is a biproduct.</li> <li>The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct 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">\mathcal{B}</annotation></semantics></math> is a biproduct.</li> <li>The functor preserves finite products and coproducts.</li> </ul> </div> <h2 id="examples">Examples</h2> <p> <div class='num_remark' id='HomFunctor'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>𝒜</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒜</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Hom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab</annotation></semantics></math> is additive in both arguments separately (using the nature of <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a> and that <a class="existingWikiWord" href="/nlab/show/hom-functors+preserve+limits">hom-functors preserve limits</a> in each variable separately).</p> </div> </p> <div class="num_example"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} = R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">N \in \mathcal{A}</annotation></semantics></math>, the functor that forms <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">(-)\otimes N \colon \mathcal{A} \to \mathcal{A}</annotation></semantics></math>.</p> </div> <p>In fact these examples are generic, see prop. <a class="maruku-ref" href="#RightExactAdditiveIsTensor"></a> below.</p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/solid+abelian+group">solid abelian group</a> is by definition an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="relation_to_enriched_functors">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enriched functors</h3> <p>An <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> canonically carries the structure of an <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a> where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enrichment structure is induced from the biproducts as described at <em><a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a></em>.</p> <div class="num_prop" id="AdditiveIsAbEnriched"> <h6 id="proposition">Proposition</h6> <p>With respect to the canonical <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a>-structure on additive categories <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">\mathcal{A}</annotation></semantics></math>, <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">\mathcal{B}</annotation></semantics></math>, additive functors <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 : \mathcal{A} \to \mathcal{B}</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enriched functor preserves all finite biproducts that exist, since finite biproducts in <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched categories</a> are <a class="existingWikiWord" href="/nlab/show/Cauchy+complete+category">Cauchy colimits</a>.</p> </div> <h3 id="CharacterizationByExactness">Characterization of right exact additive functors</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>,</mo><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R, R'</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <p>The following is the <em><a class="existingWikiWord" href="/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a></em>. See there for more.</p> <div class="num_prop" id="RightExactAdditiveIsTensor"> <h6 id="proposition_2">Proposition</h6> <p>If an additive functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">F : R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\to R'</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> is a <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a>, then there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R'</annotation></semantics></math>-<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/bimodule">bimodule</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> and a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≃</mo><mi>B</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F \simeq B \otimes_R (-) </annotation></semantics></math></div> <p>with the functor that forms the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> with <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> <p>This is (<a href="#Watts">Watts, theorem 1</a>),</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+monad">additive monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a></p> </li> <li> <p>In the context of <a class="existingWikiWord" href="/nlab/show/derived+functors+in+homological+algebra">derived functors in homological algebra</a> one considers functors that are additive and in addition left/right <a class="existingWikiWord" href="/nlab/show/exact+functors">exact functors</a>, as discussed above in <em><a href="#CharacterizationByExactness">Characterization by exactness</a></em>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/satellite">satellite</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Watts">Charles Watts, <em>Intrinsic characterizations of some additive functors</em>, Proceedings of the American Mathematical Society, <strong>11</strong> 1 (1960) 5-8 (1959) [<a href="http://www.jstor.org/stable/2032707">jstor:2032707</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 15, 2023 at 09:59:22. 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