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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> <blockquote> <p>This entry is about the mathematical notion. For the commutative diagram editor, see <em><a class="existingWikiWord" href="/nlab/show/quiver+%28editor%29">quiver (editor)</a></em>.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="graph_theory">Graph theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/graph+theory">graph theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/graph">graph</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vertex">vertex</a>, <a class="existingWikiWord" href="/nlab/show/edge">edge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/omega-graph">omega-graph</a>, <a class="existingWikiWord" href="/nlab/show/hypergraph">hypergraph</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>, <a class="existingWikiWord" href="/nlab/show/n-quiver">n-quiver</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/category+of+simple+graphs">category of simple graphs</a></p> <h3 id="properties">Properties</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/graph+distance">graph distance</a></li> </ul> <h3 id="extra_properties">Extra properties</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a>, <a class="existingWikiWord" href="/nlab/show/directed+graph">directed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bipartite+graph">bipartite</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/planar+graph">planar</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a><a class="existingWikiWord" href="/nlab/show/directed+graph">directed graph</a> + <a class="existingWikiWord" href="/nlab/show/unit+law">unital</a> <a class="existingWikiWord" href="/nlab/show/associative">associative</a> <a class="existingWikiWord" href="/nlab/show/composition">composition</a> = <a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+graph">ribbon graph</a>, <a class="existingWikiWord" href="/nlab/show/combinatorial+map">combinatorial map</a>, <a class="existingWikiWord" href="/nlab/show/topological+map">topological map</a>, <a class="existingWikiWord" href="/nlab/show/child%27s+drawing">child's drawing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vertex+coloring">vertex coloring</a>, <a class="existingWikiWord" href="/nlab/show/clique">clique</a></p> </li> </ul> </div></div> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beilinson-Bernstein+localization">Beilinson-Bernstein localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#slick_definition'>Slick definition</a></li> <li><a href='#nutsandbolts_definitions'>Nuts-and-bolts definitions</a></li> </ul> <li><a href='#remarks'>Remarks</a></li> <li><a href='#terminology'>Terminology</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_free_categories'>Relation to free categories</a></li> <li><a href='#relation_to_representation_theory_of_algebras'>Relation to representation theory of algebras</a></li> <li><a href='#Classification'>Classification</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <ul> <li><a href='#enriched_quivers'>Enriched quivers</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>A <em>quiver</em> (<a href="#Gabriel72">Gabriel 72</a>) is a collection of <a class="existingWikiWord" href="/nlab/show/edges">edges</a> which may stretch between (ordered) <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of “points”, called <em><a class="existingWikiWord" href="/nlab/show/vertices">vertices</a></em>. Hence a quiver is a kind of <a class="existingWikiWord" href="/nlab/show/graph">graph</a>.</p> <p>The term “quiver” (German: <em>Köcher</em>) — for what in <a class="existingWikiWord" href="/nlab/show/graph+theory">graph theory</a> is (and was) known as <em><a class="existingWikiWord" href="/nlab/show/directed+graphs">directed graphs</a></em> (<a class="existingWikiWord" href="/nlab/show/directed+graph">directed</a> <a class="existingWikiWord" href="/nlab/show/pseudograph">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/multigraphs">multigraphs</a>, to be precise) — is due to <a href="#Gabriel72">Gabriel 1972</a>, where it says on the first page:</p> <blockquote> <p>For such a 4-tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo></mrow><annotation encoding="application/x-tex">\big[</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><munderover><mrow></mrow><munder><mo>⟶</mo><mi>t</mi></munder><mover><mo>⟶</mo><mi>s</mi></mover></munderover><mi>V</mi></mrow><annotation encoding="application/x-tex">E \underoverset{\underset{t}{\longrightarrow}}{\overset{s}{\longrightarrow}}{} V</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex">\big]</annotation></semantics></math> we propose the term <em>quiver</em>, and not <em>graph</em>, since there are already too many notions attached to the latter word.</p> </blockquote> <p id="ConceptWithAnAttitude"> Hence the notion “quiver” is a <em><a class="existingWikiWord" href="/nlab/show/concept+with+an+attitude">concept with an attitude</a></em>, indicating that one is interested in certain special constructions with these graphs, distinct from what <a class="existingWikiWord" href="/nlab/show/graph+theory">graph theory</a> typically cares about: Namely one is interested in their <em><a class="existingWikiWord" href="/nlab/show/quiver+representations">quiver representations</a></em>. In this way, “quiver” is really a term in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>; see also <a href="#DerksenWeyman05">Derksen & Weyman 2005, ftn 1</a>:</p> <blockquote> <p>The underlying motivations of quiver theory are quite different from those in the traditional graph theory. To emphasize this distinction, it is common in our context to use the word “quivers” instead of “graphs”.