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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the notion of <em>monad</em> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>. For other notions see <em><a class="existingWikiWord" href="/nlab/show/monad+%28disambiguation%29">monad (disambiguation)</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="categorical_algebra">Categorical algebra</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>+<a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+object">group object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+object">algebra object</a> (associative, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+object">module object</a>/<a class="existingWikiWord" href="/nlab/show/action+object">action object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internal+infinity-categories+contents">more</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+site">internal site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+diagram">internal diagram</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/monads">monads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/operads">operads</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+category+theory+and+type+theory">relation between category theory and type theory</a></p> </li> </ul> </div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#Etymology'>Etymology</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#monads'>Monads</a></li> <li><a href='#BicategoryOfMonads'>The 2-category of monads</a></li> <li><a href='#Algebras'>Algebras/modules over a monad</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationBetweenAdjunctionsAndMonads'>Relation between adjunctions and monads</a></li> <ul> <li><a href='#MonadInducedByAnAdjunction'>Monad induced by an adjunction</a></li> <li><a href='#CategoryOfAdjunctionResolutionsOfAMonad'>Category of adjunction-resolutions of a monad</a></li> <li><a href='#SemanticsStructureAdjunction'>Semantics-structure adjunction</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#monads_on_'>Monads on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></a></li> <li><a href='#algebra'>Algebra</a></li> <li><a href='#topology'>Topology</a></li> <li><a href='#monads_in_cat'>Monads in Cat</a></li> <li><a href='#other_examples'>Other examples</a></li> </ul> <li><a href='#monads_in_higher_category_theory'>Monads in higher category theory</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>In <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, the notion of <em><a class="existingWikiWord" href="/nlab/show/monad">monad</a></em> (earlier: “<em>standard construction</em>” or “<em>triple</em>”) is a kind of <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of that of <em><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></em>: In their default incarnation monads are <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> on some <a class="existingWikiWord" href="/nlab/show/category">category</a> which are equipped with a <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> under <a class="existingWikiWord" href="/nlab/show/composition">composition</a>. More generally this notion makes sense for <a class="existingWikiWord" href="/nlab/show/endomorphism">endo-</a> <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> on any <a class="existingWikiWord" href="/nlab/show/object">object</a> in any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> beyond <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>: Monads are <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/endofunctor">endo</a>-<a class="existingWikiWord" href="/nlab/show/hom-categories">hom-categories</a>.</p> <p>Together with the <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> (<a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>) that they correspond to (see <a href="#RelationBetweenAdjunctionsAndMonads">below</a>) monads are among the most pervasive structures in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> (where they form the basis of <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical</a> and <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>, whence one speaks of <em><a class="existingWikiWord" href="/nlab/show/algebras+over+a+monad">algebras over a monad</a></em>) and in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> more generally (certainly in fields like <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a> and <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, where the notion originates in the guise of “<a class="existingWikiWord" href="/nlab/show/canonical+resolutions">canonical resolutions</a>”).</p> <p>Last not least, monads play a central role in <a class="existingWikiWord" href="/nlab/show/formal+logic">formal logic</a> (cf. <a class="existingWikiWord" href="/nlab/show/modal+logic">modal logic</a> and <a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>) and in <a class="existingWikiWord" href="/nlab/show/computer+science">computer science</a>, where they are understood (cf. the “<a class="existingWikiWord" href="/nlab/show/computational+trilogy">computational trilogy</a>”) as encoding “notions of computation” with “computational effects” in the framework of <a class="existingWikiWord" href="/nlab/show/functional+programming">functional programming</a>: see at <em><a class="existingWikiWord" href="/nlab/show/monad+%28computer+science%29">monads in computer science</a></em>.</p> <h2 id="Etymology">Etymology</h2> <p>The terminology “monad” was introduced in <a href="#Bénabou67">Bénabou 1967, Def. 5.4.1</a>, where, after observing (Exp. 5.4.1) that monads in 1-object 2-categories (<a href="delooping#DeloopingOfHigherCategoricalStructures">deloopings of</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>) are <em><a class="existingWikiWord" href="/nlab/show/monoids">monoids</a></em>:</p> <div style="margin: -20px 0px 20px 10px"> <img src="/nlab/files/Benabou-MonoidsAsMonads.jpg" width="700px" /> </div> <p id="BenabouFootnote"> it says in a footnote:</p> <div style="margin: -20px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/Benabou-FootnoteOnMonadTerminology.jpg" width="700px" /> <figcaption style="text-align: center">(from <a href="#Bénabou67">Bénabou 1967, p. 40</a>)</figcaption> </figure> </div> <p>Beyond this footnote, the only contemporary account that seems to exist of the terminological genesis, <a href="#Barr09">Barr 2009</a>, recalls the following exchange, on the backdrop of a widely felt dissatisfaction with the earlier terminology of “standard construction” and “triple”:</p> <blockquote> <p>In the summer (or maybe late spring, the Oberwohlfach records will show this) of 1966, there was a category meeting there. […] One day at lunch or dinner I happened to be sitting next to Jean Bénabou and he turned to me and said something like “How about `monad'?" I thought about and said it sounded pretty good to me. So Jean proposed it to the general audience and there was general agreement. It suggested "monoid" of course and it is a monoid in a functor category.</p> </blockquote> <p id="MonadTerminologyFurtherDiscussion"> Further discussion on the issue in 2023 has recollections by <a class="existingWikiWord" href="/nlab/show/Jir%C3%AD+Ad%C3%A1mek">Jirí Adámek</a> of recollections by <a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a> to the extent that:</p> <blockquote> <p>it was in the common room of the old castle at Oberwolfach when Sammy [<a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>] came out from behind the piano and announced the change.</p> </blockquote> <p>But <a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a> clarifies that:</p> <blockquote> <p>I am more than willing to believe that it was Bénabou sitting next to me who proposed monad. It is entirely possible that Sammy came down and pronounced it “official”. And it was certainly in the old castle.