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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 category </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/4223/#Item_19" 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='#properties'>Properties</a></li> <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" id="AdditiveCategory"> <h6 id="definition_2">Definition</h6> <p>An <strong>additive category</strong> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> which is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a>;</p> <p>(sometimes called a <a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a>–this means that each <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> carries the structure of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and composition is <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear</a>)</p> </li> <li> <p>which admits <a class="existingWikiWord" href="/nlab/show/finite+limit">finite</a> <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a></p> <p>(and hence, by prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a> below, finite <a class="existingWikiWord" href="/nlab/show/products">products</a> which coincide with the coproducts, hence finite <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a>).</p> </li> </ol> <p>The natural <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <em>between</em> additive categories are <a class="existingWikiWord" href="/nlab/show/additive+functors">additive functors</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a> is an additive category which also has <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> and <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a>. Equivalently, it is an Ab-enriched category with all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and finite <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. An especially important sort of additive category is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, which is a pre-abelian one satisfying the extra exactness property that all <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> are <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> and all <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> are <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a>. See at <em><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></em> for more.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-enrichment of an additive category does not have to be given a priori. Every <a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive category</a> (a category with finite <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a>) is automatically <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> (as described at <em><a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a></em>), so an additive category may be defined as a category with finite biproducts whose <a class="existingWikiWord" href="/nlab/show/hom-object">hom-monoids</a> happen to be <a class="existingWikiWord" href="/nlab/show/groups">groups</a>. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire <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 structure follows automatically for <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian categories</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Some authors use <strong>additive category</strong> to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi></mrow><annotation encoding="application/x-tex">CMon</annotation></semantics></math>-enriched (commutative monoid enriched) category, with or without assumptions of products.</p> </div> <h2 id="properties">Properties</h2> <div class="num_prop" id="ProductsAreBiproducts"> <h6 id="proposition">Proposition</h6> <p>In an <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a> (or even just a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi></mrow><annotation encoding="application/x-tex">CMon</annotation></semantics></math>-enriched category), a <a class="existingWikiWord" href="/nlab/show/finite+product">finite</a> <a class="existingWikiWord" href="/nlab/show/product">product</a> is also a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, and dually (hence a <a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a>).</p> <p>This statement includes the zero-ary case: any <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is also an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, hence a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> (and dually), hence every additive category has a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>.</p> <p>More precisely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> of objects in an Ab-enriched category, the unique morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>⟶</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> \underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j </annotation></semantics></math></div> <p>whose components are identities for <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 = j</annotation></semantics></math> and are <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero</a> otherwise is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Consider first the nullary (i.e., zero-ary) case. Given a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>, the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mo>*</mo></msub><mo>:</mo><mo>*</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">id_\ast: \ast \to \ast</annotation></semantics></math> is the zero morphism <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> in its hom-object. For any object <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>, the zero morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">0_A: \ast \to A</annotation></semantics></math> must equal any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f: \ast \to A</annotation></semantics></math> on account of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>f</mi><msub><mi>id</mi> <mo>*</mo></msub><mo>=</mo><mi>f</mi><mn>0</mn><mo>=</mo><msub><mn>0</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">f = f id_\ast = f 0 = 0_A</annotation></semantics></math> where the last equation is by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi></mrow><annotation encoding="application/x-tex">CMon</annotation></semantics></math>-enrichment. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> is initial. (N.B.: this argument applies more generally to categories <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, and is self-<a class="existingWikiWord" href="/nlab/show/formal+duality">dual</a>.)