</p> </blockquote> <p>A key result in <a class="existingWikiWord" href="/nlab/show/quiver+representation">quiver representation</a>-theory is <em><a class="existingWikiWord" href="/nlab/show/Gabriel%27s+theorem">Gabriel's theorem</a></em>, also form <a href="#Gabriel72">Gabriel 1972</a>.</p> <h2 id="definitions">Definitions</h2> <h3 id="slick_definition">Slick definition</h3> <p>The <strong><a class="existingWikiWord" href="/nlab/show/walking+structure">walking</a> quiver</strong><sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category">category</a> with</p> <ul> <li> <p>one <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>, called the object of <em>vertices</em>;</p> </li> <li> <p>one object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math>, called the object of <em>edges</em>;</p> </li> <li> <p>two <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s, t\colon X_1 \to X_0</annotation></semantics></math>, called the <em>source</em> and <em>target</em>;</p> </li> <li> <p>together with <a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a>.</p> </li> </ul> <p>A <strong>quiver</strong> is 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>G</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">G\colon X \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>More generally, a <strong>quiver <a class="existingWikiWord" href="/nlab/show/internalization">in</a> a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></strong> is 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>G</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon X \to C</annotation></semantics></math>.</p> <p>The category of quivers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/Quiv">Quiv</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C)</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">C^{X}</annotation></semantics></math>, where:</p> <ul> <li> <p>objects are functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon X \to C</annotation></semantics></math>,</p> </li> <li> <p>morphisms are <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> between such functors.</p> </li> </ul> <p>In the basic case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the category Quiv(Set) is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">X^{op}</annotation></semantics></math>. So: the category of quivers, <a class="existingWikiWord" href="/nlab/show/Quiv">Quiv</a>, is the category of presheaves on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">X^{op}</annotation></semantics></math>.</p> <h3 id="nutsandbolts_definitions">Nuts-and-bolts definitions</h3> <p>A <strong>quiver</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> consists of two sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> (the set of <em>edges</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (the set of <em>vertices</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>) and two functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⇉</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">s, t\colon E \rightrightarrows V</annotation></semantics></math></div> <p>(the source and target functions). More generally, a <strong>quiver internal to a category</strong> (more simply, <em>in</em> a category) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> consists of two objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⇉</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">s, t\colon E \rightrightarrows V</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>V</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = (E, V, s, t)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>′</mo><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>′</mo><mo>,</mo><mi>V</mi><mo>′</mo><mo>,</mo><mi>s</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G' = (E', V', s', t')</annotation></semantics></math> are two quivers in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a <strong>morphism</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>G</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g\colon G \to G'</annotation></semantics></math> is a pair of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g_0\colon V \to V'</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g_1\colon E \to E'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>′</mo><mo>∘</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>0</mn></msub><mo>∘</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">s' \circ g_1 = g_0 \circ s</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>′</mo><mo>∘</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>0</mn></msub><mo>∘</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">t' \circ g_1 = g_0 \circ t</annotation></semantics></math>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/graph+theory">graph theory</a>, a quiver is often (cf. <a href="#DG2nd">p. 4</a> or <a href="#White84">Figure 2-2</a>) called a <strong>directed pseudograph</strong> (or some variation on that theme), but category theorists often just call them <strong>directed graphs</strong>.</p> <h2 id="remarks">Remarks</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>=</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_0 = G(X_0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mo>=</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_1 = G(X_1)</annotation></semantics></math>.</p> <ul> <li> <p>A quiver in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">X^{op}</annotation></semantics></math> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </li> <li> <p>A quiver is a <a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a> which is concentrated in the first two degrees.