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bimodule in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> bimodule </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a 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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#over_a_ring'>Over a ring</a></li> <ul> <li><a href='#with_a_left_action_and_a_right_action'>With a left action and a right action</a></li> <li><a href='#with_a_biaction'>With a biaction</a></li> </ul> <li><a href='#over_a_monoid_in_a_monoidal_category'>Over a monoid in a monoidal category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#biactions_left_actions_and_right_actions'>Biactions, left actions, and right actions</a></li> <li><a href='#linear_maps'>Linear maps</a></li> <li><a href='#tensor_product_of_bimodules'>Tensor product of bimodules</a></li> <li><a href='#twosided_ideals_of_a_ring'>Two-sided ideals of a ring</a></li> <li><a href='#rings_over_a_ring'>Rings over a ring</a></li> </ul> <li><a href='#categories_of_bimodules'>Categories of bimodules</a></li> <ul> <li><a href='#the_1category_of_bimodules_and_intertwiners'>The 1-category of bimodules and intertwiners</a></li> <li><a href='#AsMorphismsInA2Category'>The 2-category of rings, bimodules, and intertwiners</a></li> <li><a href='#Infinity2CategoryOfInfinityAlgebrasAndBimodules'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bimodules</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>bimodule</em> is a <a class="existingWikiWord" href="/nlab/show/module">module</a> in two compatible ways over two <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <h2 id="definition">Definition</h2> <h3 id="over_a_ring">Over a ring</h3> <h4 id="with_a_left_action_and_a_right_action">With a left action and a right action</h4> <p>Given two <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/bilinear+function">bilinear</a> <a class="existingWikiWord" href="/nlab/show/left+action">left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha_R:R \times B \to B</annotation></semantics></math> and a bilinear <a class="existingWikiWord" href="/nlab/show/right+action">right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo>:</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha_S:B \times S \to B</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math>.</p> <h4 id="with_a_biaction">With a biaction</h4> <p>Equivalently, given two <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/multilinear+function">trilinear</a> <a class="existingWikiWord" href="/nlab/show/biaction"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-biaction</a>, a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times B \times S \to B</annotation></semantics></math> such that</p> <ul> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">1_R b 1_S = b</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_1 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_2 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_2 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_1 (r_2 b s_1) s_2 = (r_1 \cdot_R r_2) b (s_1 \cdot_S s_2)</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_1 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_2 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mi>s</mi><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mi>b</mi><mi>s</mi><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">(r_1 + r_2) b s = r_1 b s + r_2 b s</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_1 \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>2</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_2 \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>s</mi><mo>=</mo><mi>r</mi><msub><mi>b</mi> <mn>1</mn></msub><mi>s</mi><mo>+</mo><mi>r</mi><msub><mi>b</mi> <mn>2</mn></msub><mi>s</mi></mrow><annotation encoding="application/x-tex">r (b_1 + b_2) s = r b_1 s + r b_2 s</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_2 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>b</mi><msub><mi>s</mi> <mn>1</mn></msub><mo>+</mo><mi>r</mi><mi>b</mi><msub><mi>s</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">r b (s_1 + s_2) = r b s_1 + r b s_2</annotation></semantics></math></p> </li> </ul> <p>representing simultaneous left multiplication by scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and right multiplication by scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>.</p> <h3 id="over_a_monoid_in_a_monoidal_category">Over a monoid in a monoidal category</h3> <p>We can define in more generality what is 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>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B)</annotation></semantics></math>-bimodule in a monoidal category <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>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,I)</annotation></semantics></math> where <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><msup><mo>∇</mo> <mi>A</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>A</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\nabla^{A},\eta^{A})</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>B</mi><mo>,</mo><msup><mo>∇</mo> <mi>B</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>B</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,\nabla^{B},\eta^{B})</annotation></semantics></math> are two monoids. It is given by:</p> <ul> <li>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math></li> <li>A left-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">l:A \otimes X \rightarrow X</annotation></semantics></math></li> <li>A right-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi>X</mi><mo>⊗</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">r:X \otimes B \rightarrow X</annotation></semantics></math></li> </ul> <p>such that:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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id="biactions_left_actions_and_right_actions">Biactions, left actions, and right actions</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/ring">ring</a>s, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule.