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class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</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/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="frobenius_algebra">Frobenius algebra</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#AlgebraCoalgebra'>As associative algebra with coalgebra structure</a></li> <li><a href='#AsAssociativeAlgebraWithLinearForm'>As associative algebra with linear form</a></li> <li><a href='#FurtherDefinitions'>Further definitions</a></li> </ul> <li><a href='#types_of_frobenius_algebras'>Types of Frobenius algebras</a></li> <ul> <li><a href='#commutative_frobenius_algebras'>Commutative Frobenius algebras</a></li> <li><a href='#symmetric_frobenius_algebras'>Symmetric Frobenius algebras</a></li> <li><a href='#special_frobenius_algebras'>Special Frobenius algebras</a></li> <li><a href='#frobenius_algebras'>†-Frobenius algebras</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#props_for_frobenius_algebras'>PROPs for Frobenius algebras</a></li> <li><a href='#RelationTo2DTQFT'>Classification of 2d TQFT</a></li> <li><a href='#NormalFormAndSpiderTheorem'>Normal form and “Spider theorem”</a></li> <li><a href='#frobenius_algebras_in_polycategories'>Frobenius algebras in polycategories</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>Frobenius algebra</em> is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> equipped with the <a class="existingWikiWord" href="/nlab/show/structures">structures</a> both of an <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and of an a <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> in a compatible way, where the compatibility is different from (more “topological” than) that in a <a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>/<a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>:</p> <p>The <em>Frobenius property</em> on an algebra/coalgebra <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> states that all ways of using <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> product operations and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> coproduct operations to map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup><mo>→</mo><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex">A^{\otimes^{n+1}} \to A^{\otimes^{m+1}}</annotation></semantics></math> are equal.</p> <p>More generally, Frobenius algebras can be defined <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (and even in any <a class="existingWikiWord" href="/nlab/show/polycategory">polycategory</a>) in which case they are sometimes called <em>Frobenius monoids</em>. (For example a Frobenius monoid in an <a class="existingWikiWord" href="/nlab/show/endofunctor">endo</a>-<a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> is a <em><a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a></em>.)</p> <p>After their original introduction in pure <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> in the 1930s, Frobenius algebras later attracted much attention for the role they play in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, in fact for two rather different roles they play there:</p> <ul> <li> <p><strong>in <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a></strong> Frobenius algebras encode the structure of <a class="existingWikiWord" href="/nlab/show/2-dimensional+TQFTs">2-dimensional TQFTs</a>:</p> <p>Her the underlying vector space of the Frobenius algebra is identified (cf. <em><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></em>) with the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> over a connected 1-manifold (a <a class="existingWikiWord" href="/nlab/show/circle">circle</a> for the “<a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a>” states or a <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> for the “<a class="existingWikiWord" href="/nlab/show/open+string">open string</a>” states), the product and coproduct operations are the <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a> on a <a class="existingWikiWord" href="/nlab/show/trinion">trinion</a> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> (“<a class="existingWikiWord" href="/nlab/show/pair+of+pants">pair of pants</a>”) read in either direction, and the Frobenius property reflects the <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> between <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> obtained by gluing <a class="existingWikiWord" href="/nlab/show/trinions">trinions</a> along their boundary components.</p> <p>Beware that this is similar to but subtly different from the “non-compact” <a class="existingWikiWord" href="/nlab/show/2d+TQFTs">2d TQFTs</a> commonly known as the “<a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>” (the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> of <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a>-fame) which are <em>not</em> described by Frobenius algebras (because here one ex-cludes the “cap” cobordism and thus the <a class="existingWikiWord" href="/nlab/show/trace">trace</a>-operation hence the finite-dimensionality on the algebra of states) but by “<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+objects">Calabi-Yau objects</a>” (for more see <em><a href="TCFT#Definition">here</a></em> at <em><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></em>.)</p> <p>On the other hand, the <a class="existingWikiWord" href="/nlab/show/FRS-theorem">FRS-theorem</a> shows that all <em><a class="existingWikiWord" href="/nlab/show/rational+2d+conformal+field+theories">rational 2d conformal field theories</a></em> – hence the compact-target-space sectors of the <em>physical</em> <a class="existingWikiWord" href="/nlab/show/string">string</a> (such as <a class="existingWikiWord" href="/nlab/show/WZW+models">WZW models</a> and <a class="existingWikiWord" href="/nlab/show/Gepner+models">Gepner models</a>) – <em>are</em> described by <a class="existingWikiWord" href="/nlab/show/Frobenius+monoids">Frobenius monoids</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a> (MTCs): Here the ambient MTC encodes the local non-topological (“chiral”) component of the <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> (its spaces of <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a>) while the internal Frobenius monoid encodes the remaining topological “<a class="existingWikiWord" href="/nlab/show/sewing+constraints">sewing constraints</a>”.</p> </li> </ul> <p>Later</p> <ul> <li> <p><strong>in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a></strong> (such as now in the <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a>) Frobenius algebras encode aspects of the <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>-process, including its associated <a class="existingWikiWord" href="/nlab/show/wavefunction+collapse">wavefunction collapse</a> (see <a href="quantum+information+theory+via+dagger-compact+categories#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams">here</a> at <em><a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">quantum information theory via dagger-compact categories</a></em>).</p> <p>One way to understand how this comes about (not the original way, though) is to observe that in the <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a> which is relevant for <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> the <a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a> and <a class="existingWikiWord" href="/nlab/show/coreader+comonad">coreader comonad</a> for a given <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/possible+worlds">possible</a> measurement outcomes merge to a single <a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a> (the <em><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></em>, see there for more) which is in turn identified with the <a class="existingWikiWord" href="/nlab/show/writer+monad">writer monad</a>/<a class="existingWikiWord" href="/nlab/show/cowriter+comonad">cowriter comonad</a> for a canonical Frobenius algebra structure on the <a class="existingWikiWord" href="/nlab/show/linear+span">linear span</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> </li> </ul> <p>Curiously, these two different roles that Frobenius algebras play in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> are not closely related, in fact they seem somewhat orthogonal to each other.