CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> module over a monad in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <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> module over a monad </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/647/#Item_24" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</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='#modules'>Modules</a></li> <li><a href='#bimodules'>Bimodules</a></li> <li><a href='#AlgebrasForMonadsInCat'>Algebras for monads in Cat</a></li> <li><a href='#in_a_virtual_double_category'>In a virtual double category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#colimits'>Colimits</a></li> <li><a href='#tensor_product'>Tensor product</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Just as the notion of a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> in a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> generalizes that of a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">modules over monoids</a> generalize easily to modules over monads.</p> <p>Beware that modules over monads in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> are often called <a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">*algebras* for the monad</a> (see there for more), since they literally are algebras in the sense of <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>, see <a href="#AlgebrasForMonadsInCat">below</a>. By extension, one might speak of modules over monads in any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> as “algebras for the monad”.</p> <p>The <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a> concept is that of <em><a class="existingWikiWord" href="/nlab/show/coalgebra+over+a+comonad">coalgebra over a comonad</a></em>.</p> <h2 id="definition">Definition</h2> <h3 id="modules">Modules</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">t \colon a \to a</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with structure 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo lspace="verythinmathspace">:</mo><mi>t</mi><mi>t</mi><mo>⇒</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\mu \colon t t \Rightarrow t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msub><mn>1</mn> <mi>a</mi></msub><mo>⇒</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\eta \colon 1_a \Rightarrow t</annotation></semantics></math>. Then a <strong>left <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>-module</strong> (or <strong><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>-algebra</strong>) is given by a 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x \colon b \to a</annotation></semantics></math> and a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo lspace="verythinmathspace">:</mo><mi>t</mi><mi>x</mi><mo>⇒</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\lambda \colon t x \Rightarrow x</annotation></semantics></math>, where</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>t</mi><mi>t</mi><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>μ</mi><mi>x</mi></mrow></mover></mtd> <mtd><mi>t</mi><mi>x</mi></mtd></mtr> <mtr><mtd><mi>t</mi><mi>λ</mi><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>λ</mi></mtd></mtr> <mtr><mtd><mi>t</mi><mi>x</mi></mtd> <mtd><munder><mo>→</mo><mi>λ</mi></munder></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow><mspace width="2em"></mspace><mspace width="2em"></mspace><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>η</mi><mi>x</mi></mrow></mover></mtd> <mtd><mi>t</mi><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mn>1</mn><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo><mi>λ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ t t x & \overset{\mu x}{\to} & t x \\ t\lambda\downarrow & & \downarrow \lambda \\ t x & \underset{\lambda}{\to} & x } \qquad \qquad \array{ x & \overset{\eta x}{\to} & t x \\ & 1\searrow & \downarrow \lambda \\ & & x } </annotation></semantics></math></div> <p>commute. Similarly, a <strong>right <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>-module</strong> (or <strong><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>-opalgebra</strong>) is given by a 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">y \colon a \to c</annotation></semantics></math> and a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mi>t</mi><mo>⇒</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\rho \colon y t \Rightarrow y</annotation></semantics></math>, with commuting diagrams as above with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> on the left instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> on the right.</p> <p>Clearly, a right <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>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is the same thing as a left <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>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi mathvariant="normal">op</mi></msup></mrow><annotation encoding="application/x-tex">K^{\mathrm{op}}</annotation></semantics></math>. A <strong>left <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>-comodule</strong> or <strong>coalgebra</strong> is then a left <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>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi mathvariant="normal">co</mi></msup></mrow><annotation encoding="application/x-tex">K^{\mathrm{co}}</annotation></semantics></math>, and a <strong>right <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>-comodule</strong> is a left <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>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi mathvariant="normal">coop</mi></msup></mrow><annotation encoding="application/x-tex">K^{\mathrm{coop}}</annotation></semantics></math>.