</p> </blockquote> <p>Interestingly Bénabou’s footnote <a href="#BenabouFootnote">above</a> gives a second motivation for “monad”:</p> <div class="float_right_image" style="margin: -20px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/Benabou-MonadDefinition.jpg" width="650px" /> <figcaption style="text-align: center">(from <a href="#Bénabou67">Bénabou 1967, p. 39</a>)</figcaption> </figure> </div> <p id="ItIsStriking"> It is striking that <a href="#Bénabou67">Bénabou 1967, Def. 5.4.1</a> <em>defines</em> a monad to be a <a class="existingWikiWord" href="/nlab/show/lax+2-functor">lax 2-functor</a> from the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a> <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> to the <a class="existingWikiWord" href="/nlab/show/Cat">2-category of categories</a> (and more generally to whatever given ambient <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>S</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{S}</annotation></semantics></math>”) and then proceeds to unwind the equivalence of this definition to the traditional one:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Monads</mi><mo stretchy="false">(</mo><munder><mi>S</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">{</mo><mn>1</mn><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>lax</mi><mspace width="thickmathspace"></mspace></mrow></mover><munder><mi>S</mi><mo>̲</mo></munder><mo maxsize="1.8em" minsize="1.8em">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Monads(\underline{S}) \;\;\; \simeq \;\;\; \Big\{ 1 \xrightarrow{\; lax \;} \underline{S} \Big\} \,. </annotation></semantics></math></div> <p>In this sense, monads are <em><a class="existingWikiWord" href="/nlab/show/global+element">point-like elements</a></em> in a <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theoretic</a> sense (say in the <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>), which squares well with <a href="monad+terminology#HistoricalOrigins">Euclid’s ancient notion of monads</a> as indivisible building blocks. In fact, as discussed there, “monad” (both in ancient and still in modern Greek) just means “unit” in the sense of the unit <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <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>, and <a href="#Bénabou67">Bénabou 1967, Def. 5.4.1</a> literally identifies monads with the (lax) units <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><munder><mi>S</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">1 \to \underline{S}</annotation></semantics></math> in the ambient 2-category.</p> <p>In generalization of this situation, one may consider lax functors out of <a class="existingWikiWord" href="/nlab/show/codiscrete+groupoids">codiscrete groupoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CoDisc</mi><mo stretchy="false">(</mo><msup><mn>1</mn> <mrow><msub><mo>⊔</mo> <mi>n</mi></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CoDisc(1^{\sqcup_n})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> objects, which <a href="#Bénabou67">Bénabou 1967, Def. 5.5</a> calls <em>polyads</em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Polyads</mi><mo stretchy="false">(</mo><munder><mi>S</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">{</mo><mi>CoDisc</mi><mo stretchy="false">(</mo><msup><mn>1</mn> <mrow><msub><mo>⊔</mo> <mi>n</mi></msub></mrow></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>lax</mi><mspace width="thickmathspace"></mspace></mrow></mover><munder><mi>S</mi><mo>̲</mo></munder><mo maxsize="1.8em" minsize="1.8em">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Polyads(\underline{S}) \;\;\; \simeq \;\;\; \Big\{ CoDisc(1^{\sqcup_n}) \xrightarrow{\; lax \;} \underline{S} \Big\} \,. </annotation></semantics></math></div> <p>On the other hand (as maybe alluded to in the first line of <a href="#Barr09">Barr 2009</a>), just a few years earlier the ancient <a class="existingWikiWord" href="/nlab/show/monad+terminology">monad terminology</a> had already been adopted in <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> as the term for <em><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhoods">infinitesimal neighbourhoods</a></em> (<a href="infinitesimal+neighborhood#Robinson66">Robinson 1966, p. 57</a> and <a href="#infinitesimal+neighborhood#Luxemburg66">Luxembourg 1966</a>, compare also <a href="infinitesimal+neighborhood#Keisler76">Keisler 1976, Def. 1.2</a>, <a href="infinitesimal+neighborhood#Kutateladze11">Kutateladze 2011</a> and, speaking <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetically</a>: <a href="infinitesimal+neighborhood#Kock80">Kock 1980</a>).</p> <p>Now it so happens — in the <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theoretic</a> formulation of <a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a> via <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> — that the construction of <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhoods">infinitesimal neighbourhoods</a> <em>is</em> (see <a href="infinitesimal+disk+bundle#MonadicityAdjointToJetBundles">here</a>) a monad in the sense of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>! – namely the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+monad">adjoint monad</a> to the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> (<a href="infinitesimal+disk+bundle#KhavkineSchreiber17">Khavkine & Schreiber 2017, p. 23</a>).</p> <h2 id="definition">Definition</h2> <h3 id="monads">Monads</h3> <p>A <strong>monad</strong> 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> is given by</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><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 \to 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></p> <p>(the <em><a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a></em> or <em>return</em> operation)</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <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><mo>∘</mo><mi>t</mi><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\mu \colon t \circ t \to t</annotation></semantics></math></p> <p>(the <em>multiplication</em> or <em>join</em> operation)</p> </li> </ol> <p>such that the diagrams <a href="https://q.uiver.app/#q=WzAsNCxbMCwwLCJ0Il0sWzEsMCwidHQiXSxbMiwwLCJ0Il0sWzEsMSwidCJdLFswLDEsIlxcZXRhIHQiXSxbMiwxLCJ0XFxldGEiLDJdLFswLDMsIiIsMix7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzIsMywiIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMSwzLCJcXG11IiwxXV0="><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="172.118pt" height="86.275pt" viewBox="0 0 172.118 86.275" version="1.2"> <defs> 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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 -25.202589 30.27547 L 30.007616 30.27547 " transform="matrix(0.98933,0,0,-0.98933,56.378982,44.593046)"></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 -2.487583 2.869682 C -2.033519 1.148189 -1.018785 0.334822 -0.000103189 -0.000790196 C -1.018785 -0.336403 -2.033519 -1.145821 -2.487583 -2.867314 " transform="matrix(0.98933,0,0,-0.98933,86.30479,14.639843)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#ZisIvSfgTlrsxSOw2xCmRtB9ybo=-glyph1-2" x="53.951167" y="9.245286"></use> <use xlink:href="#ZisIvSfgTlrsxSOw2xCmRtB9ybo=-glyph1-1" x="57.733037" y="9.245286"></use> </g> </g> </svg></a> commute (where certain <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s have been omitted).</p> <p>The name “monad” and the terms “unit”, “multiplication” and “associativity” bear a clear analogy with <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> (but see also at <em><a class="existingWikiWord" href="/nlab/show/monad+%28disambiguation%29">monad (disambiguation)</a></em>). Indeed, one can define a monad on an 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> of 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> as just a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in the endomorphism 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>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a,a)</annotation></semantics></math>. Alternatively, monads can be taken as more fundamental, and a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal 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> can be defined as a monad in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} C</annotation></semantics></math>, the one-object bicategory corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>A third and somewhat less obvious definition says that 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 a <strong><a class="existingWikiWord" href="/nlab/show/lax+2-functor">lax 2-functor</a></strong> from the terminal bicategory <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> to <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 unique object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> of <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 sent to 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>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">1_a</annotation></semantics></math> becomes <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>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> arise from the coherent 2-cells expressing lax functoriality. This in turn is equivalent to saying that a monad is a <a class="existingWikiWord" href="/nlab/show/category+enriched+in+a+bicategory">category enriched in a bicategory</a> with a single object and single morphism. Among higher-category theorists, it’s tempting to suggest that this is the most fundamental definition, and the most basic reason for the ubiquity and importance of monads. Regardless of this, however, the earlier more elementary definitions are both practically and pedagogically essential.