</p> </li> <li> <p>A quiver can have distinct edges <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>,</mo><mi>e</mi><mo>′</mo><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e,e'\in G_1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s(e) = s(e')</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>e</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t(e) = t(e')</annotation></semantics></math>. A quiver can also have loops, namely, edges with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s(e) = t(e)</annotation></semantics></math>.</p> </li> <li> <p>A quiver is <strong><a class="existingWikiWord" href="/nlab/show/complete+graph">complete</a></strong> if for any pair of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>v</mi><mo>′</mo><mo>∈</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">v,v'\in G_0</annotation></semantics></math>, there exists a unique directed edge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e\in G_1</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>v</mi><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>v</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">s(e) = v, t(e) = v'</annotation></semantics></math>.</p> </li> </ul> <h2 id="terminology">Terminology</h2> <p>Saying <em>quiver</em> instead of <em>directed (multi)graph</em> indicates focus on a certain set of operation intended on that graph. Notably there is the notion of a <a class="existingWikiWord" href="/nlab/show/quiver+representation">quiver representation</a>.</p> <p>Now, one sees that a <em>representation</em> of a graph <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in the sense of quiver representation is nothing but 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>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>:</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\rho\colon Q := F(G) \to Vect</annotation></semantics></math> from the <em>free category</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(G)</annotation></semantics></math> on the quiver <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <p>Given a graph <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with collection of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0</annotation></semantics></math> and collection of edges <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>, there is the free category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(G)</annotation></semantics></math> on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>s of edges in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> that fit together head-to-tail (also known as <em><a class="existingWikiWord" href="/nlab/show/path">path</a>s</em>). The composition operation in this free category is the concatenation of sequences of edges.</p> <p>Here we are taking advantage of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> (the category of small categories) and <a class="existingWikiWord" href="/nlab/show/Quiv">Quiv</a> (the category of directed graphs). Namely, any category has an underlying directed graph:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>→</mo><mi>Quiv</mi></mrow><annotation encoding="application/x-tex">U\colon Cat \to Quiv</annotation></semantics></math></div> <p>and the left adjoint of this functor gives the free category on a directed graph:</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 lspace="verythinmathspace">:</mo><mi>Quiv</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F\colon Quiv\to Cat </annotation></semantics></math></div> <p>Since this is the central operation on quivers that justifies their distinction from the plain concept of directed graph, we may adopt here the point of view that <em>quiver</em> is synonymous with <em>free category</em>.</p> <p>So a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of a quiver <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q = F(G)</annotation></semantics></math> is a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">R\colon Q \to Vect </annotation></semantics></math></div> <p>Concretely, such a thing assigns a vector space to each vertex of the graph <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, and a linear operator to each edge. Representations of quivers are fascinating things, with connections to ADE theory, quantum groups, string theory, and more.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_free_categories">Relation to free categories</h3> <p>It may be handy to <em>identify</em> a quiver with its free category. This can be justified in the sense that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>Quiv</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F\colon Quiv \to Cat</annotation></semantics></math> is an embedding (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/k-surjectivity">surjective</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k \gt 0</annotation></semantics></math>) on the <a class="existingWikiWord" href="/nlab/show/cores">cores</a>. In other words, <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> between quivers may be identified with <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalences</a> between free categories with no ambiguity.</p> <p>However, at the level of noninvertible morphisms, this doesn't work; while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a>, it is <em>not</em> <a class="existingWikiWord" href="/nlab/show/full+functor">full</a>. In other words, there are many <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between free categories that are not morphisms of quivers.