</p> <p>A <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>-module of any of these sorts is <em>a fortiori</em> an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+endomorphism">algebra over the underlying endomorphism</a> <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>.</p> <h3 id="bimodules">Bimodules</h3> <p>Given monads <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> on <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 <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> 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>, an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">s,t</annotation></semantics></math>-bimodule</strong> is given by a 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x\colon b \to a</annotation></semantics></math>, together with the structures of 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>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mi>s</mi><mo>⇒</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\rho \colon x s \Rightarrow x</annotation></semantics></math> and a left <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>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo lspace="verythinmathspace">:</mo><mi>t</mi><mi>x</mi><mo>⇒</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\lambda \colon t x \Rightarrow x</annotation></semantics></math> that are compatible in the sense that the diagram</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>t</mi><mi>x</mi><mi>s</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>t</mi><mi>ρ</mi></mrow></mover></mtd> <mtd><mi>t</mi><mi>x</mi></mtd></mtr> <mtr><mtd><mi>λ</mi><mi>s</mi><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>λ</mi></mtd></mtr> <mtr><mtd><mi>x</mi><mi>s</mi></mtd> <mtd><munder><mo>→</mo><mi>ρ</mi></munder></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ t x s & \overset{t\rho}{\to} & t x \\ \lambda s \downarrow & & \downarrow \lambda \\ x s & \underset{\rho}{\to} & x } </annotation></semantics></math></div> <p>commutes. Such a bimodule may be written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>s</mi><mi>⇸</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x \colon s &#8696; t</annotation></semantics></math>.</p> <p>A <strong>morphism</strong> of left <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>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo>,</mo><mi>λ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,\lambda) \to (x',\lambda')</annotation></semantics></math> is given by a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>⇒</mo><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha \colon x \Rightarrow x'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><mo>∘</mo><mi>t</mi><mi>α</mi><mo>=</mo><mi>α</mi><mo>∘</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda' \circ t\alpha = \alpha \circ \lambda</annotation></semantics></math>. Similarly, a morphism of right <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>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>y</mi><mo>′</mo><mo>,</mo><mi>ρ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y,\rho) \to (y',\rho')</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mo>⇒</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\beta \colon y \Rightarrow y'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>′</mo><mo>∘</mo><mi>α</mi><mi>s</mi><mo>=</mo><mi>α</mi><mo>∘</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho' \circ \alpha s = \alpha \circ \rho</annotation></semantics></math>. A morphism of bimodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo>,</mo><mi>λ</mi><mo>′</mo><mo>,</mo><mi>ρ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,\lambda,\rho) \to (x',\lambda',\rho')</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>⇒</mo><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha \colon x \Rightarrow x'</annotation></semantics></math> that is a morphism of both left and right modules.</p> <p>More abstractly, the monads <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> and <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> give rise to ordinary monads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>s</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">s^*</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">t_*</annotation></semantics></math> on the hom-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(b,a)</annotation></semantics></math>, by pre- and post-composition. The associativity isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> then gives rise to an invertible <a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a> between these, so that the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>s</mi> <mo>*</mo></msup><msub><mi>t</mi> <mo>*</mo></msub><mo>≅</mo><msub><mi>t</mi> <mo>*</mo></msub><msup><mi>s</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>↦</mo><mi>t</mi><mi>x</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">s^* t_* \cong t_* s^* \colon x \mapsto t x s</annotation></semantics></math> is again a monad. Then the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod_K(s,t)</annotation></semantics></math> of bimodules from <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> to <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> is the ordinary <a class="existingWikiWord" href="/nlab/show/Eilenberg--Moore+category">Eilenberg–Moore category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>s</mi> <mo>*</mo></msup><msub><mi>t</mi> <mo>*</mo></msub></mrow></msup></mrow><annotation encoding="application/x-tex">K(b,a)^{s^* t_*}</annotation></semantics></math>.