</p> <h2 id="definition">Definition</h2> <p>There are a number of equivalent definitions of the concept of Frobenius algebra.</p> <p>The original definition is an associative algebra with a suitable <a class="existingWikiWord" href="/nlab/show/linear+form">linear form</a> on it.</p> <ul> <li><a href="#AsAssociativeAlgebraWithLinearForm">As an associative algebra with linear form</a>.</li> </ul> <p>In the context of <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> what crucially matters is that this is equivalent to an associative algebra structure with a compatible coalgebra structure</p> <ul> <li><a href="#AlgebraCoalgebra">As an associative algebra with compatible coalgebra structure</a>.</li> </ul> <p>There are</p> <ul> <li><a href="#FurtherDefinition">Further equivalent definitions</a></li> </ul> <h3 id="AlgebraCoalgebra">As associative algebra with coalgebra structure</h3> <p> <div class='num_defn' id='FrobeniusAlgebra'> <h6>Definition</h6> <p>A <em>Frobenius algebra</em> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</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>,</mo><mo>⊗</mo><mo>,</mo><mi>𝟙</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, \mathbb{1})</annotation></semantics></math> (for instance <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> with the usual <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>) is</p> <ol> <li> <p>an <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><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/unit">unit</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>𝟙</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta \,\colon\, \mathbb{1} \to A</annotation></semantics></math>,</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/counit">counit</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>→</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\epsilon \,\colon\, A \to \mathbb{1}</annotation></semantics></math>,</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mu \,\colon\, A \otimes A \to A</annotation></semantics></math>.</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/comultiplication">comultiplication</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>→</mo><mi>A</mi><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\delta \,\colon\, A \to A \otimes A</annotation></semantics></math>,</p> </li> </ul> </li> </ol> <p>such that:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \mu, \eta)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} = Vect</annotation></semantics></math>),</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \delta, \epsilon)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/comonoid">comonoid</a> (a <a class="existingWikiWord" href="/nlab/show/coassociative+coalgebra">coassociative coalgebra</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} = Vect</annotation></semantics></math>),</p> </li> <li> <p>the <strong>Frobenius laws</strong> hold: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>A</mi></msub><mo>⊗</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>δ</mi><mo>⊗</mo><msub><mi>id</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><mo>∘</mo><mi>μ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>⊗</mo><msub><mi>id</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>A</mi></msub><mo>⊗</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(id_A \otimes \mu) \circ (\delta \otimes id_A) = \delta \circ \mu = (\mu \otimes id_A) \circ (id_A \otimes \delta)</annotation></semantics></math>.</p> </li> </ol> <p></p> </div> </p> <p>In terms of <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>, Def. <a class="maruku-ref" href="#FrobeniusAlgebra"></a> says:</p> <p><img src="/nlab/files/frobenius_algebra.jpg" alt="string diagrams for the Frobenius algebra axioms" /></p> <p>The first line here shows the associative law and left/right unit laws for a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>. The second line shows the coassociative law and left/right counit laws for a <a class="existingWikiWord" href="/nlab/show/comonoid">comonoid</a>. The third line shows the Frobenius laws.</p> <p>In fact, although this seems rarely to be remarked, the two Frobenius laws can be replaced by the single axiom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>δ</mi><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1 \otimes \mu) \circ (\delta \otimes 1) = (\mu \otimes 1) \circ (1 \otimes \delta)</annotation></semantics></math>. Here is a proof in string diagram notation that this axiom implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>δ</mi><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu</annotation></semantics></math> (taken from <a href="#PastroStreet2008">Pastro-Street 2008</a>):</p> <p><img src="/nlab/files/frobenius_axioms_one_to_two.png" alt="Proof that one Frobenius axiom implies both the usual ones" /></p> <p>Here the first and fifth steps use the counitality property of the comonoid structure, the second and fourth steps use the assumed axiom (once in each direction), and the third step uses coassociativity.</p> <h3 id="AsAssociativeAlgebraWithLinearForm">As associative algebra with linear form</h3> <p>Frobenius algebras were originally formulated in the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> with the following equivalent definition:</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>Frobenius algebra</strong> is a <a class="existingWikiWord" href="/nlab/show/unitality">unital</a>, <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \mu, \eta)</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/linear+form">linear form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\epsilon \colon A \rightarrow k</annotation></semantics></math> – to becalled a <strong>Frobenius form</strong> – such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\epsilon \circ \mu</annotation></semantics></math> is a non-degenerate <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>. I.e. the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>u</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>v</mi><mo>↦</mo><mi>ϵ</mi><mo>∘</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>v</mi><mo>⊗</mo><mi>u</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> u \;\mapsto\; \big( v \mapsto \epsilon \circ \mu(v \otimes u) \big) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <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> with its <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math>.</p> </div> <p>From this definition it is easy to see that every Frobenius algebra <a class="existingWikiWord" href="/nlab/show/internalization">in</a> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> is necessarily <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a>.</p> <h3 id="FurtherDefinitions">Further definitions</h3> <p>There are about a dozen equivalent definitions of a Frobenius algebra. <a href="#Street2004">Ross Street (2004)</a> lists most of them.</p> <h2 id="types_of_frobenius_algebras">Types of Frobenius algebras</h2> <h3 id="commutative_frobenius_algebras">Commutative Frobenius algebras</h3> <p>We can define ‘commutative’ Frobenius algebras in any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. Namely, a Frobenius algebra is <strong>commutative</strong> if its associated monoid is commutative — or equivalently, if its associated comonoid is cocommutative.</p> <h3 id="symmetric_frobenius_algebras">Symmetric Frobenius algebras</h3> <p>We can define ‘commutative’ or ‘symmetric’ Frobenius algebras in any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. A Frobenius algebra <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> is <strong>symmetric</strong> if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mi>μ</mi><mo>∘</mo><msub><mi>S</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>=</mo><mi>ϵ</mi><mi>μ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\epsilon \mu \circ S_{A,A} = \epsilon \mu \, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">S_{A,A} : A \otimes A \to A \otimes A</annotation></semantics></math> is the symmetry, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mi>μ</mi></mrow><annotation encoding="application/x-tex">\epsilon\mu</annotation></semantics></math> is the nondegenerate pairing induced as above from the multiplication and the counit. Any commutative Frobenius algebra is symmetric, but not conversely: for example the algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> matrices with entries in a field, with its usual trace as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, is symmetric but not commutative when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math>.