</p> <p>Nevertheless, if we fix a quiver <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, then a <a class="existingWikiWord" href="/nlab/show/quiver+representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is precisely a functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(G)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (or equivalently a quiver morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(D)</annotation></semantics></math>), and we may well want to think of this as being a morphism (a <a class="existingWikiWord" href="/nlab/show/heteromorphism">heteromorphism</a>) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. As long as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is not itself a free category, this is unlikely to cause confusion.</p> <h3 id="relation_to_representation_theory_of_algebras">Relation to representation theory of algebras</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> a quiver, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">k Q</annotation></semantics></math> for the <em>path algebra</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> over a ground field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">k Q</annotation></semantics></math> is an algebra with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-basis given by finite composable sequences of arrows in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, including a “lazy path” of length zero at each vertex. The product of two paths composable paths is their composite, and the product of non-composable paths is zero.</p> <p>A module over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">k Q</annotation></semantics></math> is the same thing as a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, so the theory of representations of quivers can be viewed within the broader context of representation theory of (associative) algebras.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is acyclic, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">k Q</annotation></semantics></math> is finite-dimensional as a vector space, so in studying representations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, we are really studying representations of a finite dimensional algebra, for which many interesting tools exist (Auslander-Reiten theory, tilting, etc.).</p> <h3 id="Classification">Classification</h3> <p><a class="existingWikiWord" href="/nlab/show/Gabriel%27s+theorem">Gabriel's theorem</a> (<a href="#Gabriel72">Gabriel 72</a>) says that connected quivers with a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of <a class="existingWikiWord" href="/nlab/show/indecomposable+object">indecomposable</a> <a class="existingWikiWord" href="/nlab/show/quiver+representations">quiver representations</a> over an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a>are precisely the <em><a class="existingWikiWord" href="/nlab/show/Dynkin+quivers">Dynkin quivers</a></em>: those whose underlying <a class="existingWikiWord" href="/nlab/show/undirected+graph">undirected graph</a> is a <a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a> in the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE series</a>, and that the <a class="existingWikiWord" href="/nlab/show/indecomposable+object">indecomposable</a> <a class="existingWikiWord" href="/nlab/show/quiver+representations">quiver representations</a> are in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with the positive <a class="existingWikiWord" href="/nlab/show/root+%28in+representation+theory%29">roots</a> in the <a class="existingWikiWord" href="/nlab/show/root+system">root system</a> of the Dynkin diagram. (<a href="#Gabriel72">Gabriel 72</a>).</p> <h2 id="generalizations">Generalizations</h2> <h3 id="enriched_quivers">Enriched quivers</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> (or a <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>). A quiver <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched quiver</strong> if it has a collection of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(Q)</annotation></semantics></math> and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo>:</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">Mor: Ob(Q) \times Ob(Q) \to V</annotation></semantics></math> for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a, b \in Ob(Q)</annotation></semantics></math>. Quivers in the usual sense are enriched in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, while <a class="existingWikiWord" href="/nlab/show/graph">loop directed graphs</a>/<a class="existingWikiWord" href="/nlab/show/binary+relation">binary</a> <a class="existingWikiWord" href="/nlab/show/relation">endorelations</a> are quivers enriched in the category of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>. <a class="existingWikiWord" href="/nlab/show/n-quiver"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-quivers</a> are quivers enriched in the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-quivers.