</p> <p>Given a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_R</annotation></semantics></math> and a right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_S</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule, the biaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times B \times S \to B</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mi>s</mi><mo>≔</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r b s \coloneqq \alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math></div> <p>The biaction is trilinear because the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action are bilinear.</p> <p>On the other hand, given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule, the <a class="existingWikiWord" href="/nlab/show/left+action">left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-action</a> is defined from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>r</mi><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_R(r, b) \coloneqq r b 1_S</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>. It is a left action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\alpha_R(1_R, b) = 1_R b 1_S = m</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>α</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>S</mi></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r_1, \alpha_L(r_2, b)) = r_1 (r_2 b 1_S) 1_S = (r_1 \cdot_R r_2) b (1_S \cdot_S 1_S) = (r_1 \cdot_R r_2) b 1_S = \alpha_R(r_1 \cdot_R r_2, b)</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/right+action">right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-action</a> is defined from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\alpha_S(b, s) \coloneqq 1_R b s</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>. It is a right action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\alpha_S(b, 1_S) = 1_R b 1_S = m</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>S</mi></msub><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_S(\alpha_S(b, s_1), s_2) = 1_R (1_R, b, s_1) s_2 = (1_R \cdot_R 1_R) b (s_1 \cdot_S s_2) = 1_S b (s_1 \cdot_S s_2) = \alpha_S(b, s_1 \cdot_S s_2)</annotation></semantics></math></div> <p>The left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action satisfy the following identity:</p> <ul> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math>.</li> </ul> <p>This is because when expanded out, the identity becomes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha(r, \alpha(1_R, b, s), 1_S) = \alpha(1_R, \alpha(r, b, 1_S), s)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><mi>s</mi><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><msub><mo>⋅</mo> <mi>R</mi></msub><mi>r</mi><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>S</mi></msub><msub><mo>⋅</mo> <mi>S</mi></msub><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r \cdot_R 1_R) b (s \cdot_S 1_S) = (1_R \cdot_R r) b (1_S \cdot_S s)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mi>s</mi><mo>=</mo><mi>r</mi><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">r b s = r b s</annotation></semantics></math></div> <p>The left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action are bilinear because the original biaction is trilinear.</p> <h3 id="linear_maps">Linear maps</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/rings">rings</a>. A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule homomorphism</strong> between two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodules <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is an abelian group <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</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>f</mi><mo stretchy="false">(</mo><mi>r</mi><mi>a</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">f(r a s) = r f(a) s</annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/monic">monic</a> or an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule monomorphism</strong> if for every other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:C \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">k:C \to A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>h</mi><mo>=</mo><mi>f</mi><mo>∘</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f \circ h = f \circ k</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">h = k</annotation></semantics></math>.</p> <p>A <strong>sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule</strong> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <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 <a class="existingWikiWord" href="/nlab/show/monic">monic</a> linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i:A \hookrightarrow B</annotation></semantics></math>.</p> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/invertible">invertible</a> or an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule isomorphism</strong> if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">g:B \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">g \circ f = id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">f \circ g = id_B</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">id_B</annotation></semantics></math> are the identity linear maps 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> respectively.</p> <h3 id="tensor_product_of_bimodules">Tensor product of bimodules</h3> <p>Given <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">R, S, T</annotation></semantics></math> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodule <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> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, the tensor product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is formed as a <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes_N B</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\otimes B</annotation></semantics></math>. This is a special case of a more general construction:</p> <p>Given three monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>,</mo><mi>N</mi><mo>,</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">M,N,P</annotation></semantics></math> in a monoidal category <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>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,I)</annotation></semantics></math>, a <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>-<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>-bimodules <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> and a <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, we denote the monoid actions as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mi>A</mi></msup><mo>:</mo><mi>M</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\lambda^{A}:M \otimes A \rightarrow A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>A</mi></msup><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>N</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\rho^{A}:A \otimes N \rightarrow N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mi>B</mi></msup><mo>:</mo><mi>N</mi><mo>⊗</mo><mi>B</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\lambda^{B}:N \otimes B \rightarrow N</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>B</mi></msup><mo>:</mo><mi>B</mi><mo>⊗</mo><mi>P</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho^{B}:B \otimes P \rightarrow P</annotation></semantics></math>. 