</p> <p>A theorem of <a href="#Eilenberg1955">Eilenberg & Nakayama (1955)</a> says that in the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <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>, an algebra <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> can be equipped with the structure of a symmetric Frobenius algebra if (but not only if) it is <strong><a class="existingWikiWord" href="/nlab/show/separable+algebra">separable</a></strong>, meaning that for any 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> <a class="existingWikiWord" href="/nlab/show/field+extension">extending</a> <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>K</mi></mrow><annotation encoding="application/x-tex">A \otimes_k K</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/semisimple+algebra">semisimple</a> algebra over <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>.</p> <h3 id="special_frobenius_algebras">Special Frobenius algebras</h3> <p> <div class='num_defn' id='SpecialFrobeniusAlgebras'> <h6>Definition</h6> <p></p> <p>The term “special Frobenius algebra” is not used consistently in the literature, but the main point is to require that the product is the <a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a> to the coproduct, possibly up to a non-vanishing element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\beta_A</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>:</p> <div class="maruku-equation" id="eq:ProdLeftInverseToCoprodUpToScaling"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>prod</mi><mo>∘</mo><mi>coprod</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mi>β</mi> <mi>A</mi></msub><mo>⋅</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> prod \circ coprod \,=\, \beta_A \cdot id </annotation></semantics></math></div> <p>This condition removes the <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a>-contribution in normal forms of “connected maps”, up to rescaling, see <a href="#NormalFormAndSpiderTheorem">below</a>.</p> <p>But then it is natural to also require the counit to be left inverse to the unit, up to (another) non-vanishing ground field element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\beta_1</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:CounitLeftInverseToUnitUpTOScaling"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>counit</mi><mo>∘</mo><mi>unit</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mi>β</mi> <mn>1</mn></msub><mo>⋅</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> counit \circ unit \,=\, \beta_1 \cdot id </annotation></semantics></math></div> <p>which also removes the disconnected “vaccum”-contribution, up to scaling.</p> <p>The two conditions <a class="maruku-eqref" href="#eq:ProdLeftInverseToCoprodUpToScaling">(1)</a> and <a class="maruku-eqref" href="#eq:CounitLeftInverseToUnitUpTOScaling">(2)</a> on a Frobenius algebra are taken to be the definition of “special Frobenius” in <a href="#FRS-theorem+on+rational+2d+CFT#FuchsRunkelSchweigert02">Fuchs, Runkel & Schweigert 2002, Def. 3.4 (i)</a>:</p> <div style="margin: -20px 10px 20px 10px"> <img src="/nlab/files/FRS-SpecialFrobeniusDiagram.jpg" width="400px" /> </div> <p>However, other authors take the condition for “special Frobenius” to be the single constraint that</p> <div class="maruku-equation" id="eq:CoprodLetfInverseToProd"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>prod</mi><mo>∘</mo><mi>coprod</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>id</mi></mrow><annotation encoding="application/x-tex"> prod \circ coprod \,=\, id </annotation></semantics></math></div> <p>(eg. <a href="quantum+information+theory+via+dagger-compact+categories#CoeckePavlović08">Coecke & Pavlović 2008, p. 17</a>, <a href="https://arxiv.org/abs/1202.6380">Fauser 2012, Def. 3.7</a>)</p> <p>and may instead call <a class="maruku-eqref" href="#eq:ProdLeftInverseToCoprodUpToScaling">(1)</a> the condition to be <em>trivially connected</em>.</p> <p>Beware that both of these conventions are adhered to by no small number of authors.</p> </div> </p> <p>In the category of vector spaces, any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> of an associative unital algebra gives a left multiplication map</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>L</mi> <mi>a</mi></msub><mo>:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>a</mi><mi>b</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ L_a : &A &\to& A \\ &b &\mapsto& a b } </annotation></semantics></math></div> <p>which in turn gives a bilinear pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">g: A \times A \to k</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>a</mi></msub><msub><mi>L</mi> <mi>b</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g(a,b) = tr(L_a L_b) </annotation></semantics></math></div> <p>One can show that the algebra <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> can be equipped with the structure of a special Frobenius algebra if and only if <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> is nondegenerate, i.e., if there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">A \to A^*</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><mi>g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a \mapsto g(a, -) </annotation></semantics></math></div> <p>In this case, there is just one way to make <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> into a special Frobenius algebra, namely by taking the counit to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \epsilon(a) = tr(L_a) </annotation></semantics></math></div> <p>(In any Frobenius algebra, the unit, multiplication and counit determine the comultiplication.)</p> <p>In fact, all the results of the previous paragraph generalize to Frobenius algebras in any symmetric monoidal category, since the proofs can be done using string diagrams.</p> <p>An associative unital algebra for which the bilinear pairing <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> is nondegenerate is called <strong>strongly separable</strong>. So, any strongly separable algebra becomes a special Frobenius algebra in a unique way. For more details, see <a class="existingWikiWord" href="/nlab/show/separable+algebra">separable algebra</a> and <a href="#Aguiar2000">Aguiar (2000)</a>.</p> <p>To get a feeling for some of the concepts we are discussing, an example is helpful. The group algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> of a finite group <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> is always separable but strongly separable if and only if the order 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> is invertible in the 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>. By the results mentioned, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> can always be made into a symmetric Frobenius algebra, but only into a special Frobenius algebra when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|G|</annotation></semantics></math> is invertible in <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>.</p> <p>To see this, we can check that the group algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> becomes a symmetric Frobenius algebra if we define the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\epsilon: k[G] \to k</annotation></semantics></math> to pick out the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">1 \in G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msub><mi>a</mi> <mi>g</mi></msub><mspace width="thinmathspace"></mspace><mi>g</mi><mo>↦</mo><msub><mi>a</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon : \sum_{g \in G} a_g \, g \mapsto a_1 \,. </annotation></semantics></math></div> <p>But when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|G|</annotation></semantics></math> is invertible in <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>, we can check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> becomes a <em>special</em> symmetric Frobenius algebra if we normalize the counit as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msub><mi>a</mi> <mi>g</mi></msub><mspace width="thinmathspace"></mspace><mi>g</mi><mo>↦</mo><mfrac><mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon : \sum_{g \in G} a_g \, g \mapsto \frac{a_1}{|G|} \, .