</p> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with <a class="existingWikiWord" href="/nlab/show/type+of+types">universes</a>, a <a class="existingWikiWord" href="/nlab/show/type">type</a> equipped with an <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> is a quiver type enriched in a universe <a class="existingWikiWord" href="/nlab/show/Type">Type</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quiver+representation">quiver representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/McKay+quiver">McKay quiver</a>, <a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quiver+gauge+theory">quiver gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knots-quivers+correspondence">knots-quivers correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/zigzag+persistence">zigzag persistence</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a></th><th><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magma">magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magmoid">magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>/<a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magma">unital magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroupoid">quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loop</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loopoid">loopoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroup">associative quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroupoid">associative quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/group">group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magma">flexible magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magma">alternative magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+category">absorption category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+category">cancellative category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rig">rig</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/CMon">CMon</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+category">C-star category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebroid">differential algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+algebra">flexible algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+algebra">alternative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+poset">monoidal poset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a></td></tr> <tr><td style="text-align: left;"><span class="newWikiWord">strict monoidal groupoid<a href="/nlab/new/strict+monoidal+groupoid">?</a></span></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-category">strict (2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>/<a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The concept of quiver in the context of <a class="existingWikiWord" href="/nlab/show/quiver+representation">quiver representation</a> (and their classification in <a class="existingWikiWord" href="/nlab/show/Gabriel%27s+theorem">Gabriel's theorem</a>) originates with</p> <ul> <li id="Gabriel72"><a class="existingWikiWord" href="/nlab/show/Peter+Gabriel">Peter Gabriel</a>, <em>Unzerlegbare Darstellungen. I</em>, Manuscripta Mathematica 6: 71–103, (1972) (<a href="https://link.springer.com/article/10.1007/BF01298413">doi:10.1007/BF01298413</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=332887">MR332887</a> <a href="http://dx.doi.org/10.1007/BF01298413">doi</a>)</li> </ul> <p>Some general-purpose references include</p> <ul> <li id="DerksenWeyman05"> <p>Harm Derksen, Jerzy Weyman, <em>Quiver representations</em>, Notices of the AMS (Feb 2005) [<a href="http://www.ams.org/notices/200502/fea-weyman.pdf">pdf</a>, full:<a href="http://www.ams.org/notices/200502/200502FullIssue.pdf">pdf</a>]</p> </li> <li> <p>William Crawley-Boevey, <em>Lectures on quiver representations</em> (<a href="https://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf">pdf</a>).</p> </li> <li> <p>Alistair Savage, <em>Finite-dimensional algebras and quivers</em> (<a href="http://www.arxiv.org/abs/math/0505082">arXiv:math/0505082</a>), <em>Encyclopedia of Mathematical Physics</em>, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320.</p> </li> </ul> <p>Quivers (referred to as <em>directed pseudographs</em>) were a tool in parts of the work of Ringel and Youngs in the second half the twentieth century to prove Heawood’s formula for every finite genus, cf. e.g. Fig. 2.3 the monograph</p> <ul> <li id="Ringel1974">Gerhard Ringel: <em>Map Color Theorem</em>. Springer. Grundlehren Band 209. 1974</li> </ul> <p>Beware that, strictly speaking, for Ringel, “quiver” means “embedded quiver” (into a given surface); in particular the author distinguishes between the two possible orientations of an embedded loop.</p> <p>Quivers embedded in surfaces are studied in:</p> <ul> <li id="White84">Arthur T. White: <em>Graphs, Groups and Surfaces</em>. North Holland. Completely revised and enlarged edition (1985)</li> </ul> <p>A special kind of quiver (finite, no loops, no parallel arcs) is treated in</p> <ul> <li id="DG2nd"> <p>Gregory Gutin, Jørgen Bang-Jensen: <em>Digraphs: Theory, Algorithms and Applications</em>. Springer Monographs in Mathematics. Second Edition (2009)</p> </li> <li id="GeneralizedGraphs"> <p><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>: <em>Qualitative Distinctions Between Some Toposes of Generalized Graphs</em>, Contemporary Mathematics 92 (1989)</p> </li> </ul> <p>Some introductory material on the relation between quivers (there called <em>multigraphs</em>) and categories can be found in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a>, <em>Notes on Category Theory with examples from basic mathematics</em>. (<a href="https://arxiv.org/abs/1912.10642">arXiv</a>)</li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>Also called “the elementary ”parallel process“ ” by <a class="existingWikiWord" href="/nlab/show/William+Lawvere">Lawvere</a> in <a href="#GeneralizedGraphs">p. 272</a>, “a category that has no compositions in it” by <a class="existingWikiWord" href="/nlab/show/William+Lawvere">Lawvere</a>, in his lecture at <a class="existingWikiWord" href="/nlab/show/Como">Como</a>. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on November 14, 2023 at 09:29:54. 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