</p> <p>Consider now the case of binary (co-)products. Using <a class="existingWikiWord" href="/nlab/show/zero+morphisms">zero morphisms</a>, in addition to its canonical <a class="existingWikiWord" href="/nlab/show/projection">projection</a> maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i \colon X_1 \times X_2 \to X_i</annotation></semantics></math>, any binary <a class="existingWikiWord" href="/nlab/show/product">product</a> also admits “injection” maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_i \to X_1 \times X_2</annotation></semantics></math>, and dually for the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</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><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msub><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>⊔</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow \\ X_1 && && X_2 } \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\; \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow \\ X_1 && && X_2 } \,. </annotation></semantics></math></div> <p>Observe some basic compatibility of 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 with the product:</p> <p>First, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>β</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>β</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2</annotation></semantics></math> then</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><mo>⋆</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>β</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>β</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>α</mi> <mn>2</mn></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>β</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>β</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\star) \;\;\;\;\;\; (\alpha_1,\beta_1) + (\alpha_2, \beta_2) = (\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2) </annotation></semantics></math></div> <p>(using that the projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> are linear and by the universal property of the product).</p> <p>Second, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(id,0) \circ p_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(0,id) \circ p_2</annotation></semantics></math> are two projections on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1\times X_2</annotation></semantics></math> whose sum is the identity:</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><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\star\star) \;\;\;\;\;\; (id, 0) \circ p_1 + (0, id) \circ p_2 = id_{X_1 \times X_2} \,. </annotation></semantics></math></div> <p>(We may check this, via the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> on <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(\alpha, \beta) \colon R \to X_1\times X_2</annotation></semantics></math> any morphism, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta)</annotation></semantics></math>, so the statement follows with equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math>.)</p> <p>Now observe that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f_i \;\colon\; X_i \to Q</annotation></semantics></math> any two morphisms, the sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>⟶</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex"> \phi \;\coloneqq\; f_1 \circ p_1 + f_2 \circ p_2 \;\colon\; X_1 \times X_2 \longrightarrow Q </annotation></semantics></math></div> <p>gives a morphism of <a class="existingWikiWord" href="/nlab/show/cocones">cocones</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><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Q</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & && \\ X_1 && \downarrow^{\mathrlap{\phi}} && X_2 \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && Q } \,. </annotation></semantics></math></div> <p>Moreover, this is unique: suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi'</annotation></semantics></math> is another morphism filling this diagram, then, by using equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star \star)</annotation></semantics></math>, we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>ϕ</mi></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>∘</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>∘</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><mi>ϕ</mi><mo>∘</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><mi>ϕ</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>′</mo><mo>∘</mo><msub><mi>id</mi> <mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \phi & = \phi \circ id_{X_1 \times X_2} \\ &= \phi \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ & = \phi \circ (id_{X_1}, 0) \circ p_1 + \phi \circ (0, id_{X_2}) \circ p_2 \\ & = f_1 \circ p_1 + f_2 \circ p_2 \\ & = \phi' \circ (id_{X_1}, 0) \circ p_1 + \phi' \circ (0, id_{X_2}) \circ p_2 \\ & = \phi' \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ &= \phi' \circ id_{X_1 \times X_2} \\ &= \phi' \end{aligned} \,. </annotation></semantics></math></div> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1\times X_2</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>.</p> <p>By a <a class="existingWikiWord" href="/nlab/show/formal+dual">dual</a> argument, the binary coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>⊔</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \sqcup X_2</annotation></semantics></math> is seen to also satisfy the universal property of the binary product. By <a class="existingWikiWord" href="/nlab/show/induction">induction</a>, this implies the statement for all finite (co-)products. (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Such products which are also coproducts as in prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a> are sometimes called <em><a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a></em> or <em><a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a></em>; they are <a class="existingWikiWord" href="/nlab/show/absolute+limit">absolute limits</a> for <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</a>.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The coincidence of products with biproducts in prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a> does <em>not</em> extend to infinite products and coproducts.) In fact, an <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a> is <a class="existingWikiWord" href="/nlab/show/Cauchy+complete+category">Cauchy complete</a> just when it is additive and moreover its <a class="existingWikiWord" href="/nlab/show/idempotents">idempotents</a> split.</p> </div> <p>Conversely:</p> <div class="num_defn" id="SemiadditiveCategory"> <h6 id="definition_3">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive category</a></strong> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> that has all <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a> which, moreover, are <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a> in that they coincide with finite <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> as in def. <a class="maruku-ref" href="#ProductsAreBiproducts"></a>.