</annotation></semantics></math></div> <p>We should warn the reader that <a href="#Rosebrugh2005">Rosebrugh et al (2005)</a> call a special Frobenius algebra ‘separable’. This usage conflicts with the standard definition of a <a class="existingWikiWord" href="/nlab/show/separable+algebra">separable algebra</a> in the category of vector spaces over a field, so we suggest avoiding it.</p> <h3 id="frobenius_algebras">†-Frobenius algebras</h3> <p>If a Frobenius algebra lives in a monoidal <a class="existingWikiWord" href="/nlab/show/%E2%80%A0-category">†-category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>δ</mi><msup><mo stretchy="false">)</mo> <mo>†</mo></msup><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">(\delta)^\dagger = \mu</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mo>†</mo></msup><mo>=</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">(\epsilon)^\dagger = \eta</annotation></semantics></math>, then it is said to be a <strong>†-Frobenius algebra</strong>. These crop up in the theory of 2d <a class="existingWikiWord" href="/nlab/show/TQFTs">TQFTs</a>, and also in the foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">quantum theory</a>.</p> <h2 id="Examples">Examples</h2> <p> <div class='num_remark' id='VectorSpaceWithLinearBasis'> <h6>Example</h6> <p><br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">W \,\in\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/FiniteSets">FiniteSets</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mo>≡</mo><munder><mo>⊕</mo><mi>W</mi></munder><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W \equiv \underset{W}{\oplus} \mathbb{1}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\mathbb{1}</annotation></semantics></math> – with its inherited <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> – becomes a <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> by taking the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/counit">counit</a> to be the canonical <a class="existingWikiWord" href="/nlab/show/sum">sum</a>-operation;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> to be map induced by the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> on indices.</p> </li> </ul> <p>In components, if we choose the canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Span</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">{</mo><mrow><mo>|</mo><mi>w</mi><mo>⟩</mo></mrow><msub><mo maxsize="1.2em" minsize="1.2em">}</mo> <mrow><mi>w</mi><mo lspace="verythinmathspace">:</mo><mi>W</mi></mrow></msub><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \mathrm{Q}W \;\simeq\; Span \Big( \big\{ \left\vert w \right\rangle \big\}_{w \colon W} \Big) </annotation></semantics></math></div> <p>then on basis vectors these structure maps are expressed as follows (where “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math>” is the <a class="existingWikiWord" href="/nlab/show/Kronecker+delta">Kronecker delta</a>):</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="90.199pt" height="41.624pt" viewBox="0 0 90.199 41.624" version="1.2"> <defs> <g> <symbol overflow="visible" id="IQRqErJhtSNuOfcYVu9WwmDcTn4=-glyph0-0"> <path style="stroke:none;" d=""></path> 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width="thinmathspace"></mspace><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{K} \,=\, \mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, the corresponding <a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mi>Mod</mi><mo>→</mo><mi>ℂ</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W \otimes (\text{-}) \,\colon\, \mathbb{C}Mod \to \mathbb{C}Mod</annotation></semantics></math> (or rather <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>FinDimHilb</mi><mo>→</mo><mi>FinDimHilb</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W \otimes (\text{-}) \,\colon\, FinDimHilb \to FinDimHilb</annotation></semantics></math>) has been proposed [<a href="quantum+measurement#CoeckePavlović08">Coecke & Pavlović 2008</a>] to reflect aspects of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> in the context of <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> and is used as such in the <em><a class="existingWikiWord" href="/nlab/show/zxCalculus">zxCalculus</a></em> (where the Frobenius property is embodied by “spider diagrams”). Various authors discuss the Frobenius algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W</annotation></semantics></math> in this context, see the references <a href="quantum+information+theory+via+dagger-compact+categories#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams">there</a>.</p> <p>From another perspective on the same phenomenon (discussed at <em><a class="existingWikiWord" href="/nlab/show/quantum+circuits+via+dependent+linear+types">quantum circuits via dependent linear types</a></em>): The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/writer+monad">writer monad</a> underlying the <a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi><mi>Mod</mi><mo>→</mo><mi>𝕂</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathrm{Q}W \otimes (-) \,\colon\, \mathbb{K}Mod \to \mathbb{K}Mod</annotation></semantics></math> is a linear version of the <a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a>, as such discussed at <em><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></em>. Regarded as a <a class="existingWikiWord" href="/nlab/show/monad+in+computer+science">computational effect</a> its <a class="existingWikiWord" href="/nlab/show/Kleisli+morphisms">Kleisli morphisms</a> encode <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> with an “indefiniteness”-effect (in the sense of <a href="necessity+and+possibility#ModalQuantumLogic">quantum modal logic</a>) whose “<a href="monad+in+computer+science#MonadModulesInIdeaSection">handling</a>” is the process of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>.</p> <p>In particular, the <a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebras</a> (hence also the corresponding <a class="existingWikiWord" href="/nlab/show/coalgebra+over+a+comonad">coalgebras</a>) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Q</mi><mi>W</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Q}W \otimes (\text{-})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/dependent+linear+types"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>W</mi> </mrow> <annotation encoding="application/x-tex">W</annotation> </semantics> </math>-dependent <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">\mathbb{K}</annotation> </semantics> </math>-linear types</a>, namely the <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">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>, hence the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/indexed+sets">indexed sets</a> of <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">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, understood as residual <a class="existingWikiWord" href="/nlab/show/spaces+of+quantum+states">spaces of quantum states</a> after <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">collapse</a> following the measurement result <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math>.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A Frobenius algebra <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> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is an <a class="existingWikiWord" href="/nlab/show/dual+object">object dual</a> to itself.