</p> <h3 id="AlgebrasForMonadsInCat">Algebras for monads in Cat</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">K =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,\eta,\mu)</annotation></semantics></math> is a monad on a <a class="existingWikiWord" href="/nlab/show/category">category</a> <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>, then a left <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>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \colon 1 \to C</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>, is usually called an <strong><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over <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></a></strong> or <strong><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>-algebra</strong> (see there): it is given by an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \in C</annotation></semantics></math> together with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\alpha \colon T A \to A</annotation></semantics></math>, such that</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>T</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>T</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>α</mi></mtd></mtr> <mtr><mtd><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>α</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { T(T(A)) & \stackrel{\mu_A}\rightarrow & T(A) \\ T(\alpha) \downarrow & & \downarrow \alpha \\ T(A) & \stackrel{\alpha}\rightarrow & A } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>η</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mi>id</mi> <mi>A</mi></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo><mi>α</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { A & \stackrel{\eta_A}\rightarrow & T(A) \\ & id_A \searrow & \downarrow \alpha \\ & & A } </annotation></semantics></math></div> <p>commute.</p> <p>In particular, every algebra over a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,\eta,\mu)</annotation></semantics></math> in <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> has the structure of an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+endofunctor">algebra over the underlying endofunctor</a> <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>.</p> <p><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>-algebras can also be defined as left <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over <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> <em>qua</em> monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">End(C)</annotation></semantics></math>. There the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is represented by the constant endofunctor at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> of <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> is the category of these algebras. It has a universal property that allows the notion of <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+object">Eilenberg-Moore object</a> to be defined in any bicategory.</p> <h3 id="in_a_virtual_double_category">In a virtual double category</h3> <p>The notion of (bi)module makes sense in <a class="existingWikiWord" href="/nlab/show/virtual+double+categories">virtual double categories</a>, generalizing the previous definition.</p> <p>A <em>monad</em> in a virtual double category is a <a class="existingWikiWord" href="/nlab/show/loose+morphism">loose</a> endomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><mi>a</mi><mo>↛</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">t : a \nrightarrow a</annotation></semantics></math> together with a binary map (i.e. a square in the virtual double category) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\mu:t,t \to t</annotation></semantics></math> giving the multiplication and a nullary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\eta:() \to t</annotation></semantics></math> giving the unit:</p>  <p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="194.575pt" height="50.278pt" viewBox="0 0 194.575 50.278" version="1.2"> <defs> <g> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1"> <path style="stroke:none;" d="M 3.546875 -1.40625 C 3.484375 -1.203125 3.484375 -1.171875 3.328125 -0.953125 C 3.0625 -0.625 2.546875 -0.125 1.984375 -0.125 C 1.515625 -0.125 1.234375 -0.546875 1.234375 -1.25 C 1.234375 -1.890625 1.609375 -3.21875 1.828125 -3.71875 C 2.234375 -4.53125 2.78125 -4.953125 3.234375 -4.953125 C 4.015625 -4.953125 4.171875 -4 4.171875 -3.90625 C 4.171875 -3.890625 4.140625 -3.734375 4.125 -3.71875 Z M 4.296875 -4.421875 C 4.171875 -4.71875 3.859375 -5.203125 3.234375 -5.203125 C 1.90625 -5.203125 0.46875 -3.46875 0.46875 -1.734375 C 0.46875 -0.5625 1.15625 0.125 1.953125 0.125 C 2.609375 0.125 3.15625 -0.390625 3.484375 -0.78125 C 3.609375 -0.078125 4.15625 0.125 4.515625 0.125 C 4.859375 0.125 5.140625 -0.09375 5.359375 -0.515625 C 5.546875 -0.921875 5.71875 -1.640625 5.71875 -1.6875 C 5.71875 -1.75 5.671875 -1.796875 5.59375 -1.796875 C 5.484375 -1.796875 5.484375 -1.734375 5.4375 -1.5625 C 5.25 -0.859375 5.03125 -0.125 4.546875 -0.125 C 4.203125 -0.125 4.1875 -0.421875 4.1875 -0.65625 C 4.1875 -0.9375 4.21875 -1.0625 4.328125 -1.515625 C 4.40625 -1.8125 4.46875 -2.078125 4.5625 -2.421875 C 5 -4.1875 5.09375 -4.609375 5.09375 -4.671875 C 5.09375 -4.84375 4.96875 -4.96875 4.796875 -4.96875 C 4.421875 -4.96875 4.328125 -4.5625 4.296875 -4.421875 Z M 4.296875 -4.421875 "></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-1"> <path style="stroke:none;" d="M 1.734375 -3.125 L 2.5 -3.125 C 2.65625 -3.125 2.75 -3.125 2.75 -3.28125 C 2.75 -3.390625 2.65625 -3.390625 2.515625 -3.390625 L 1.796875 -3.390625 L 2.078125 -4.5 C 2.109375 -4.625 2.109375 -4.65625 2.109375 -4.671875 C 2.109375 -4.828125 1.984375 -4.90625 1.859375 -4.90625 C 1.59375 -4.90625 1.53125 -4.703125 1.453125 -4.34375 L 1.203125 -3.390625 L 0.453125 -3.390625 C 0.296875 -3.390625 0.203125 -3.390625 0.203125 -3.234375 C 0.203125 -3.125 0.296875 -3.125 0.4375 -3.125 L 1.140625 -3.125 L 0.671875 -1.234375 C 0.625 -1.046875 0.546875 -0.765625 0.546875 -0.65625 C 0.546875 -0.1875 0.9375 0.078125 1.359375 0.078125 C 2.1875 0.078125 2.671875 -1.03125 2.671875 -1.125 C 2.671875 -1.203125 2.59375 -1.21875 2.546875 -1.21875 C 2.46875 -1.21875 2.453125 -1.1875 2.40625 -1.078125 C 2.25 -0.703125 1.859375 -0.140625 1.375 -0.140625 C 1.203125 -0.140625 1.109375 -0.25 1.109375 -0.515625 C 1.109375 -0.65625 1.140625 -0.75 1.15625 -0.84375 Z M 1.734375 -3.125 "></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-2"> <path style="stroke:none;" d="M 1.90625 -2.78125 C 1.9375 -2.921875 2 -3.1875 2 -3.21875 C 2 -3.390625 1.875 -3.46875 1.75 -3.46875 C 1.484375 -3.46875 1.421875 -3.203125 1.390625 -3.09375 L 0.296875 1.265625 C 0.265625 1.390625 0.265625 1.4375 0.265625 1.453125 C 0.265625 1.640625 0.421875 1.6875 0.515625 1.6875 C 0.546875 1.6875 0.734375 1.6875 0.828125 1.484375 C 0.859375 1.421875 1.078125 0.484375 1.234375 -0.15625 C 1.375 -0.046875 1.640625 0.078125 2.046875 0.078125 C 2.6875 0.078125 3.109375 -0.453125 3.140625 -0.484375 C 3.28125 0.0625 3.8125 0.078125 3.90625 0.078125 C 4.265625 0.078125 4.453125 -0.21875 4.515625 -0.359375 C 4.671875 -0.640625 4.78125 -1.09375 4.78125 -1.125 C 4.78125 -1.171875 4.75 -1.21875 4.65625 -1.21875 C 4.5625 -1.21875 4.546875 -1.171875 4.5 -0.984375 C 4.390625 -0.546875 4.234375 -0.140625 3.9375 -0.140625 C 3.75 -0.140625 3.671875 -0.296875 3.671875 -0.515625 C 3.671875 -0.640625 3.765625 -0.984375 3.8125 -1.203125 L 4.03125 -2.015625 C 4.078125 -2.21875 4.109375 -2.375 4.171875 -2.625 C 4.21875 -2.796875 4.296875 -3.09375 4.296875 -3.140625 C 4.296875 -3.34375 4.140625 -3.390625 4.046875 -3.390625 C 3.765625 -3.390625 3.71875 -3.1875 3.625 -2.84375 L 3.46875 -2.1875 L 3.21875 -1.203125 L 3.140625 -0.890625 C 3.125 -0.84375 2.921875 -0.546875 2.734375 -0.390625 C 2.59375 -0.296875 2.375 -0.140625 2.078125 -0.140625 C 1.71875 -0.140625 1.46875 -0.34375 1.46875 -0.828125 C 1.46875 -1.03125 1.53125 -1.265625 1.578125 -1.453125 Z M 1.90625 -2.78125 "></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-3"> <path style="stroke:none;" d="M 4.09375 -2.171875 C 4.125 -2.28125 4.15625 -2.421875 4.15625 -2.59375 C 4.15625 -3.1875 3.75 -3.46875 3.1875 -3.46875 C 2.546875 -3.46875 2.140625 -3.078125 1.90625 -2.78125 C 1.859375 -3.21875 1.515625 -3.46875 1.109375 -3.46875 C 0.828125 -3.46875 0.625 -3.28125 0.5 -3.046875 C 0.328125 -2.6875 0.234375 -2.28125 0.234375 -2.265625 C 0.234375 -2.1875 0.296875 -2.15625 0.359375 -2.15625 C 0.453125 -2.15625 0.46875 -2.1875 0.515625 -2.390625 C 0.625 -2.796875 0.765625 -3.25 1.078125 -3.25 C 1.28125 -3.25 1.34375 -3.0625 1.34375 -2.875 C 1.34375 -2.734375 1.296875 -2.578125 1.234375 -2.328125 C 1.21875 -2.265625 1.09375 -1.796875 1.0625 -1.6875 L 0.78125 -0.515625 C 0.75 -0.390625 0.703125 -0.203125 0.703125 -0.171875 C 0.703125 0.015625 0.84375 0.078125 0.953125 0.078125 C 1.09375 0.078125 1.203125 -0.015625 1.265625 -0.109375 C 1.28125 -0.15625 1.359375 -0.421875 1.390625 -0.59375 L 1.578125 -1.28125 C 1.59375 -1.40625 1.671875 -1.703125 1.703125 -1.828125 C 1.8125 -2.25 1.8125 -2.25 1.984375 -2.515625 C 2.25 -2.90625 2.625 -3.25 3.140625 -3.25 C 3.421875 -3.25 3.59375 -3.078125 3.59375 -2.71875 C 3.59375 -2.5625 3.53125 -2.265625 3.53125 -2.265625 L 2.640625 1.265625 C 2.609375 1.390625 2.609375 1.4375 2.609375 1.453125 C 2.609375 1.640625 2.765625 1.6875 2.859375 1.6875 C 3.140625 1.6875 3.203125 1.4375 3.21875 1.34375 Z M 4.09375 -2.171875 "></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-1"> <path style="stroke:none;" d="M 1.546875 -4.671875 C 1.546875 -4.859375 1.546875 -5.078125 1.3125 -5.078125 C 1.078125 -5.078125 1.078125 -4.859375 1.078125 -4.65625 L 1.078125 -0.15625 C 1.078125 0.046875 1.078125 0.265625 1.3125 0.265625 C 1.546875 0.265625 1.546875 0.046875 1.546875 -0.15625 Z M 1.546875 -4.671875 "></path> </symbol> </g> <clipPath id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip1"> <path d="M 106 0 L 192.988281 0 L 192.988281 49.566406 L 106 49.566406 Z M 106 0 "></path> </clipPath> <clipPath id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip2"> <path d="M 120 0 L 186 0 L 186 49.566406 L 120 49.566406 Z M 120 0 "></path> </clipPath> <clipPath id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip3"> <path d="M 126 0 L 192.988281 0 L 192.988281 49.566406 L 126 49.566406 Z M 126 0 "></path> </clipPath> <clipPath id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip4"> <path d="M 140 0 L 192.988281 0 L 192.988281 49.566406 L 140 49.566406 Z M 140 0 "></path> </clipPath> </defs> <g id="BiGu3peNhT_xJMSWV0QG_rxII_Q=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="6.241929" y="13.089039"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="50.145467" y="13.089039"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="94.049005" y="13.089039"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="6.241929" y="42.369566"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="94.049005" y="42.369566"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="137.952544" y="42.369566"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph0-1" x="181.856082" y="42.369566"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -80.649939 -14.433979 L -8.895845 -14.433979 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.488176 2.870358 C -2.032507 1.146738 -1.018147 0.334458 0.000175904 0.00162067 C -1.018147 -0.335179 -2.032507 -1.147459 -2.488176 -2.871079 " transform="matrix(0.985843,0,0,-0.985843,88.542795,39.423473)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-1" x="51.667609" y="47.028661"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-1" x="51.864777" y="42.369566"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -80.649939 15.267754 L -53.432595 15.267754 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486445 2.870651 C -2.030775 1.147032 -1.020377 0.334751 0.00190758 0.00191403 C -1.020377 -0.334885 -2.030775 -1.147166 -2.486445 -2.870786 " transform="matrix(0.985843,0,0,-0.985843,44.638744,10.142512)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-1" x="29.71584" y="7.369177"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-1" x="29.913008" y="13.089039"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -36.117151 