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y="93.463883"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="361.370513" y="95.137224"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-3" x="368.26855" y="95.137224"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="371.102112" y="95.137224"></use> </g> </g> </svg> <p>Assuming all requisite <a class="existingWikiWord" href="/nlab/show/reflexive+coequalizer">(reflective) coequalizers</a> exist, universal property arguments guarantee associativity isomorphisms of type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mi>B</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>P</mi></msub><mi>C</mi><mo>→</mo><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>B</mi><msub><mo>⊗</mo> <mi>P</mi></msub><mi>C</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">(A \otimes_N B) \otimes_P C \to A \otimes_N (B \otimes_P C).</annotation></semantics></math></div> <p>In fact, this tensor product defines composition in a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> where objects or 0-cells are monoids in a monoidal category, where 1-cells <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> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodules, and where 2-cells from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodules.</p> <p>This in turn can be seen as a special case of a bicategory of <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with suitably nice cocompleteness properties – see <a class="existingWikiWord" href="/nlab/show/monoidally+cocomplete+category">monoidally cocomplete category</a> and <a class="existingWikiWord" href="/nlab/show/Benabou+cosmos">Benabou cosmos</a>.</p> <h3 id="twosided_ideals_of_a_ring">Two-sided ideals of a ring</h3> <p>Every ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule, with the <a class="existingWikiWord" href="/nlab/show/biaction">biaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times R \times R \to R</annotation></semantics></math> defined by the ternary product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</mi><mi>c</mi><mo>≔</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a b c \coloneqq a \cdot b \cdot c</annotation></semantics></math> for elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">b \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">c \in R</annotation></semantics></math>.</p> <p>Given a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/two-sided+ideal">two-sided ideal</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <h3 id="rings_over_a_ring">Rings over a ring</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>. An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-ring</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule with a <a class="existingWikiWord" href="/nlab/show/bilinear+function">bilinear function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">(-)\cdot(-):S \times S \to S</annotation></semantics></math> and an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">1 \in S</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo>⋅</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, \cdot, 1)</annotation></semantics></math> forms a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>.</p> <h2 id="categories_of_bimodules">Categories of bimodules</h2> <h3 id="the_1category_of_bimodules_and_intertwiners">The 1-category of bimodules and intertwiners</h3> <div class="num_defn" id="1CategoryOfBimodulesAndIntertwiners"> <h6 id="definition_2">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BMod</mi></mrow><annotation encoding="application/x-tex">BMod</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(R,S,B)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/rings">rings</a> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g, \phi)</annotation></semantics></math> consisting of two ring <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \colon R \to R'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>S</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \colon S \to S'</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">S'</annotation></semantics></math>-bimodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>⋅</mo><mi>g</mi><mo>→</mo><mi>f</mi><mo>⋅</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi \colon B \cdot g \to f \cdot B'</annotation></semantics></math>. This we may depict as 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>R</mi></mtd> <mtd><mover><mo>→</mo><mi>B</mi></mover></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇓</mo> <mi>ϕ</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>R</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>B</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>S</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ R &\stackrel{B}{\to}& S \\ {}^{\mathllap{f}}\downarrow &\Downarrow_{\phi}& \downarrow^{\mathrlap{g}} \\ R' &\stackrel{B'}{\to}& S' } \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>As this notation suggests, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BMod</mi></mrow><annotation encoding="application/x-tex">BMod</annotation></semantics></math> is naturally the vertical category of a <a class="existingWikiWord" href="/nlab/show/pseudo+double+category">pseudo double category</a> whose horizontal composition is given by tensor product of bimodules.