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/monoidal+unit">monoidal unit</a>. To say <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> is dual to itself means there are maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">e: I \to A \otimes A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">p: A \otimes A \to I</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a> hold. The maps are defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>=</mo><mo stretchy="false">(</mo><mi>I</mi><mover><mo>→</mo><mi>η</mi></mover><mi>A</mi><mover><mo>→</mo><mi>δ</mi></mover><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="2em"></mspace><mi>p</mi><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mover><mo>→</mo><mi>μ</mi></mover><mi>A</mi><mover><mo>→</mo><mi>ϵ</mi></mover><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e = (I \stackrel{\eta}{\to} A \stackrel{\delta}{\to} A \otimes A), \qquad p = (A \otimes A \stackrel{\mu}{\to} A \stackrel{\epsilon}{\to} I)</annotation></semantics></math></div> <p>and one of the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a> uses one of the Frobenius laws and unit and counit axioms to derive the following <a class="existingWikiWord" href="/nlab/show/commutative+diagram">commutative diagram</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mn>1</mn><mo>⊗</mo><mi>η</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mn>1</mn><mo>⊗</mo><mi>δ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mn>1</mn></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mi>μ</mi></mpadded></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mrow><mi>μ</mi><mo>⊗</mo><mn>1</mn></mrow></mpadded></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>δ</mi></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mn>1</mn></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mrow><mi>ϵ</mi><mo>⊗</mo><mn>1</mn></mrow></mpadded></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ A & \stackrel{1 \otimes \eta}{\to} & A \otimes A & \stackrel{1 \otimes \delta}{\to} & A \otimes A \otimes A \\ & {}_1\searrow & \downarrow \mathrlap{\mu} & & \downarrow \mathrlap{\mu \otimes 1} \\ & & A & \overset{\delta}{\to} & A \otimes A \\ & & & {}_{1}\searrow & \downarrow \mathrlap{\epsilon \otimes 1} \\ & & & & A } </annotation></semantics></math></div> <p>The other <a class="existingWikiWord" href="/nlab/show/triangle+identity">triangle identity</a> uses the other Frobenius law and unit and counit axioms.</p> </div> <p>As a result, we see that in the monoidal category <a class="existingWikiWord" href="/nlab/show/Mod"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Mod</mi> <mi>k</mi></msub> </mrow> <annotation encoding="application/x-tex">Mod_k</annotation> </semantics> </math></a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <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>, Frobenius algebras <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> considered as modules over <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> are <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> and <a class="existingWikiWord" href="/nlab/show/projective+module">projective</a>. This is because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo>−</mo></mrow><annotation encoding="application/x-tex">A \otimes_k -</annotation></semantics></math>, being <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint</a> to itself, is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">therefore preserves all colimits</a>. That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo>−</mo></mrow><annotation encoding="application/x-tex">A \otimes_k -</annotation></semantics></math> preserves arbitrary small coproducts means <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> is finitely generated over <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>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo>−</mo></mrow><annotation encoding="application/x-tex">A \otimes_k-</annotation></semantics></math> preserves coequalizers means <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> is projective over <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>.</p> <ul> <li> <p>Every Frobenius algebra <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> is a <a class="existingWikiWord" href="/nlab/show/quasi-Frobenius+algebra">quasi-Frobenius algebra</a>: projective and injective left (right) modules over <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> coincide.</p> </li> <li> <p>Every Frobenius algebra <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> is a <span class="newWikiWord">pseudo-Frobenius algebra<a href="/nlab/new/pseudo-Frobenius+algebra">?</a></span>: <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> is an injective cogenerator in the category of left (right) <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>-modules.</p> </li> </ul> <p>Frobenius algebras are closely connected with <a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunctions">ambidextrous adjunctions</a>. For example, a <strong><a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a></strong> on 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> is by definition a Frobenius monoid in the monoidal category of <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> (with monoidal product given by endofunctor composition), and if we have a pair of <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">F \dashv U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊣</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">U \dashv F</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><mi>U</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">M = U F</annotation></semantics></math> carries a monad structure and a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> structure and the Frobenius laws are satisfied, a fact most easily seen by using <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>.</p> <h3 id="props_for_frobenius_algebras">PROPs for Frobenius algebras</h3> <p>Certain kinds of Frobenius algebras have nice <a class="existingWikiWord" href="/nlab/show/PROPs">PROPs</a> or <a class="existingWikiWord" href="/nlab/show/PRO">PROs</a>. The PRO for Frobenius algebras is the monoidal category of planar thick tangles, as noted by <a href="#Lauda2006">Lauda (2006)</a> and illustrated here:</p> <p><img src="/nlab/files/frobenius_algebra.jpg" alt="string diagrams for the Frobenius algebra axioms" /></p> <p>Lauda and Pfeiffer <a href="#Lauda2008">Lauda (2008)</a> showed that the PROP for <em>symmetric</em> Frobenius algebras is the category of ‘topological open strings’, since it obeys this extra axiom:</p> <p><img src="/nlab/files/symmetric_frobenius_algebra_law.jpg" alt="string diagram for the "symmetric" law in a Frobenius algebra" /></p> <p>The PROP for <em>commutative</em> Frobenius algebras is <span class="newWikiWord">2Cob<a href="/nlab/new/2Cob">?</a></span>, as noted by many people and formally proved in <a href="#Abrams96">Abrams (1996)</a>. This means that any commutative Frobenius algebra gives a 2d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>. See <a href="#Kock2006">Kock (2006)</a> for a history of this subject and <a href="#Kock2004">Kock (2004)</a> for a detailed introduction. In 2Cob, the circle is a Frobenius algebra. The monoid laws look like this:</p> <p><img src="/nlab/files/monoid_laws.jpg" alt="diagrams for the monoid laws in 2Cob" /></p> <p>The comonoid laws look like this:</p> <p><img src="/nlab/files/comonoid_laws.jpg" alt="diagrams for the comonoid laws in 2Cob" /></p> <p>The Frobenius laws look like this:</p> <p><img src="/nlab/files/frobenius_laws.jpg" alt="diagrams for the Frobenius laws in 2Cob" /></p> <p>and the commutative law looks like this:</p> <p><img src="/nlab/files/commutative_law.jpg" alt="diagrams for the commutative law in 2Cob" /></p> <p>The PROP for <em>special</em> commutative Frobenius algebras is Cospan(FinSet), as proved by Rosebrugh, Sabadini and Walters. This is worth comparing to the PROP for commutative <a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebras</a>, which is Span(FinSet). For details, see <a href="#Rosebrugh2005">Rosebrugh et al (2005)</a>, and also <a href="#Lack2004">Lack (2004)</a>.