15.267754 L -8.895845 15.267754 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.488176 2.870651 C -2.032507 1.147032 -1.018147 0.334751 0.000175904 0.00191403 C -1.018147 -0.334885 -2.032507 -1.147166 -2.488176 -2.870786 " transform="matrix(0.985843,0,0,-0.985843,88.542795,10.142512)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-1" x="73.619378" y="7.369177"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-1" x="73.816547" y="13.089039"></use> </g> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -89.069921 7.703639 L -89.069921 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M -89.069921 7.703639 L -89.069921 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.000381658 7.703639 L -0.000381658 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.000381658 7.703639 L -0.000381658 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 52.952388 -14.433979 L 80.169732 -14.433979 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487657 2.870358 C -2.031988 1.146738 -1.02159 0.334458 0.000694892 0.00162067 C -1.02159 -0.335179 -2.031988 -1.147459 -2.487657 -2.871079 " transform="matrix(0.985843,0,0,-0.985843,176.350877,39.423473)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-1" x="161.426455" y="47.028661"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph2-1" x="161.623623" y="42.369566"></use> </g> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -44.53317 7.703639 L -44.53317 -8.086304 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M -44.53317 7.703639 L -44.53317 -8.086304 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.551595 3.068185 C -0.79875 1.447586 1.226008 0.0568028 2.386975 0.00133002 C 1.226008 -0.0581051 -0.79875 -1.448888 -1.551595 -3.069487 " transform="matrix(0,0.985843,0.985843,0,53.17447,33.162442)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-2" x="45.379901" y="29.075472"></use> </g> <g clip-path="url(#BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip1)" clip-rule="nonzero"> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 63.860722 11.341071 L 49.580433 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> </g> <g clip-path="url(#BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip2)" clip-rule="nonzero"> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 63.860722 11.341071 L 49.580433 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> </g> <g clip-path="url(#BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip3)" clip-rule="nonzero"> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 69.744804 11.341071 L 84.021131 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> </g> <g clip-path="url(#BiGu3peNhT_xJMSWV0QG_rxII_Q=-clip4)" clip-rule="nonzero"> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 69.744804 11.341071 L 84.021131 -7.701956 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> </g> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 66.800782 5.563973 L 66.800782 -7.087793 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 66.800782 5.563973 L 66.800782 -7.087793 " transform="matrix(0.985843,0,0,-0.985843,97.078501,25.192236)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.553826 3.069839 C -0.800981 1.44924 1.22774 0.0584572 2.384744 -0.000978003 C 1.22774 -0.0564508 -0.800981 -1.447234 -1.553826 -3.067833 " transform="matrix(0,0.985843,0.985843,0,162.934558,32.180266)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#BiGu3peNhT_xJMSWV0QG_rxII_Q=-glyph1-3" x="155.689833" y="28.493824"></use> </g> </g> </svg> This data satisfies strict unitality and associativity equations.</p> <p>A <em>left module</em> <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> over a <em>monad</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><mi>a</mi><mo>↛</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">t:a \nrightarrow a</annotation></semantics></math> is another loose cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>a</mi><mo>↛</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">m:a \nrightarrow b</annotation></semantics></math> together with a binary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊳</mo><mo>:</mo><mi>t</mi><mo>,</mo><mi>m</mi><mo>→</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\rhd : t,m \to m</annotation></semantics></math>:</p>  <p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="104.34pt" height="51.963pt" viewBox="0 0 104.34 51.963" version="1.2"> <defs> <g> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-1"> <path style="stroke:none;" d="M 3.578125 -1.40625 C 3.515625 -1.21875 3.515625 -1.1875 3.34375 -0.96875 C 3.09375 -0.625 2.5625 -0.125 2 -0.125 C 1.515625 -0.125 1.25 -0.5625 1.25 -1.265625 C 1.25 -1.90625 1.609375 -3.234375 1.84375 -3.734375 C 2.25 -4.578125 2.796875 -5 3.265625 -5 C 4.046875 -5 4.203125 -4.03125 4.203125 -3.9375 C 4.203125 -3.921875 4.171875 -3.765625 4.15625 -3.734375 Z M 4.328125 -4.453125 C 4.203125 -4.765625 3.890625 -5.234375 3.265625 -5.234375 C 1.921875 -5.234375 0.46875 -3.5 0.46875 -1.75 C 0.46875 -0.5625 1.15625 0.125 1.96875 0.125 C 2.625 0.125 3.1875 -0.390625 3.515625 -0.78125 C 3.640625 -0.078125 4.1875 0.125 4.546875 0.125 C 4.90625 0.125 5.1875 -0.09375 5.40625 -0.515625 C 5.59375 -0.921875 5.765625 -1.65625 5.765625 -1.703125 C 5.765625 -1.75 5.71875 -1.8125 5.640625 -1.8125 C 5.53125 -1.8125 5.515625 -1.75 5.46875 -1.5625 C 5.296875 -0.859375 5.078125 -0.125 4.578125 -0.125 C 4.234375 -0.125 4.21875 -0.421875 4.21875 -0.671875 C 4.21875 -0.9375 4.25 -1.0625 4.359375 -1.53125 C 4.4375 -1.828125 4.5 -2.09375 4.59375 -2.4375 C 5.03125 -4.21875 5.140625 -4.640625 5.140625 -4.71875 C 5.140625 -4.875 5.015625 -5.015625 4.828125 -5.015625 C 4.453125 -5.015625 4.359375 -4.59375 4.328125 -4.453125 Z M 4.328125 -4.453125 "></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-2"> <path