</p> <p>Finally, a monad can be defined in terms of the “Kleisli operation” taking any map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>→</mo><mi>T</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">a \to T b</annotation></semantics></math> to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>a</mi><mo>→</mo><mi>T</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">T a \to T b</annotation></semantics></math>; see <a class="existingWikiWord" href="/nlab/show/extension+system">extension system</a>.</p> <p>We can picture 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> as an image of the <a class="existingWikiWord" href="/nlab/show/oriental">third oriental</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>. See the remarks at <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</p> <p>The data of and axioms for a monad can be expressed graphically as <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">T \colon C \to C, \eta, \mu</annotation></semantics></math> for the monad in question (this notation being the standard one when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">K = Cat</annotation></semantics></math>), these data can be represented as</p> <p><img src="/nlab/files/monad-data-labeled.png" alt="String diagrams of the monad data (for "Monad")" /></p> <p>Thanks to the distinctive shapes, one can usually omit the labels:</p> <p><img src="/nlab/files/monad-data-unlabeled.png" alt="String diagrams of the monad data, unlabeled (for "Monad")" /></p> <p>The axioms then appear as:</p> <p><img src="/nlab/files/monad-axioms-unlabeled.png" alt="String diagrams of the monad axioms, unlabeled (for "Monad")" /></p> <h3 id="BicategoryOfMonads">The 2-category of monads</h3> <p>Given the equivalence between monads in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</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> and <a class="existingWikiWord" href="/nlab/show/lax+functors">lax functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">1 \to K</annotation></semantics></math> [<a href="#Bénabou67">Bénabou 1967, pp. 39</a>] it is straightforward to define the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mnd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mnd(K)</annotation></semantics></math> of monads 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> to be the <a class="existingWikiWord" href="/nlab/show/lax+functor">lax</a> <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>K</mi><msub><mo stretchy="false">]</mo> <mi>ℓ</mi></msub></mrow><annotation encoding="application/x-tex">[1,K]_\ell</annotation></semantics></math>, which consists of <a class="existingWikiWord" href="/nlab/show/lax+functors">lax functors</a>, <a class="existingWikiWord" href="/nlab/show/lax+transformations">lax transformations</a> and their<a class="existingWikiWord" href="/nlab/show/modifications">modifications</a>.</p> <p>Spelling this out [<a href="#Maranda66">Maranda 1966</a>, <a href="#Maranda68">Maranda 1968</a>, <a href="#Frei69">Frei 1969 p. 269</a>, <a href="#Pumplün70">Pumplün 1970 p 330 & 334</a>, <a href="#Coppey70">Coppey 1970</a>, <a href="#Street72">Street 1972 p. 150-151</a>, review in <a href="#Leinster04">Leinster 2004 pp. 148</a>]:</p> <p> <div class='num_defn' id='TwoCategoryOfMonads'> <h6>Definition</h6> <p><strong>(2-category of monads)</strong> <br /> 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/2-category">2-category</a>.</p> <ol> <li> <p>An <a class="existingWikiWord" href="/nlab/show/object">object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mnd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mnd(K)</annotation></semantics></math> is a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,t,\eta,\mu)</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>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,t) \to (b,s)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mnd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mnd(K)</annotation></semantics></math> (“monad functor” or “<a class="existingWikiWord" href="/nlab/show/monad+transformation">monad transformation</a>”)</p> <p>is given by</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">x \colon a \to b</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></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo lspace="verythinmathspace">:</mo><mi>s</mi><mi>x</mi><mo>→</mo><mi>x</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\lambda \colon s x \to x 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></p> </li> </ol> <p>making the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>η</mi> <mi>s</mi></msup><mi>x</mi></mrow></mover></mtd> <mtd><mi>s</mi><mi>x</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>x</mi><msup><mi>η</mi> <mi>t</mi></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mi>λ</mi></mpadded></mtd></mtr> <mtr><mtd><mi>x</mi><mi>t</mi></mtd> <mtd><mover><mo>⟶</mo><mn>1</mn></mover></mtd> <mtd><mi>x</mi><mi>t</mi></mtd></mtr></mtable></mrow><mspace width="2em"></mspace><mspace width="2em"></mspace><mrow><mtable><mtr><mtd><mi>s</mi><mi>s</mi><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>s</mi><mi>λ</mi></mrow></mover></mtd> <mtd><mi>s</mi><mi>x</mi><mi>t</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>λ</mi><mi>t</mi></mrow></mover></mtd> <mtd><mi>x</mi><mi>t</mi><mi>t</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mi>μ</mi> <mi>s</mi></msup><mi>x</mi></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><mi>x</mi><msup><mi>μ</mi> <mi>t</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>s</mi><mi>x</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>λ</mi></mover></mtd> <mtd></mtd> <mtd><mi>x</mi><mi>t</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ x & \stackrel{\eta^s x}{\to} & s x \\ \mathllap{x \eta^t} \big\downarrow & & \big\downarrow \mathrlap{\lambda} \\ x t & \overset{1}{\longrightarrow} & x t } \qquad \qquad \array{ s s x & \overset{s \lambda}{\to} & s x t & \overset{\lambda t}{\longrightarrow} & x t t \\ \mathllap{\mu^s x} \big\downarrow & & & & \big\downarrow \mathrlap{x \mu^t} \\ s x & & \overset{\lambda}{\longrightarrow} & & x t } </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <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>y</mi><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,\lambda) \Rightarrow (y, \kappa)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mnd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mnd(K)</annotation></semantics></math> is given by</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>⇒</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">m \colon x \Rightarrow y</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></li> </ul> <p>making the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>s</mi><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>s</mi><mi>m</mi></mrow></mover></mtd> <mtd><mi>s</mi><mi>y</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mi>λ</mi></mpadded><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mi>κ</mi></mpadded></mtd></mtr> <mtr><mtd><mi>x</mi><mi>t</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>m</mi><mi>t</mi></mrow></mover></mtd> <mtd><mi>y</mi><mi>t</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ s x & \stackrel{s m}{\to} & s y \\ \mathllap{\lambda} \downarrow & & \downarrow \mathrlap{\kappa} \\ x t & \stackrel{m t}{\to} & y t } </annotation></semantics></math></div></li> </ol> <p></p> </div> </p> <p> <div class='num_remark' id='HandednessOfMonadMorphism'> <h6>Remark</h6> <p><strong>(handedness of the underlying natural transformation)</strong> <br /> Beware that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> in Def. <a class="maruku-ref" href="#TwoCategoryOfMonads"></a> is oriented oppositely to what one might expect. This need not be so but is a possible choice, see <a href="#Pumplün70">Pumplün 1970 p 334</a>, <a href="#Street72">Street 1972 pp 158</a>.