</p> </div> <h3 id="AsMorphismsInA2Category">The 2-category of rings, bimodules, and intertwiners</h3> <p>Consider bimodules over <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <div class="num_prop" id="AlgebrasAndBimodules"> <h6 id="proposition">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/rings">rings</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are bimodules;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/intertwiners">intertwiners</a>.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of 1-morphisms is given by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> over the middle algebra.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> from the above 2-category of rings and bimodules to <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> which</p> <ul> <li> <p>sends an ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>;</p> </li> <li> <p>sends a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>B</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Mod</mi> <mi>R</mi></msub><mo>→</mo><msub><mi>Mod</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex"> (-)\otimes_{R} B \;\colon\; Mod_{R} \to Mod_{S} </annotation></semantics></math></div></li> <li> <p>sends an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> to the evident <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of the above functors.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>This construction has as its image precisely the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a>.</p> </div> <p>This is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In the context of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> one may interpret this as says that the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of those <a class="existingWikiWord" href="/nlab/show/2-modules">2-modules</a> over the given ring which are equivalent to a <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> is that of rings, bimodules and intertwiners. See also at <em><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>The 2-category of rings and bimodules is an archtypical example for a <a class="existingWikiWord" href="/nlab/show/2-category+with+proarrow+equipment">2-category with proarrow equipment</a>, hence for a <a class="existingWikiWord" href="/nlab/show/pseudo+double+category">pseudo double category</a> with niche-fillers. Or in the language of <a class="existingWikiWord" href="/nlab/show/internal+%28infinity%2C1%29-category">internal (infinity,1)-category</a>-theory: it naturally induces the structure of a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><msub><mi>X</mi> <mn>1</mn></msub><mover><munder><mo>→</mo><mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></munder><mover><mo>→</mo><mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></mover></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi>Cat</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}} </annotation></semantics></math></div> <p>which satisfies the <a class="existingWikiWord" href="/nlab/show/Segal+conditions">Segal conditions</a>. Here</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>Ring</mi></mrow><annotation encoding="application/x-tex"> X_0 = Ring </annotation></semantics></math></div> <p>is the category of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> and <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between them, while</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>BMod</mi></mrow><annotation encoding="application/x-tex"> X_1 = BMod </annotation></semantics></math></div> <p>is the category of def. <a class="maruku-ref" href="#1CategoryOfBimodulesAndIntertwiners"></a>, whose objects are pairs consisting of two rings <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and an <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> bimodule <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> between them, and whose morphisms are pairs consisting of two ring homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \colon A \to A'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \colon B \to B'</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>N</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">N \cdot (g) \to (f) \cdot N'</annotation></semantics></math>.</p> </div> <h3 id="Infinity2CategoryOfInfinityAlgebrasAndBimodules">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bimodules</h3> <p>The above has a generalization to <em><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-bimodules">(infinity,1)-bimodules</a></em>. See there for more.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monad">module over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule+object">bimodule object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+bimodule">quotient bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+bimodule">Hilbert bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+bimodule">Noetherian bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Artinian+bimodule">Artinian bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/two-sided+ideal">two-sided ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+two-sided+ideals+in+a+ring">category of two-sided ideals in a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/biaction">biaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/amplimorphism">amplimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-bimodule">(infinity,1)-bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule+category">bimodule category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>, <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a></p> </li> </ul> <h2 id="references">References</h2> <p>The 2-category of bimodules in its incarnation as a <a class="existingWikiWord" href="/nlab/show/2-category+with+proarrow+equipment">2-category with proarrow equipment</a> appears as example 2.3 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>Framed bicategories and monoidal fibrations</em> (<a href="http://arxiv.org/abs/0706.1286">arXiv:0706.1286</a>)</li> </ul> <p>Bimodules in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> are discussed in section 4.3 of</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>For more on that see at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-bimodule">(∞,1)-bimodule</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 20, 2024 at 13:10:19. 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