style="stroke:none;" d="M 2.75 -7.9375 C 2.75 -7.984375 2.78125 -8.0625 2.78125 -8.125 C 2.78125 -8.234375 2.65625 -8.234375 2.640625 -8.234375 C 2.625 -8.234375 2.203125 -8.203125 1.984375 -8.1875 C 1.78125 -8.171875 1.609375 -8.140625 1.390625 -8.140625 C 1.109375 -8.109375 1.015625 -8.09375 1.015625 -7.890625 C 1.015625 -7.765625 1.140625 -7.765625 1.265625 -7.765625 C 1.859375 -7.765625 1.859375 -7.65625 1.859375 -7.546875 C 1.859375 -7.453125 1.765625 -7.109375 1.71875 -6.90625 L 1.4375 -5.765625 C 1.3125 -5.28125 0.640625 -2.59375 0.59375 -2.375 C 0.53125 -2.078125 0.53125 -1.875 0.53125 -1.71875 C 0.53125 -0.515625 1.21875 0.125 1.984375 0.125 C 3.359375 0.125 4.78125 -1.65625 4.78125 -3.375 C 4.78125 -4.46875 4.171875 -5.234375 3.28125 -5.234375 C 2.65625 -5.234375 2.109375 -4.71875 1.875 -4.484375 Z M 2 -0.125 C 1.609375 -0.125 1.203125 -0.40625 1.203125 -1.328125 C 1.203125 -1.71875 1.234375 -1.953125 1.453125 -2.78125 C 1.484375 -2.9375 1.671875 -3.6875 1.71875 -3.84375 C 1.75 -3.9375 2.453125 -5 3.25 -5 C 3.78125 -5 4.015625 -4.484375 4.015625 -3.859375 C 4.015625 -3.296875 3.6875 -1.953125 3.390625 -1.328125 C 3.09375 -0.6875 2.546875 -0.125 2 -0.125 Z M 2 -0.125 "></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-1"> <path style="stroke:none;" d="M 1.578125 -1.296875 C 1.609375 -1.421875 1.6875 -1.71875 1.71875 -1.84375 C 1.734375 -1.921875 1.78125 -2.109375 1.796875 -2.1875 C 1.8125 -2.21875 2.078125 -2.734375 2.421875 -3 C 2.6875 -3.203125 2.953125 -3.265625 3.171875 -3.265625 C 3.46875 -3.265625 3.625 -3.09375 3.625 -2.734375 C 3.625 -2.546875 3.578125 -2.359375 3.5 -2 C 3.4375 -1.796875 3.296875 -1.265625 3.25 -1.046875 L 3.140625 -0.578125 C 3.09375 -0.4375 3.046875 -0.203125 3.046875 -0.171875 C 3.046875 0.015625 3.1875 0.078125 3.296875 0.078125 C 3.4375 0.078125 3.5625 -0.015625 3.609375 -0.109375 C 3.640625 -0.15625 3.703125 -0.421875 3.734375 -0.59375 L 3.921875 -1.296875 C 3.9375 -1.421875 4.03125 -1.71875 4.046875 -1.84375 C 4.15625 -2.265625 4.15625 -2.28125 4.34375 -2.53125 C 4.609375 -2.921875 4.96875 -3.265625 5.5 -3.265625 C 5.796875 -3.265625 5.953125 -3.109375 5.953125 -2.734375 C 5.953125 -2.296875 5.625 -1.390625 5.46875 -1 C 5.390625 -0.796875 5.375 -0.75 5.375 -0.59375 C 5.375 -0.140625 5.75 0.078125 6.078125 0.078125 C 6.859375 0.078125 7.1875 -1.03125 7.1875 -1.140625 C 7.1875 -1.21875 7.125 -1.234375 7.0625 -1.234375 C 6.96875 -1.234375 6.953125 -1.1875 6.9375 -1.09375 C 6.734375 -0.4375 6.40625 -0.140625 6.109375 -0.140625 C 5.984375 -0.140625 5.921875 -0.21875 5.921875 -0.40625 C 5.921875 -0.59375 5.984375 -0.765625 6.0625 -0.953125 C 6.171875 -1.265625 6.53125 -2.171875 6.53125 -2.609375 C 6.53125 -3.203125 6.109375 -3.5 5.546875 -3.5 C 5 -3.5 4.546875 -3.203125 4.1875 -2.71875 C 4.125 -3.34375 3.625 -3.5 3.203125 -3.5 C 2.84375 -3.5 2.359375 -3.359375 1.921875 -2.796875 C 1.875 -3.265625 1.484375 -3.5 1.109375 -3.5 C 0.84375 -3.5 0.640625 -3.328125 0.5 -3.0625 C 0.3125 -2.6875 0.234375 -2.296875 0.234375 -2.28125 C 0.234375 -2.203125 0.296875 -2.171875 0.359375 -2.171875 C 0.453125 -2.171875 0.46875 -2.203125 0.515625 -2.421875 C 0.625 -2.796875 0.765625 -3.265625 1.09375 -3.265625 C 1.296875 -3.265625 1.34375 -3.078125 1.34375 -2.90625 C 1.34375 -2.75 1.3125 -2.609375 1.25 -2.34375 C 1.234375 -2.28125 1.109375 -1.8125 1.078125 -1.703125 L 0.78125 -0.515625 C 0.75 -0.390625 0.703125 -0.203125 0.703125 -0.171875 C 0.703125 0.015625 0.859375 0.078125 0.953125 0.078125 C 1.09375 0.078125 1.21875 -0.015625 1.28125 -0.109375 C 1.296875 -0.15625 1.359375 -0.421875 1.40625 -0.59375 Z M 1.578125 -1.296875 "></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-2"> <path style="stroke:none;" d="M 1.75 -3.15625 L 2.53125 -3.15625 C 2.671875 -3.15625 2.765625 -3.15625 2.765625 -3.296875 C 2.765625 -3.421875 2.671875 -3.421875 2.53125 -3.421875 L 1.8125 -3.421875 L 2.09375 -4.546875 C 2.125 -4.65625 2.125 -4.703125 2.125 -4.703125 C 2.125 -4.875 2 -4.953125 1.875 -4.953125 C 1.59375 -4.953125 1.546875 -4.734375 1.453125 -4.375 L 1.21875 -3.421875 L 0.453125 -3.421875 C 0.296875 -3.421875 0.203125 -3.421875 0.203125 -3.265625 C 0.203125 -3.15625 0.296875 -3.15625 0.4375 -3.15625 L 1.15625 -3.15625 L 0.671875 -1.25 C 0.625 -1.046875 0.546875 -0.78125 0.546875 -0.671875 C 0.546875 -0.1875 0.9375 0.078125 1.359375 0.078125 C 2.203125 0.078125 2.6875 -1.03125 2.6875 -1.140625 C 2.6875 -1.21875 2.625 -1.234375 2.578125 -1.234375 C 2.484375 -1.234375 2.484375 -1.203125 2.421875 -1.078125 C 2.265625 -0.703125 1.875 -0.140625 1.390625 -0.140625 C 1.21875 -0.140625 1.125 -0.25 1.125 -0.515625 C 1.125 -0.671875 1.15625 -0.75 1.171875 -0.859375 Z M 1.75 -3.15625 "></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph2-1"> <path style="stroke:none;" d="M 1.5625 -4.703125 C 1.5625 -4.90625 1.5625 -5.125 1.3125 -5.125 C 1.078125 -5.125 1.078125 -4.890625 1.078125 -4.6875 L 1.078125 -0.15625 C 1.078125 0.046875 1.078125 0.265625 1.3125 0.265625 C 1.5625 0.265625 1.5625 0.046875 1.5625 -0.15625 Z M 1.5625 -4.703125 "></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph3-1"> <path style="stroke:none;" d="M 5.34375 -1.796875 C 5.46875 -1.859375 5.5 -1.90625 5.5 -1.96875 C 5.5 -2.03125 5.46875 -2.09375 5.34375 -2.15625 L 0.953125 -4.21875 C 0.859375 -4.265625 0.84375 -4.265625 0.828125 -4.265625 C 0.65625 -4.265625 0.65625 -4.125 0.65625 -3.984375 L 0.65625 0.046875 C 0.65625 0.171875 0.65625 0.3125 0.8125 0.3125 C 0.84375 0.3125 0.859375 0.3125 0.953125 0.265625 Z M 4.96875 -1.984375 L 0.96875 -0.09375 L 0.96875 -3.859375 Z M 4.96875 -1.984375 "></path> </symbol> </g> </defs> <g id="0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-1" x="5.114647" y="13.339467"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-1" x="49.368719" y="13.339467"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-2" x="93.622791" y="13.339467"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-1" x="5.114647" y="45.988943"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph0-2" x="93.622791" y="45.988943"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -35.530251 -17.590805 L 36.221541 -17.590805 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487107 2.869245 C -2.031116 1.147485 -1.020859 0.333777 0.00118997 -0.000354881 C -1.020859 -0.334486 -2.031116 -1.148195 -2.487107 -2.869955 " transform="matrix(0.993714,0,0,-0.993714,88.073036,43.019179)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-1" x="48.700943" y="49.223483"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph2-1" x="51.101757" y="45.988943"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -35.530251 15.264149 L -8.312292 15.264149 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.48644 2.867909 C -2.030449 1.146149 -1.020193 0.336372 0.00185677 -0.00169052 C -1.020193 -0.335822 -2.030449 -1.149531 -2.48644 -2.86736 " transform="matrix(0.993714,0,0,-0.993714,43.818467,10.369414)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-2" x="28.775978" y="7.573937"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph2-1" x="28.974721" y="13.339467"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 9.003582 15.264149 L 36.221541 15.264149 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487107 2.867909 C -2.031116 1.146149 -1.020859 0.336372 0.00118997 -0.00169052 C -1.020859 -0.335822 -2.031116 -1.149531 -2.487107 -2.86736 " transform="matrix(0.993714,0,0,-0.993714,88.073036,10.369414)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph1-1" x="70.826986" y="7.573937"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph2-1" x="73.228793" y="13.339467"></use> </g> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -43.950365 7.700985 L -43.950365 -10.857072 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M -43.950365 7.700985 L -43.950365 -10.857072 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 44.53552 7.700985 L 44.53552 -7.700512 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 44.53552 7.700985 L 44.53552 -7.700512 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.583468 7.700985 L 0.583468 -11.238375 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.583468 7.700985 L 0.583468 -11.238375 " transform="matrix(0.993714,0,0,-0.993714,51.842074,25.539297)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.549952 3.069637 C -0.799139 1.446151 1.229235 0.0585226 2.384937 -0.000441771 C 1.229235 -0.0594062 -0.799139 -1.447035 -1.549952 -3.070521 " transform="matrix(0,0.993714,0.993714,0,52.422314,36.708179)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#0Wh6YG8Qmpx4Weme6XXLSvtbdLw=-glyph3-1" x="43.46705" y="29.683086"></use> </g> </g> </svg> This again satisfies standard equations stating compatibility with the monad structure on <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>.</p> <p>A <em>right module</em> is equipped with a binary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊲</mo><mo>:</mo><mi>m</mi><mo>,</mo><mi>t</mi><mo>→</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\lhd : m, t \to m</annotation></semantics></math> instead, and a bimodule has a ternary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊳</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊲</mo><mo>:</mo><mi>t</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>t</mi><mo>→</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\rhd - \lhd : t,m,t \to m</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="colimits">Colimits</h3> <p>see at <em><a class="existingWikiWord" href="/nlab/show/colimits+in+categories+of+algebras">colimits in categories of algebras</a></em></p> <h3 id="tensor_product">Tensor product</h3> <p>Given bimodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>r</mi><mi>⇸</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">x' \colon r &#8696; s</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>s</mi><mi>⇸</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x \colon s &#8696; t</annotation></semantics></math>, where <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> are monads on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">c,b,a</annotation></semantics></math> respectively, we may be able to form the <strong>tensor product</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>⊗</mo> <mi>s</mi></msub><mi>x</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>r</mi><mi>⇸</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x \otimes_s x' \colon r &#8696; t</annotation></semantics></math> just as in the case of bimodules over rings. If the hom-categories of the bicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> have <a class="existingWikiWord" href="/nlab/show/reflexive+coequalizer">reflexive coequalizer</a>s that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(c,a)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi><mi>s</mi><mi>x</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mo>→</mo></mover></mtd> <mtd><mi>x</mi><mi>x</mi><mo>′</mo></mtd> <mtd><mo>→</mo><mi>x</mi><msub><mo>⊗</mo> <mi>s</mi></msub><mi>x</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x' } </annotation></semantics></math></div> <p>where the parallel arrows are the two induced actions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\rho x'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>λ</mi></mrow><annotation encoding="application/x-tex">x \lambda</annotation></semantics></math> on <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>. Indeed, under the hypothesis on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>r</mi> <mo>*</mo></msup><msub><mi>t</mi> <mo>*</mo></msub></mrow></msup><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/colimits+in+categories+of+algebras">reflects reflexive coequalizers</a>, because the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mo>*</mo></msup><msub><mi>t</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">r^* t_*</annotation></semantics></math> preserves them, and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>⊗</mo> <mi>s</mi></msub><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x \otimes_s x'</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>,</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">r,t</annotation></semantics></math>-bimodule.