</p> <p>One issue is that the functor between <a class="existingWikiWord" href="/nlab/show/Kleisli+categories">Kleisli categories</a> induced by a monad morphism goes in the direction <em>opposite</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> as defined above (as generally for <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> of <a class="existingWikiWord" href="/nlab/show/module+objects">module objects</a> along a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>), so that for authors who adopt the opposite of the above convention (such as <a href="#Frei69">Frei 1969 p. 269</a>, <a href="#Pumplün70">Pumplün 1970 p 330</a>, <a href="#BarrWells85">Barr & Wells 1985 §6.1</a> and in our Exp. <a class="maruku-ref" href="#TransformationOfMonadsOnFixedCategory"></a> below) the association of monad morphisms to functors between Kleisli catgeories is contravariant (eg. <a href="#Frei69">Frei 1969, Thm. 2</a>, <a href="#BarrWells85">Barr & Wells 1985 Thm. 6.3</a>).</p> </div> </p> <p> <div class='num_remark' id='TransformationOfMonadsOnFixedCategory'> <h6>Example</h6> <p><strong>(transformation of monads on a fixed category)</strong> <br /> This example is the simpler but important special case of the general Def. <a class="maruku-ref" href="#TwoCategoryOfMonads"></a> where the monads all act on the same fixed object – in particular the same category if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">K = Cat</annotation></semantics></math> (stated in this form for instance in <a href="#BarrWells85">Barr & Wells 1985 §6.1</a>), of relevance notably for <a class="existingWikiWord" href="/nlab/show/monads+in+computer+science">monads in computer science</a> (where this is <a href="monad+in+computer+science#Moggi89Abstract">Moggi 1989 Def. 4.0.11</a>) which typically all act on the same category of <a class="existingWikiWord" href="/nlab/show/data+types">data</a> <a class="existingWikiWord" href="/nlab/show/types">types</a>:</p> <p>For a pair of monads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><msup><mi>ret</mi> <mi>ℰ</mi></msup><mo>,</mo><msup><mi>join</mi> <mi>ℰ</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{E}, ret^{\mathcal{E}}, join^{\mathcal{E}})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℰ</mi><mo>′</mo><mo>,</mo><msup><mi>ret</mi> <mrow><mi>ℰ</mi><mo>′</mo></mrow></msup><mo>,</mo><msup><mi>join</mi> <mrow><mi>ℰ</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{E}', ret^{\mathcal{E}'}, join^{\mathcal{E}'})</annotation></semantics></math> on a fixed category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>,</mo><mi>ℰ</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⟶</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{E}, \mathcal{E}' \;\colon\; \mathbf{C} \longrightarrow \mathbf{C} \,, </annotation></semantics></math></div> <p>a morphism between them is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>trans</mi> <mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">trans^{\mathcal{E} \to \mathcal{E}'} \,\colon\, \mathcal{E} \to \mathcal{E}'</annotation></semantics></math> between the 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style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#WCHIbF5aRNr5dJvByYStEkVrYy4=-glyph3-1" x="101.867073" y="84.930247"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#WCHIbF5aRNr5dJvByYStEkVrYy4=-glyph3-2" x="106.327368" y="84.930247"></use> <use xlink:href="#WCHIbF5aRNr5dJvByYStEkVrYy4=-glyph3-1" x="113.564024" y="84.930247"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#WCHIbF5aRNr5dJvByYStEkVrYy4=-glyph3-3" x="118.018698" y="82.445991"></use> </g> </g> </svg> <p>in that it makes these <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</a>.</p> <p></p> </div> </p> <p> <div class='num_remark' id='ExtensionOfModalesAlongMonadTransformation'> <h6>Remark</h6> <p>A monad transformation as in Exp. <a class="maruku-ref" href="#TransformationOfMonadsOnFixedCategory"></a> <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariantly</a> induces a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+categories">Eilenberg-Moore categories</a> of <a class="existingWikiWord" href="/nlab/show/module+over+a+monad">modales</a> by <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> [<a href="#Frei69">Frei 1969, Thm. 2</a>, <a href="#BarrWells85">Barr & Wells 1985 thm. 6.3</a>]:</p> <div style="margin: -20px 0px 20px 10px"> <img src="/nlab/files/FunctorOnModalesFromMonadMorphism-230924.jpg" width="450px" /> </div> <p>Since this <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> is the identity on <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, it cannot in general restrict to a functor on <a class="existingWikiWord" href="/nlab/show/Kleisli+categories">Kleisli categories</a>.</p> <p id="ExtensionOfFreeModalesAlsoIsomorphicMonadTransformation"> However, when the <a class="existingWikiWord" href="/nlab/show/monad+transformation">monad transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">tr \,\colon\, \mathcal{E} \to \mathcal{E}'</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>tr</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">tr^\ast</annotation></semantics></math> does take <a href="algebra+over+a+monad#FreeAlgebras">free modales</a> to free modales up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. This is seen from the following 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>ℰ</mi><mi>ℰ</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ℰ</mi><mo stretchy="false">(</mo><msub><mi>tr</mi> <mi>D</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>ℰ</mi><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>trans</mi> <mrow><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow> <mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow></msubsup></mrow></mover></mtd> <mtd><mi>ℰ</mi><mo>′</mo><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><msubsup><mi>join</mi> <mi>D</mi> <mi>ℰ</mi></msubsup></mrow></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><msup><mi>trans</mi> <mo>*</mo></msup><msubsup><mi>join</mi> <mi>D</mi> <mrow><mi>ℰ</mi><mo>′</mo></mrow></msubsup></mrow></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><msubsup><mi>join</mi> <mi>D</mi> <mrow><mi>ℰ</mi><mo>′</mo></mrow></msubsup></mrow></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>ℰ</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>trans</mi> <mi>D</mi> <mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder></mtd> <mtd><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{E} \mathcal{E}(D) & \overset{ \mathcal{E}(tr_D) }{\longrightarrow} & \mathcal{E} \mathcal{E}'(D) & \overset{ trans ^{\mathcal{E} \to \mathcal{E}'} _{\mathcal{E}'(D)} }{\longrightarrow} & \mathcal{E}' \mathcal{E}'(D) \\ \Big\downarrow\mathrlap{ {}^{ join^{\mathcal{E}}_D } } && \Big\downarrow\mathrlap{ {}^{ trans^\ast join^{\mathcal{E}'}_D } } && \Big\downarrow\mathrlap{ {}^{ join^{\mathcal{E}'}_D } } \\ \mathcal{E}(D) & \underset{ \;\; trans^{\mathcal{E} \to \mathcal{E}'}_D \;\; }{\longrightarrow} & \mathcal{E}'(D) &=& \mathcal{E}'(D) } </annotation></semantics></math></div> <p>Here the middle vertical morphism is the nominal image under extension of the free modale on the right along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>trans</mi> <mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">trans^{\mathcal{E} \to \mathcal{E}'}</annotation></semantics></math>, but the square on the left, which commutes by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>trans</mi> <mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">trans^{\mathcal{E} \to \mathcal{E}'}</annotation></semantics></math>, exhibits an isomorphism from the middle modale to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>-free modale on the left.</p> </div> </p> <p> <div class='num_remark' id='MonadTransformers'> <h6>Remark</h6> <p><strong>(monad transfomers)</strong> <br /> When monads are used to model <a class="existingWikiWord" href="/nlab/show/computational+effects">computational effects</a> in <a class="existingWikiWord" href="/nlab/show/functional+programming">functional programming</a>, a common concern is to <em>combine</em> different effects, such that previous effects are subsumed among the newly combined effects. This is formalized by “<a class="existingWikiWord" href="/nlab/show/monad+transformers">monad transformers</a>” which are systems of morphisms of monads as in Exp. <a class="maruku-ref" href="#TransformationOfMonadsOnFixedCategory"></a>, forming a <a class="existingWikiWord" href="/nlab/show/pointed+endofunctor">pointed endofunctor</a> on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mnd</mi></mrow><annotation encoding="application/x-tex">Mnd</annotation></semantics></math> of all monads.</p> </div> </p> <p> <div class='num_remark' id='IdentityMonadIsInitial'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the category of monads on a fixed category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> (Exp. <a class="maruku-ref" href="#TransformationOfMonadsOnFixedCategory"></a>) is the identity monad.