</p> <p>A special commutative Frobenius algebra gives a 2d TQFT that is insensitive to the genus of a 2-manifold, since in terms of pictures, the ‘specialness’ axioms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∘</mo><mi>δ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">m \circ \delta = 1</annotation></semantics></math> says that</p> <p><img src="/nlab/files/special_law.jpg" alt="string diagram for the "special" law in a Frobenius algebra" /></p> <h3 id="RelationTo2DTQFT">Classification of 2d TQFT</h3> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> (“<a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>”)</th><th><a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a></th><th><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> structure on <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>+(<a href="2d+TQFT#Abrams96">Abrams 96</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/trace">trace</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \to Z(A)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/Cardy+condition">Cardy condition</a></td><td style="text-align: left;">(<a href="#2d+TQFT#Lazaroiu00">Lazaroiu 00</a>, <a href="2d+TQFT#MooreSegal02">Moore-Segal 02</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+algebra">Calabi-Yau A-∞ algebra</a></td><td style="text-align: left;">(<a href="2d+TQFT#Kontsevich95">Kontsevich 95</a>, <a href="2d+TQFT#Costello04">Costello 04</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with various <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with various <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and with <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk</a> sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a> with <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a>(<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math>) over <a class="existingWikiWord" href="/nlab/show/field">field</a> <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></td><td style="text-align: left;">separable symmetric <a class="existingWikiWord" href="/nlab/show/Frobenius+algebras">Frobenius algebras</a></td><td style="text-align: left;">(<a href="2d+TQFT#SchommerPries11">SchommerPries 11</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a>(<a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+algebra">Calabi-Yau A-∞ algebra</a></td><td style="text-align: left;">(<a href="2d+TQFT#Lurie09">Lurie 09, section 4.2</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{S})</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math></td><td style="text-align: left;">(<a href="2d+TQFT#Lurie09">Lurie 09, section 4.2</a>)</td></tr> </tbody></table> </div> <h3 id="NormalFormAndSpiderTheorem">Normal form and “Spider theorem”</h3> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(normal form for 2d cobordisms)</strong> <br /> Every <em><a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a></em> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> with</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>in</mi></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n_{in} \gt 0</annotation></semantics></math> ingoing boundary components</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>out</mi></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n_{out} \gt 0</annotation></semantics></math> outgoing boundary components</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> <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> </li> </ul> <p>is equivalent (<a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a>) to the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>in</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{in}-1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/trinions">trinions</a> regarded as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \times S^1 \to S^1</annotation></semantics></math></p> </li> <li> <p><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> 2-punctured <a class="existingWikiWord" href="/nlab/show/tori">tori</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \to S^1</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>out</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{out}-1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/trinions">trinions</a> regared as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \to S^1 \times S^1</annotation></semantics></math>,</p> </li> </ul> <div style="margin: -20px 10px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/Abrams-2dCobordismNormalForm.jpg" width="600px" /> <figcaption style="text-align: center">(from <a href="#Abrams96">Abrams 1996, Fig. 3</a>)</figcaption> </figure> </div> <p>as shown in the following example for</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>in</mi></msub><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n_{in} = 5</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">g = 4</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>out</mi></msub><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n_{out} = 4</annotation></semantics></math></p> </li> </ul> <div style="margin: -20px 10px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/2dCObordismNormalForm.jpg" width="600px" /> <figcaption style="text-align: center">(from <a href="#Kock2004">Kock 2004, p. 64</a>)</figcaption> </figure> </div> <p></p> </div> This is discussed in <a href="#Abrams96">Abrams 1996, Prop. 12</a>; <a href="#Kock2004">Kock 2004, §1.4.16</a>.</p> <p>By the <a href="#RelationTo2DTQFT">relation</a> between 2d cobordism and Frobenius algebras this means equivalently that:</p> <p> <div class='num_cor' id='NormalFormForFrobeniusMaps'> <h6>Corollary</h6> <p><strong>(normal form for Frobenius maps)</strong> <br /> Given a Frobenius algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>unit</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>counit</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>prod</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>coprod</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big( A ,\, unit ,\, counit ,\, prod ,\, coprod \big) </annotation></semantics></math></div> <p>then every linear map of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>in</mi></msub></mrow></msup></mrow></msup><mo>⟶</mo><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>out</mi></msub></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> A^{\otimes^{n_{in}}} \longrightarrow A^{\otimes^{n_{out}}} </annotation></semantics></math></div> <p>which is</p> <ol> <li> <p>entirely the composite of the algebra’s structure maps, ie. of the (co)unit and the (co)product,</p> </li> <li> <p><em>connected</em>, in that it does not decompose as a tensor product of such morphisms,</p> </li> </ol> <p>is equal to the composite, in this order, of</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>in</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{in}-1</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>prod</mi></mrow><annotation encoding="application/x-tex">prod</annotation></semantics></math></p> </li> <li> <p>some number (<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>) of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>prod</mi><mo>∘</mo><mi>coprod</mi></mrow><annotation encoding="application/x-tex">prod \circ coprod</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>out</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{out}-1</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coprod</mi></mrow><annotation encoding="application/x-tex">coprod</annotation></semantics></math>.</p> </li> </ul> <p></p> </div> </p> <p> <div class='num_cor' id='SpiderTheorem'> <h6>Corollary</h6> <p><strong>(Spider theorem)</strong> <br /> If the Frobenius algebra in Cor. <a class="maruku-ref" href="#NormalFormForFrobeniusMaps"></a> is <em>special</em> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>prod</mi><mo>∘</mo><mi>coprod</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">prod \circ coprod = id</annotation></semantics></math> <a class="maruku-eqref" href="#eq:CoprodLetfInverseToProd">(3)</a> then every linear map of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>in</mi></msub></mrow></msup></mrow></msup><mo>⟶</mo><msup><mi>A</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>out</mi></msub></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> A^{\otimes^{n_{in}}} \longrightarrow A^{\otimes^{n_{out}}} </annotation></semantics></math></div> <p>which is</p> <ol> <li> <p>entirely the composite of the algebra’s structure maps, ie. of the (co)unit and the (co)product,</p> </li> <li> <p><em>connected</em> in that it does not decompose as a tensor product of such morphisms,</p> </li> </ol> <p>is equal to the composite, in this order, of</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>in</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{in}-1</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>prod</mi></mrow><annotation encoding="application/x-tex">prod</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>out</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_{out}-1</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coprod</mi></mrow><annotation encoding="application/x-tex">coprod</annotation></semantics></math>.