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> satisfies the above conditions then there is a bicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(K)</annotation></semantics></math> consisting of monads, bimodules and bimodule morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. The identity module on a monad <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> is <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> itself, and the unit and associativity conditions follow from the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the above coequalizer. There is a <a class="existingWikiWord" href="/nlab/show/lax+functor">lax</a> forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">Mod(K) \to K</annotation></semantics></math>, with comparison morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>a</mi></msub><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">1_a \to t</annotation></semantics></math> the unit of <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>x</mi><mo>′</mo><mo>→</mo><mi>x</mi><msub><mo>⊗</mo> <mi>s</mi></msub><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x x' \to x \otimes_s x'</annotation></semantics></math> the coequalizer map.</p> <h2 id="examples">Examples</h2> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>Span</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K = Span(Set)</annotation></semantics></math>, the bicategory of <a class="existingWikiWord" href="/nlab/show/span">span</a>s of sets, then a monad in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is precisely a small category. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Prof</mi></mrow><annotation encoding="application/x-tex">Mod(K) = Prof</annotation></semantics></math>, the category of small categories, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a>s and natural transformations.</p> <p>More generally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mi>Span</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(Span(C))</annotation></semantics></math>, for <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> any category with coequalizers and pullbacks that preserve them, consists of <a class="existingWikiWord" href="/nlab/show/internal+category">internal categories</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, together with <a class="existingWikiWord" href="/nlab/show/internal+profunctor">internal profunctors</a> between them and transformations between those.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>algebra over a monad</strong>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+endofunctor">algebra over an endofunctor</a>, <a class="existingWikiWord" href="/nlab/show/coalgebra+over+an+endofunctor">coalgebra over an endofunctor</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+a+profunctor">algebra over a profunctor</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> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">model structure on algebras over a monad</a></li> </ul> </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> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a> / <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a></li> </ul> </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> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+object">Eilenberg-Moore object</a>, <a class="existingWikiWord" href="/nlab/show/Kleisli+object">Kleisli object</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Isbell">John Isbell</a>, <em>Generic algebras</em> Transactions of the AMS, vol 275, number 2 (<a href="http://www.ams.org/journals/tran/1983-275-02/S0002-9947-1983-0682715-8/S0002-9947-1983-0682715-8.pdf">pdf</a>)</p> </li> <li> <p>H. Lindner, <em>Commutative monads</em> in <em>Deuxiéme colloque sur l’algébre des catégories</em>. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.</p> </li> <li> <p>R. Guitart, <em>Tenseurs et machines</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.</p> </li> <li> <p>A. Kock. <em>Closed categories generated by commutative monads</em>, Journal of the Australian Mathematical Society, 12(04):405-424, 1971.</p> </li> <li> <p>G. J. Seal. <em>Tensors, monads and actions</em>, Theory Appl. Categ., 28:No. 15, 403-433, 2013.</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on categories of <a class="existingWikiWord" href="/nlab/show/coalgebras">coalgebras</a> over <a class="existingWikiWord" href="/nlab/show/comonads">comonads</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>The homotopy theory of coalgebras over a comonad</em> (<a href="http://arxiv.org/abs/1205.3979">arXiv:1205.3979</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 27, 2024 at 17:41:19. See the <a href="/nlab/history/module+over+a+monad" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/module+over+a+monad" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/647/#Item_24">Discuss</a><span class="backintime"><a href="/nlab/revision/module+over+a+monad/38" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/module+over+a+monad" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/module+over+a+monad" accesskey="S" class="navlink" id="history" rel="nofollow">History (38 revisions)</a> <a href="/nlab/show/module+over+a+monad/cite" style="color: black">Cite</a> <a href="/nlab/print/module+over+a+monad" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/module+over+a+monad" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>