</p> </div> <div class='proof'> <h6>Proof</h6> <p>We need to show that for every monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> there is a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>trans</mi> <mrow><mi>Id</mi><mo>→</mo><mi>ℰ</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Id</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">trans^{Id \to \mathcal{E}} \,\colon\, Id \to \mathcal{E}</annotation></semantics></math> which makes, first of all, this square commute:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="157.752pt" height="105.475pt" viewBox="0 0 157.752 105.475" version="1.2"> <defs> <g> <symbol overflow="visible" id="GoyIMVhmeM3K9O3iYNJ5rRzx-q8=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="GoyIMVhmeM3K9O3iYNJ5rRzx-q8=-glyph0-1"> <path style="stroke:none;" d="M 1.859375 -0.875 C 1.765625 -0.46875 1.734375 -0.34375 0.90625 -0.34375 C 0.671875 -0.34375 0.5625 -0.34375 0.5625 -0.125 C 0.5625 0 0.625 0 0.875 0 L 4.640625 0 C 7.046875 0 9.375 -2.484375 9.375 -5.140625 C 9.375 -6.84375 8.359375 -8.125 6.65625 -8.125 L 2.84375 -8.125 C 2.609375 -8.125 2.515625 -8.125 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xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph6-1" x="269.893345" y="64.830996"></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 -119.909544 -43.154338 L 93.776428 -43.154338 " transform="matrix(0.999605,0,0,-0.999605,140.389522,58.835365)"></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 -2.486193 2.869072 C -2.032889 1.145735 -1.020771 0.332914 -0.000836706 0.000751923 C -1.020771 -0.335318 -2.032889 -1.148139 -2.486193 -2.867569 " transform="matrix(0.999605,0,0,-0.999605,234.368024,101.973408)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph4-1" x="119.062949" y="111.679482"></use> <use xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph4-2" x="122.362064" y="111.679482"></use> <use xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph4-3" x="126.124059" y="111.679482"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph5-1" x="129.415858" y="108.867593"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#hA3RXyIWM1mhP_srab2lPi3ExAg=-glyph6-1" x="129.415858" y="114.384413"></use> </g> </g> </svg> <p>which is the case by the unitality clause on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>, as indicated.</p> </div> </p> <p><br /></p> <h3 id="Algebras">Algebras/modules over a monad</h3> <p>Given that a monad in a bicategory <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">\mathcal{B}</annotation></semantics></math> is nothing but a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in a hom-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}(a,a)</annotation></semantics></math>, it is natural to consider a <a class="existingWikiWord" href="/nlab/show/module">module</a> over this monoid: a <a class="existingWikiWord" href="/nlab/show/module+for+a+monad">module for a monad</a>. This notion of module is more general than a module in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, however, since it need not live in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}(a,a)</annotation></semantics></math> but can be in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}(b,a)</annotation></semantics></math> (for left modules) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}(a,c)</annotation></semantics></math> (for right modules).</p> <p>In a <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-like <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, left modules over a monad are usually known as <em><a class="existingWikiWord" href="/nlab/show/algebras+over+a+monad">algebras over the monad</a></em>. This terminology is confusing from the point of view of monads as monoids, but is justified because in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> itself, such algebras with domain <a class="existingWikiWord" href="/nlab/show/terminal+category">1</a> are just algebras for a monad in the classical sense. Such algebras are a powerful tool to encode general algebraic structures; this is the topic of <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>. The algebras over a monad form its <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, which is characterized by a <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>.</p> <p>Some monads arise from <a class="existingWikiWord" href="/nlab/show/operad">operad</a>s, in which case algebras for the monad are the same as algebras for the operad. A <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> is another special sort of monad 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>.</p> <h2 id="properties">Properties</h2> <div> <h3 id="RelationBetweenAdjunctionsAndMonads">Relation between adjunctions and monads</h3> <p>There is a close relation between <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> (<a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>) and <a class="existingWikiWord" href="/nlab/show/monads">monads</a>:</p> <ul> <li> <p><a href="#MonadInducedByAnAdjunction">Monad induced by an adjunction</a></p> </li> <li> <p><a href="#CategoryOfAdjunctionResolutionsOfAMonad">Category of adjunction-resolutions of a monad</a></p> </li> <li> <p><a href="#SemanticsStructureAdjunction">Semantics-Structure adjunction</a></p> </li> </ul> <h4 id="MonadInducedByAnAdjunction">Monad induced by an adjunction</h4> <p>Every <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">R \circ L</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">L \circ R</annotation></semantics></math>.</p> <p>(<a href="monad#Huber61">Huber 1961, §4</a>; see eg. <a href="monad#MacLane71">MacLane 1971, §VI.1 (p 134)</a>; <a href="monad#Borceux94">Borceux 1994, vol. 2, prop. 4.2.1</a>).</p> <p>In detail:</p> <p> <div class="num_prop"> <h6>Proposition</h6> <p></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>U</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\mathcal{D},F,U,\eta,\epsilon)</annotation></semantics></math> be a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> ie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">F \dashv U</annotation></semantics></math> are adjoint functors where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \rightarrow \mathcal{D}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">U \colon\mathcal{D} \rightarrow \mathcal{C}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_{A}:A \rightarrow U(F(A))</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\epsilon_{B} \colon F(U(B)) \rightarrow B</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a>. Then:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">U \circ F</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, with <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> and multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">F \circ U</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>, with <a class="existingWikiWord" href="/nlab/show/counit+of+a+comonad">counit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> and comultiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>η</mi> <mrow><mi>U</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\eta_{U(B)}):F(U(B)) \rightarrow F(U(F(U(B))))</annotation></semantics></math>.</p> </li> </ul> <p></p> </div> <div class="proof"> <h6>Proof</h6> <p></p> <p>We verify that we obtain a monad, the argument for the comonad is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p><strong>(1)</strong> We know that this diagram commutes:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="189.709pt" height="97.661pt" viewBox="0 0 189.709 97.661" version="1.1"> <defs> <g> <symbol overflow="visible" id="6LFu8Ju15STMSdfwxekX4iJ4afM=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" 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<p>which is exactly the associativity of the multiplication of the monad.</p> <p><strong>(4)</strong> The naturality of the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))</annotation></semantics></math> is obtained by two <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> of the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>B</mi></msub><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\epsilon_{B}:U(F(B)) \rightarrow B</annotation></semantics></math>.</p> </div> </p> <h4 id="CategoryOfAdjunctionResolutionsOfAMonad">Category of adjunction-resolutions of a monad</h4> <p>An adjunction inducing 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> (as <a href="#MonadInducedByAnAdjunction">above</a>) is also called a <em>resolution 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></em>.