</p> </div> <div class="num_prop" id="SemiAdditivityInducesAbelianMonoidEnrichment"> <h6 id="proposition_2">Proposition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive category</a>, def. <a class="maruku-ref" href="#SemiadditiveCategory"></a>, the <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> acquire the structure of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> by defining the sum of two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \;\colon\; X \longrightarrow Y</annotation></semantics></math> to be</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>g</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mo>≃</mo><mi>X</mi><mo>⊕</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mi>f</mi><mo>⊕</mo><mi>g</mi></mrow></mover><mi>Y</mi><mo>⊕</mo><mi>Y</mi><mo>≃</mo><mi>Y</mi><mo>⊔</mo><mi>Y</mi><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>Y</mi></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f + g \;\coloneqq\; X \overset{\Delta_X}{\to} X \times X \simeq X \oplus X \overset{f \oplus g}{\longrightarrow} Y \oplus Y \simeq Y \sqcup Y \overset{\nabla_Y}{\to} Y \,. </annotation></semantics></math></div> <p>With respect to this operation, <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and commutativity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math> follows directly from the corresponding properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math>. Bilinearity of composition follows from <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_X</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_X</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>W</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>W</mi></msub></mrow></mover></mtd> <mtd><mi>W</mi><mo>×</mo><mi>W</mi></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>W</mi><mo>⊕</mo><mi>W</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>e</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>e</mi><mo>×</mo><mi>e</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>e</mi><mo>⊕</mo><mi>e</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>X</mi><mo>⊕</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo>⊕</mo><mi>g</mi></mrow></mover></mtd> <mtd><mi>Y</mi><mo>⊕</mo><mi>Y</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Y</mi><mo>⊔</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>h</mi><mo>⊕</mo><mi>h</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>h</mi><mo>⊔</mo><mi>h</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi><mo>⊕</mo><mi>Z</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Z</mi><mo>⊔</mo><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>Z</mi></msub></mrow></mover></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ W &\overset{\Delta_W}{\longrightarrow}& W \times W &\overset{\simeq}{\longrightarrow}& W \oplus W \\ \downarrow^{\mathrlap{e}} && \downarrow^{\mathrlap{e \times e}} && \downarrow^{\mathrlap{e \oplus e}} \\ X &\overset{\Delta_X}{\to}& X \times X &\simeq& X \oplus X &\overset{f \oplus g}{\longrightarrow}& Y \oplus Y &\simeq& Y \sqcup Y &\overset{\nabla_X}{\to}& Y \\ && && && \downarrow^{\mathrlap{h \oplus h}} && \downarrow^{\mathrlap{h \sqcup h}} && \downarrow^{\mathrlap{h}} \\ && && && Z \oplus Z &\simeq& Z \sqcup Z &\overset{\nabla_Z}{\to}& Z } </annotation></semantics></math></div></div> <div class="num_prop" id="SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment"> <h6 id="proposition_3">Proposition</h6> <p>Given an additive category according to def. <a class="maruku-ref" href="#AdditiveCategory"></a>, then the enrichment in <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> which is induced on it via prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a> and prop. <a class="maruku-ref" href="#SemiAdditivityInducesAbelianMonoidEnrichment"></a> from its underlying <a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive category</a> structure coincides with the original enrichment.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the proof of prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a>, the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> on any object in an additive category is the sum of the two projections:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊕</mo><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X \,. </annotation></semantics></math></div> <p>Therefore (checking on <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a>, as in the proof of prop. <a class="maruku-ref" href="#ProductsAreBiproducts"></a>) for all morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \to Y</annotation></semantics></math> we have <a class="existingWikiWord" href="/nlab/show/commuting+squares">commuting squares</a> of the form</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>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded> <mpadded width="0"><mrow><msub><mo>∇</mo> <mi>Y</mi></msub><mo>=</mo></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>⊕</mo><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>⊕</mo><mi>g</mi></mrow></munder></mtd> <mtd><mi>Y</mi><mo>⊕</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f+g}{\longrightarrow}& Y \\ {}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}} \\ X \oplus X &\underset{f \oplus g}{\longrightarrow}& Y\oplus Y } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Prop. <a class="maruku-ref" href="#SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment"></a> says that being an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> is an extra <a class="existingWikiWord" href="/nlab/show/property">property</a> on a category, not extra <a class="existingWikiWord" href="/nlab/show/structure">structure</a>. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semiadditive+category">semiadditive 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/additive+%28%E2%88%9E%2C1%29-category">additive (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, Section 8 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006) [<a href="https://link.springer.com/book/10.1007/3-540-27950-4">doi:10.1007/3-540-27950-4</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf">pdf</a>]</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, <em><a class="existingWikiWord" href="/nlab/show/An+Introduction+to+Homological+Algebra">An Introduction to Homological Algebra</a></em>, Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9781139644136">doi:10.1017/CBO9781139644136</a>, <a href="https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li id="IntroductionToLinearCategories"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <em>Introduction to Linear Categories and Applications</em>, course lecture notes (1992) [<a href="https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lawvere-LinearCategories.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on additive categories is around def. 4.3 of</p> <ul> <li id="Beligiannis">Apostolos Beligiannis, <em>Homotopy theory of modules and Gorenstein rings</em>, Math. Scand. 89 (2001) (<a href="http://users.uoi.gr/abeligia/mathscand.pdf">pdf</a>)</li> </ul> <p>Formalization of additive categories as <a class="existingWikiWord" href="/nlab/show/univalent+categories">univalent categories</a> in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/unimath">unimath</a>, <em>Additive categories</em> [<a href="https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Additive.html">UniMath.CategoryTheory.Additive</a>]</li> </ul> <p>A characterisation of preadditive categories in terms of commutative monoids in cartesian multicategories is given in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Claudio+Pisani">Claudio Pisani</a>, <em>Sequential multicategories</em>, Theory and Applications of Categories 29.19 (2014), <a href="https://arxiv.org/abs/1402.0253">arXiv:1402.0253</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 25, 2024 at 09:12:00. 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