</p> </li> </ul> <p></p> </div> </p> <div class="float_right_image" style="margin: -20px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/FrobNormalFormAsSpiderRule.jpg" width="500px" /> <figcaption style="text-align: center">From <a href="quantum+information+theory+via+dagger-compact+categories#CoeckeDuncan11">Cocke Duncan 2011</a></figcaption> </figure> </div> <p>The “spider theorem” (Cor. <a class="maruku-ref" href="#SpiderTheorem"></a>) is called this way in <a href="quantum+information+theory+via+dagger-compact+categories#CoeckeDuncan08">Coecke & Duncan 2008, Thm. 1</a> (it appears unnamed also in <a href="quantum+information+theory+via+dagger-compact+categories#CoeckePaquette08">Coecke & Paquette 2008, p. 6</a>) and shares its name with the corresponding “spider diagrams” used in the <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a> for <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a> (where the Frobenius algebras that appear are those that realize the <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> as a <a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius</a> <a class="existingWikiWord" href="/nlab/show/writer+monad">writer monad</a>/<a class="existingWikiWord" href="/nlab/show/cowriter+comonad">cowriter comonad</a>).</p> <h3 id="frobenius_algebras_in_polycategories">Frobenius algebras in polycategories</h3> <p>In fact, Frobenius algebras can be defined in any <a class="existingWikiWord" href="/nlab/show/polycategory">polycategory</a>, and hence in any <a class="existingWikiWord" href="/nlab/show/linearly+distributive+category">linearly distributive category</a>. The essential point is that the monoidal structure used for the monoid structure could be different from the monoidal structure used for the comonoid structure, i.e. we could have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mu:A\otimes A \to A</annotation></semantics></math> but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>⅋</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\delta :A \to A \parr A</annotation></semantics></math>. The compatibility between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\parr</annotation></semantics></math> in a linearly distributive category (or between their “<a class="existingWikiWord" href="/nlab/show/multicategory">multicategorical</a>” analogues in a polycategory) is precisely what is required to write down the composites involved in the Frobenius laws. For instance, we can have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>A</mi><mover><mo>→</mo><mrow><msub><mn>1</mn> <mi>A</mi></msub><mo>⊗</mo><mi>δ</mi></mrow></mover><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⅋</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>lin</mi><mo>−</mo><mi>distrib</mi></mrow></mover><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⅋</mo><mi>A</mi><mover><mo>→</mo><mrow><mi>μ</mi><mo>⅋</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow></mover><mi>A</mi><mo>⅋</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> A\otimes A \xrightarrow{1_A \otimes \delta} A \otimes (A \parr A) \xrightarrow{lin-distrib} (A \otimes A) \parr A \xrightarrow{\mu \parr 1_A} A\parr A </annotation></semantics></math></div> <p>and one of the Frobenius laws says that this composite is equal to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>A</mi><mover><mo>→</mo><mi>μ</mi></mover><mi>A</mi><mover><mo>→</mo><mi>δ</mi></mover><mi>A</mi><mo>⅋</mo><mi>A</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \otimes A \xrightarrow{\mu} A \xrightarrow{\delta} A\parr A. </annotation></semantics></math></div> <p>This is analogous to how a <a class="existingWikiWord" href="/nlab/show/bimonoid">bimonoid</a> can be defined in any <a class="existingWikiWord" href="/nlab/show/duoidal+category">duoidal category</a>. In fact, it is a sort of <a class="existingWikiWord" href="/nlab/show/microcosm+principle">microcosm principle</a>; it is shown in <a href="#Egger2010">(Egger2010)</a> that Frobenius monoids in the linearly distributive category <a class="existingWikiWord" href="/nlab/show/Sup">Sup</a> are precisely <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">*-autonomous</a> cocomplete <a class="existingWikiWord" href="/nlab/show/posets">posets</a> (and hence, in particular, linearly distributive).</p> <p>In polycategorical language we can give another <a class="existingWikiWord" href="/nlab/show/unbiased">unbiased</a> definition of a commutative Frobenius monoid: it is equipped with exactly one morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mover><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>…</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mo>⏞</mo></mover><mi>n</mi></mover><mo>→</mo><mover><mover><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>…</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mo>⏞</mo></mover><mi>m</mi></mover></mrow><annotation encoding="application/x-tex">\overset{n}{\overbrace{(A,A,\dots,A)}} \to \overset{m}{\overbrace{(A,A,\dots,A)}}</annotation></semantics></math> of each possible (two-sided) arity, such that any (symmetric) polycategorical composite of two such morphisms is equal to another such. The monoid structure consists of the morphisms of arity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>, while the comonoid structure is the morphisms of arity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,2)</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>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0)</annotation></semantics></math>, and the Frobenius relations say that three ways to compose these to produce a morphism of arity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,2)</annotation></semantics></math> are equal. (The morphism of arity <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><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> is the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mi>η</mi></mrow><annotation encoding="application/x-tex">\epsilon\eta</annotation></semantics></math>; no axiom is required on it, because in a polycategory there is no other morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">()\to ()</annotation></semantics></math> to compare it to.) In other words, the free symmetric polycategory containing a commutative Frobenius monoid is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> symmetric polycategory. In this way Frobenius algebras are to polycategories in the same way that monoids are to multicategories.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+algebra">Calabi-Yau algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypergraph+category">hypergraph category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+pseudomonoid">Frobenius pseudomonoid</a></p> </li> </ul> <div> <table><thead><tr><th>(<a class="existingWikiWord" href="/nlab/show/comonad">co</a>)<a class="existingWikiWord" href="/nlab/show/monad">monad</a> name</th><th><a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a></th><th>(<a class="existingWikiWord" href="/nlab/show/comonad">co</a>)<a class="existingWikiWord" href="/nlab/show/monad">monad</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> induced by</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>→</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \to (\text{-})</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian</a> types</td><td style="text-align: left;"><a href="comonoid#InACartesianMonoidalCategory">unique comonoid structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coreader+comonad">coreader comonad</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>×</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \times (\text{-})</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian</a> types</td><td style="text-align: left;"><a href="comonoid#InACartesianMonoidalCategory">unique comonoid structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/writer+monad">writer monad</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \otimes (\text{-})</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">monoidal</a> types</td><td style="text-align: left;">chosen <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid structure</a> on <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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cowriter+comonad">cowriter comonad</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>→</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">monoidal</a> types</td><td style="text-align: left;">chosen <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid structure</a> on <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> <br /><br /> chosen <a class="existingWikiWord" href="/nlab/show/comonoid+object">comonoid structure</a> on <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></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius</a> (co)writer</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>→</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{A \to (\text{-}) \\ A \otimes (\text{-})}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">monoidal</a> types</td><td style="text-align: left;">chosen <a class="existingWikiWord" href="/nlab/show/Frobenius+monoid">Frobenius monoid structure</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Frobenius algebras were introduced by <a class="existingWikiWord" href="/nlab/show/Brauer">Brauer</a> and Nesbitt and were named after <a class="existingWikiWord" href="/nlab/show/Ferdinand+Frobenius">Ferdinand Frobenius</a>.</p> <p>See for instance</p> <ul id="Aguiar2000"> <li id="Eilenberg1955"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Tadasi+Nakayama">Tadasi Nakayama</a>, <em>On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings</em>, <em>Nagoya Math. J.</em> <strong>9</strong> (1955) 1-16 [<a href="https://doi.org/10.1017/S0027763000023229">doi:10.1017/S0027763000023229</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguiar">Marcelo Aguiar</a> (2000), <em>A note on strongly separable algebras</em>, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, <strong>65</strong>, 51–60. (<a href="http://www.math.cornell.edu/~maguiar/strongly.pdf">pdf</a>)</p> </li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrew+Baker">Andrew Baker</a>, <em>Frobenius algebras</em> 2010 (<a href="http://www.maths.gla.ac.uk/~ajb/dvi-ps/Frobenius-talk.pdf">pdf</a>)</li> </ul> <p>On the role of Frobenius algebras in <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a>:</p> <ul> <li id="Abrams96"> <p><a class="existingWikiWord" href="/nlab/show/Lowell+Abrams">Lowell Abrams</a>, <em>Two-dimensional topological quantum field theories and Frobenius algebra</em>, Jour. Knot. Theory and its Ramifications <strong>5</strong>, 569-587 (1996) [<a href="https://doi.org/10.1142/S0218216596000333">doi:10.1142/S0218216596000333</a>, <a href="http://home.gwu.edu/~labrams/docs/tqft.ps">ps</a>]</p> </li> <li id="BaezTWF"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, This Week’s Finds in Mathematical Physics, <a href="http://math.ucr.edu/home/baez/week268.html">week268</a> and <a href="http://math.ucr.edu/home/baez/week299.html">week299</a>.</p> </li> <li id="Kock2004"> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Kock">Joachim Kock</a>, <em>Frobenius Algebras and 2d Topological Quantum Field Theories</em>, Cambridge U. Press (2004) [<a href="https://www.cambridge.org/core/books/frobenius-algebras-and-2d-topological-quantum-field-theories/A6438118DFADFD27175779F1FC0FF7CB">doi:10.1017/CBO9780511615443</a>, <a href="http://mat.uab.cat/~kock/TQFT.html">webpage</a>, <a href="http://mat.uab.es/~kock/TQFT/FS.pdf">course notes pdf</a>, <a class="existingWikiWord" href="/nlab/files/Kock-FrobAlgTQFT-short.pdf" title="pdf">pdf</a>]</p> </li> <li id="Kock2006"> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Kock">Joachim Kock</a>, Remarks on the history of the Frobenius equation (2006) [<a href="http://mat.uab.es/~kock/TQFT.html#history">web</a>]</p> </li> <li id="Lauda2006"> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, Frobenius algebras and ambidextrous adjunctions, <em>Theory and Applications of Categories</em> <strong>16</strong> (2006) 84-122 [<a href="http://tac.mta.ca/tac/volumes/16/4/16-04abs.html">tac:16-04</a>, <a href="http://arxiv.org/abs/math/0502550">arXiv:math/0502550</a>]</p> <blockquote> <p>(relating to <a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunctions">ambidextrous adjunctions</a> and <a class="existingWikiWord" href="/nlab/show/Frobenius+monads">Frobenius monads</a>)</p> </blockquote> </li> <li id="Lauda2008"> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, <a class="existingWikiWord" href="/nlab/show/Hendryk+Pfeiffer">Hendryk Pfeiffer</a>, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, <em>Topology Appl.</em> <strong>155</strong> (2005) 623-666 [<a href="http://arxiv.org/abs/math/0510664">arXiv:0510664</a>, <a href="https://doi.org/10.1016/j.topol.2007.11.005">doi:10.1016/j.topol.2007.11.005</a>]</p> </li> </ul> <p>For applications in proof theory of classical and <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> or linguistics:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Hyland">Martin Hyland</a>, <em>Abstract Interpretation of Proofs: Classical Propositional Calculus</em>, pp. 6-21 in Marcinkowski, Tarlecki (eds.), <em>Computer Science Logic (CSL 2004)</em>, LNCS <strong>3210</strong> Springer Heidelberg 2004. (<a href="https://www.dpmms.cam.ac.uk/~jmeh1/Research/Publications/2004/aap04.pdf">preprint</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <em>Three investigations into linear logic</em> , PhD report Cambridge 2006. (<a href="http://comp.mq.edu.au/~rgarner/Thesis/Smith-Knight-Essay.pdf">pdf</a>)</p> </li> <li> <p>D. Kartsaklis, M. Sadrzadeh, S. Pulman, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <em>Reasoning about meaning in natural language with compact closed categories and Frobenius algebras</em> (2014). (<a href="https://arxiv.org/abs/1401.5980">arXiv:1401.5980</a>)</p> </li> </ul> <p>Frobenius algebras in linearly distributive categories are discussed in</p> <ul id="Egger2010"> <li><a class="existingWikiWord" href="/nlab/show/Jeff+Egger">Jeff Egger</a>, <em>The Frobenius relations meet linear distributivity</em>, 2010 <a href="http://tac.mta.ca/tac/volumes/24/2/24-02abs.html">TAC</a></li> </ul> <p>See also</p> <ul> <li> <p>Aurelio Carboni (1991), Matrices, relations, and group representations, <em>Journal of Algebra</em> <strong>136</strong> 2, 497–529. (<a href="https://www.sciencedirect.com/science/article/pii/002186939190057F">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertfried+Fauser">Bertfried Fauser</a>, <em>Some Graphical Aspects of Frobenius Structures</em>, preprint (2012). <a href="http://arxiv.org/pdf/1202.6380v1">arXiv:1202.6380</a></p> </li> <li id="Lack2004"> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a> (2004), Composing PROPs, <em>Theory and Applications of Categories</em> <strong>13</strong>, 147–163. (<a href="http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <em>Ordinal Sums and Equational Doctrines</em>, pp. 141-155 in Eckmann (ed.), <em>Seminar on Triples and Categorical Homology Theory</em>, LNM <strong>80</strong> Springer Heidelberg 1969. (<a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf">TAC Reprint of vol. 80</a>)</p> </li> <li id="PastroStreet2008"> <p>Craig Pastro and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Weak Hopf monoids in braided monoidal categories</em>, 2008, <a href="https://arxiv.org/abs/0801.4067">arXiv:0801.4067</a></p> </li> <li id="Rosebrugh2005"> <p>R. Rosebrugh, N. Sabadini and R.F.C. Walters (2005), Generic commutative separable algebras and cospans of graphs, <em>Theory and Applications of Categories</em> <strong>15</strong> (Proceedings of CT2004), 164–177. (<a href="http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html">web</a>)</p> </li> <li id="Street2004"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a> (2004), Frobenius monads and pseudomonoids, <em>J. Math. Phys.</em> <strong>45</strong>. (<a href="https://doi.org/10.1063/1.1788852">doi:10.1063/1.1788852</a>)</p> </li> <li> <p>R. F. C. Walters, R. J. Wood, <em>Frobenius objects in Cartesian cicategories</em>, TAC <strong>20</strong> no. 3 (2008) 25–47. (<a href="http://www.tac.mta.ca/tac/volumes/20/3/20-03.pdf">pdf</a>)</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/algebra">algebra</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 26, 2024 at 11:33:38. 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