</p> <p>There is in general more than one such resolution, in fact there is a <a class="existingWikiWord" href="/nlab/show/category">category</a> of adjunctions for a given monad whose morphisms are “<a class="existingWikiWord" href="/nlab/show/comparison+functors">comparison functors</a>” (eg. <a href="#MacLane71">MacLane 1971, §VI.3</a>).</p> <p>In this category:</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> is the adjunction over the <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a> of the monad;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is that over the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> of algebras, also called the <em><a class="existingWikiWord" href="/nlab/show/monadic+adjunction">monadic adjunction</a></em> (recognized by <em><a class="existingWikiWord" href="/nlab/show/monadicity+theorems">monadicity theorems</a></em>).</p> </li> </ul> <p>(e.g. <a href="monad#Borceux94">Borceux 1994, vol. 2, prop. 4.2.2</a>)</p> <h4 id="SemanticsStructureAdjunction">Semantics-structure adjunction</h4> <p>The above passage from <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> to <a class="existingWikiWord" href="/nlab/show/monads">monads</a> and back to their <a class="existingWikiWord" href="/nlab/show/monadic+adjunctions">monadic adjunctions</a> constitutes itself an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>, sometimes called the <em><a class="existingWikiWord" href="/nlab/show/semantics-structure+adjunction">semantics-structure adjunction</a></em>.</p> </div> <h2 id="examples">Examples</h2> <h3 id="monads_on_">Monads on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></h3> <p>Many of these monads also have standard usages as <a class="existingWikiWord" href="/nlab/show/monad+%28computer+science%29">monads in computer science</a>.</p> <div class="num_example" id="MaybeMonad"> <h6 id="example">Example</h6> <p>The free-forgetful <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a> and <a class="existingWikiWord" href="/nlab/show/sets">sets</a> induces an <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">(-)_* : Set \to Set</annotation></semantics></math> which adds a new disjoint point. This is called the <a class="existingWikiWord" href="/nlab/show/maybe+monad">maybe monad</a> in computer science.</p> </div> <div class="num_example" id="ListMonad"> <h6 id="example_2">Example</h6> <p>The free-forgetful <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> and <a class="existingWikiWord" href="/nlab/show/sets">sets</a> induces an <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T : Set \to Set</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TA</mi><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨆</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></munder><msup><mi>A</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">TA := \bigsqcup_{n \ge 0} A^n </annotation></semantics></math></div> <p>giving the <strong>free monoid monad</strong>. This also goes by the name <a class="existingWikiWord" href="/nlab/show/list+monad">list monad</a> or <span class="newWikiWord">Kleene-Star<a href="/nlab/new/Kleene-Star">?</a></span> in computer science. The components of the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>T</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta_A : A \to T A</annotation></semantics></math> give inclusions sending each element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to the corresponding singleton list. The components of the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>A</mi></msub><mo>:</mo><msup><mi>T</mi> <mn>2</mn></msup><mi>A</mi><mo>→</mo><mi>T</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\mu_A : T^2 A \to T A</annotation></semantics></math> are the concatenation functions, sending a list of lists to the corresponding list (Known as flattening in computer science). This monad can be defined in any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> that distribute over the monoidal product.</p> </div> <div class="num_example" id="StateMonad"> <h6 id="example_3">Example</h6> <p>For a fixed set of “states” <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>, the (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>S</mi></msup></mrow><annotation encoding="application/x-tex">S \times - \dashv (-)^S</annotation></semantics></math>)-adjunction induces a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>S</mi></msup></mrow><annotation encoding="application/x-tex">(S \times -)^S</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> called the <a class="existingWikiWord" href="/nlab/show/state+monad">state monad</a>. This is a commonly used monad in computer science. In functional programming languages such as Haskell, states can be used to model “side effects” of computations.</p> </div> <div class="num_example" id="ContinuationMonad"> <h6 id="example_4">Example</h6> <p>The contravariant <a class="existingWikiWord" href="/nlab/show/power+set">power set</a>-functor is its own <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, giving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Set</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>P</mi><mi>B</mi><mo stretchy="false">)</mo><mo>≅</mo><mo lspace="0em" rspace="thinmathspace">Set</mo><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>P</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Set(A,P B) \cong \Set (B, P A)</annotation></semantics></math>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>P</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(A, P B) = \hom(A, \hom(B,\Omega)) \cong \hom( A \times B, \Omega) = P(A \times B)</annotation></semantics></math> inducing a <strong>double power set monad</strong> taking a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">P^2 A</annotation></semantics></math>. The components of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> are the <a class="existingWikiWord" href="/nlab/show/ultrafilter">principal ultrafilter</a> functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><msup><mi>P</mi> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta_A \colon A \to P^2 A</annotation></semantics></math> which send an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> to the set of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> that contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>. The components of the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\mu_A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> function for the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><mi>P</mi><mi>A</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>P</mi><mi>A</mi><mo>→</mo><msup><mi>P</mi> <mn>3</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta_{P A} \colon P A \to P^3 A</annotation></semantics></math>; which can be painfully stated as: the function taking a set of sets of sets of subsets to the set of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with the property that one of the sets of sets of subsets is the set of all sets of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> that include that particular subset as an element.</p> <p>Replacing the two element <a class="existingWikiWord" href="/nlab/show/power+object">power object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> with any other set gives similar monads. In <a class="existingWikiWord" href="/nlab/show/monad+in+computer+science">computer science contexts</a> these are known as <em><a class="existingWikiWord" href="/nlab/show/continuation+monad">continuation monads</a></em>. This construction can also be generalised for any other <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">bi-closed monoidal category</a>. For example there is a similar <strong>double dual monad</strong> on <a class="existingWikiWord" href="/nlab/show/Vect"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mo lspace="0em" rspace="thinmathspace">Vect</mo> <mi>k</mi></msub> </mrow> <annotation encoding="application/x-tex">\Vect_k</annotation> </semantics> </math></a>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p><a class="existingWikiWord" href="/nlab/show/function+monad">function monad</a> (also “<a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a>”, cf. <a class="existingWikiWord" href="/nlab/show/coreader+comonad">coreader comonad</a>)</p> </div> <div class="num_example"> <h6 id="example_6">Example</h6> <p><a class="existingWikiWord" href="/nlab/show/possibility">possibility</a></p> </div> <div class="num_example"> <h6 id="example_7">Example</h6> <p><a class="existingWikiWord" href="/nlab/show/selection+monad">selection monad</a></p> </div> <h3 id="algebra">Algebra</h3> <div class="num_example" id="FreeRModMonad"> <h6 id="example_8">Example</h6> <p>The free-forgetful <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/sets">sets</a> and the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a>. This induces the <strong>free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module monad</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">R[-] : Set \to Set</annotation></semantics></math>. The <strong>free abelian group monad</strong> and <strong>free vector space monad</strong> are special cases.</p> </div> <div class="num_example" id="FreeGroupMonad"> <h6 id="example_9">Example</h6> <p>The free-forgetful <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/sets">sets</a> and the category of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> gives the <strong>free group monad</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F : Set \to Set</annotation></semantics></math> that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A)</annotation></semantics></math> of finite words in the letters <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> together with inverses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{-1}</annotation></semantics></math>.</p> </div> <h3 id="topology">Topology</h3> <div class="num_example" id="UltrafilterMonad"> <h6 id="example_10">Example</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mo lspace="0em" rspace="thinmathspace">Top</mo><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">U : \Top \to \Set</annotation></semantics></math> taking a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> to its underlying <a class="existingWikiWord" href="/nlab/show/set">set</a>. It is right adjoint to the discrete space functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>:</mo><mo lspace="0em" rspace="thinmathspace">Set</mo><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Top</mo></mrow><annotation encoding="application/x-tex">D: \Set \to \Top</annotation></semantics></math> taking a set to its <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a>. There is also an adjoint pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>⊣</mo><mi>U</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\beta \dashv U'</annotation></semantics></math> between the category of <a class="existingWikiWord" href="/nlab/show/compact">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+spaces">Hausdorff topological spaces</a> and the category of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Stone-Cech+compactification">Stone-Cech compactification</a>. The composites of these two adjoint pairs gives a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mo lspace="0em" rspace="thinmathspace">Set</mo><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">\beta : \Set \to \Set</annotation></semantics></math> sending a set to its underlying set of the Stone-Cech compactification of its discrete space. It is also known as the <a class="existingWikiWord" href="/nlab/show/ultrafilter">ultrafilter</a> monad as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> can be thought of as the functor taking a set to its set of ultrafilters.</p> </div> <h3 id="monads_in_cat">Monads in Cat</h3> <p>Monads are often considered in the 2-category <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> where they are given by <a class="existingWikiWord" href="/nlab/show/endofunctors">endofunctors</a> with a monoid structure on them. In particular, monads <em>in</em> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> <em>on</em> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> are equivalent to the equational theories studied in universal algebra. In this context, a monad abstracts the concept of an <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> (such as “group” or “ring”), giving a general notion of <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> on an object of a category.</p> <p>Classically, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>T</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{T}</annotation></semantics></math> is an algebraic theory (e.g. the theory of groups), a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>T</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{T}</annotation></semantics></math>-structure on a set tells us how to interpret various <em>terms</em> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a\cdot c)</annotation></semantics></math>) formed from elements of the set, subject to certain <em>axioms</em> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a\cdot (b\cdot c))=((a\cdot b)\cdot c)</annotation></semantics></math>). A monad collects this up into a functor <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>. For a set <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is the set of all terms of the theory formed from elements 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>, with terms identified if axioms force them to be equal. For groups, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is thus the (underlying set of the) <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> of formal words <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>⋯</mi><mo>⋅</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">a \cdot b \cdot \cdots \cdot s</annotation></semantics></math> from <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>; the fact that <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> gives <a class="existingWikiWord" href="/nlab/show/free+object">free</a> structures turns out to be <a class="existingWikiWord" href="/nlab/show/monadic+adjunction">typical</a>.</p> <p>To capture the theory fully, we need to include a little more data: a natural map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\eta_X : X \to T X</annotation></semantics></math> recording how each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> gives a trivial term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, and a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>X</mi></msub><mo>:</mo><mi>T</mi><mi>T</mi><mi>X</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mu_X:T T X \to T X</annotation></semantics></math> recording how further terms built from terms are already present as terms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>.</p> <p>Given a monad in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> on a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, one can always produce a <a class="existingWikiWord" href="/nlab/show/canonical+resolution">canonical resolution</a> of any object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <h3 id="other_examples">Other examples</h3> <ul> <li> <p>Monads on <a class="existingWikiWord" href="/nlab/show/partial+order">posets</a> are particularly simple (in particular, they are always <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>). In fact, monads on <a class="existingWikiWord" href="/nlab/show/power+set">power set</a>s are extremely common throughout mathematics; they are known in less categorially-inclined circles as <a class="existingWikiWord" href="/nlab/show/Moore+closure">Moore closure</a>s, and there are many examples there.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> monad on the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> of a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a category with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, then a monad in the bicategory of <a class="existingWikiWord" href="/nlab/show/spans">spans</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> is the same thing as an <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</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>.</p> </li> <li> <p>A monad in the bicategory <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a> of <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> on a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> can be identified with an identity-on-objects functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\to B</annotation></semantics></math>.</p> </li> </ul> <h2 id="monads_in_higher_category_theory">Monads in higher category theory</h2> <p>There is a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a> of monads to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a>. See <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-monad">(∞,1)-monad</a>.</p> <p>in <a href="http://arxiv.org/PS_cache/math/pdf/0702/0702299v5.pdf#page=93">section 3</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Noncommutative algebra</em> (<a href="http://arxiv.org/abs/math/0702299">arXiv</a>)</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+monad">adjoint monad</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a>, <a class="existingWikiWord" href="/nlab/show/module+over+a+monad">module over a monad</a>, <a class="existingWikiWord" href="/nlab/show/monad+with+arities">monad with arities</a>, <a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a>, <a class="existingWikiWord" href="/nlab/show/monoidal+monad">monoidal monad</a>, <a class="existingWikiWord" href="/nlab/show/cartesian+monad">cartesian monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kleisli+triple">Kleisli triple</a>, <a class="existingWikiWord" href="/nlab/show/extension+system">extension system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><strong>monad</strong> <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a>/<a class="existingWikiWord" href="/nlab/show/doctrine">doctrine</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+linguistics%29">monad (in linguistics)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a>, <a class="existingWikiWord" href="/nlab/show/strong+monad">strong monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+monad">accessible monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+monad">adjoint monad</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monad">Frobenius monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finitary+monad">finitary monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monad">enriched monad</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/additive+monad">additive monad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strong+monad">strong monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+monad">polynomial monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+monad">relative monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polymonad">polymonad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/promonad">promonad</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> <li> <p><a class="existingWikiWord" href="/nlab/show/bar+construction">bar construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a></p> </li> </ul> <h2 id="References">References</h2> <p>The notion of (<a class="existingWikiWord" href="/nlab/show/comonad">co</a>)monads was introduced under the name “standard construction” (namely what is now thought of as their induced <a class="existingWikiWord" href="/nlab/show/canonical+resolution">canonical resolution</a>) in:</p> <ul> <li id="Huber61"><a class="existingWikiWord" href="/nlab/show/Peter+J.+Huber">Peter J. Huber</a>, §2 in: <em>Homotopy theory in general categories</em>, Mathematische Annalen <strong>144</strong> (1961) 361–385 [<a href="https://doi.org/10.1007/BF01396534">doi:10.1007/BF01396534</a>]</li> </ul> <p>following:</p> <ul> <li id="Godement58"> <p><a class="existingWikiWord" href="/nlab/show/Roger+Godement">Roger Godement</a>, Appendix of: <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) [<a href="https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement">webpage</a>, <a class="existingWikiWord" href="/nlab/files/Godement-TopologieAlgebrique.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(where the monad laws appear on p. 272 as part of the structure of the induced <a class="existingWikiWord" href="/nlab/show/canonical+resolution">canonical resolution</a>, called there the “fundamental construction”).</p> </blockquote> </li> </ul> <p>In the early <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>-literature monads were called <em>triples</em>, referring to the fact that (just as for <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>) their data-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> is that of <a class="existingWikiWord" href="/nlab/show/triples">triples</a> consisting of: (1.) the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/category">category</a>, (2.) a <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> and (3.) a <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit operation</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/H.+Applegate">H. Applegate</a>, <a class="existingWikiWord" href="/nlab/show/M.+Barr">M. Barr</a>, <a class="existingWikiWord" href="/nlab/show/J.+Beck">J. Beck</a>, <a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <a class="existingWikiWord" href="/nlab/show/F.+E.+J.+Linton">F. E. J. Linton</a>, <a class="existingWikiWord" href="/nlab/show/E.+Manes">E. Manes</a>, <a class="existingWikiWord" href="/nlab/show/M.+Tierney">M. Tierney</a>, <a class="existingWikiWord" href="/nlab/show/F.+Ulmer">F. Ulmer</a>: <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, ETH 1966/67, edited by <a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a> and <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <strong><a class="existingWikiWord" href="/nlab/show/LNM+80">LNM 80</a></strong>, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories <strong>18</strong> (2008) 1-303 [<a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html">TAC:18</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf">pdf</a>]</li> </ul> <p>The modern terminology “monad” (and the definition in the generality internal to any <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>) is (cf. <a href="Barr09">Barr 2009</a>) due to:</p> <ul> <li id="Bénabou67"><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, §5.4 in: <em>Introduction to Bicategories</em>, Lecture Notes in Mathematics <strong>47</strong> Springer (1967) 1-77 [<a href="http://dx.doi.org/10.1007/BFb0074299">doi:10.1007/BFb0074299</a>]</li> </ul> <p>Further historical comments:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Hyland">Martin Hyland</a>, <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads</em>, Electronic Notes in Theor. Comp. Sci. <strong>172</strong> (2007) 437-458 [<a href="https://doi.org/10.1016/j.entcs.2007.02.019">doi:10.1016/j.entcs.2007.02.019</a>, <a href="https://www.dpmms.cam.ac.uk/~jmeh1/Research/Publications/2007/hp07.pdf">preprint</a>]</p> </li> <li id="Barr09"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Re: Where does the term monad come from?</em> (April 1, 2009) [<a href="https://www.mta.ca/~cat-dist/archive/2009/09-4">cat-dist:09-4</a>, <a href="/nlab/files/Barr-HistoryOfMonadTerminology.txt">txt</a>, <a href="/nlab/files/Barr-HistoryOfMonadTerminology.jpg">jpg</a>]</p> </li> <li id="Street09"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Re: monads</em> (April 4, 2009) [<a href="https://www.mta.ca/~cat-dist/archive/2009/09-4">cat-dist:09-4</a>, <a href="/nlab/files/Street-HistoryOfMonadTerminology.txt">txt</a>, <a href="/nlab/files/Street-HistoryOfMonadTerminology.jpg">jpg</a>]</p> </li> </ul> <p>Further original texts:</p> <ul> <li id="Maranda66"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Maranda">Jean-Marie Maranda</a>, <em>On Fundamental Constructions and Adjoint Functors</em>, Canadian Mathematical Bulletin <strong>9</strong> 5 (1966) 581-591 [<a href="https://doi.org/10.4153/CMB-1966-072-9">doi:10.4153/CMB-1966-072-9</a>]</p> </li> <li id="Maranda68"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Maranda">Jean-Marie Maranda</a>, <em>Sur les Proprietes Universelles des Foncteurs Adjoints</em>, In: <em>Études sur les Groupes abéliens</em> / <em>Studies on Abelian Groups</em> Springer (1968) [<a href="https://doi.org/10.1007/978-3-642-46146-0_16">doi:10.1007/978-3-642-46146-0_16</a>]</p> </li> <li id="Linton69"> <p><a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <em>An outline of functorial semantics</em>, in <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, Lecture Notes in Mathematics <strong>80</strong>, Springer (1969) 7-52 [<a href="https://doi.org/10.1007/BFb0083080">doi:10.1007/BFb0083080</a>]</p> </li> <li id="Linton69b"> <p><a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <em>Applied functorial semantics</em>, in <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, Lecture Notes in Mathematics <strong>80</strong>, Springer (1969) 53-74 [<a href="https://doi.org/10.1007/BFb0083080">doi:10.1007/BFb0083080</a>]</p> </li> <li id="Frei69"> <p><a class="existingWikiWord" href="/nlab/show/Armin+Frei">Armin Frei</a>, <em>Some remarks on triples</em>, Mathematische Zeitschrift <strong>109</strong> (1969) 269–272 [<a href="https://doi.org/10.1007/BF01110118">doi:10.1007/BF01110118</a>]</p> </li> <li id="Pumplün70"> <p><a class="existingWikiWord" href="/nlab/show/Dieter+Pumpl%C3%BCn">Dieter Pumplün</a>, <em>Eine Bemerkung über Monaden und adjungierte Funktoren</em>, Mathematische Annalen <strong>185</strong> (1970) 329-337 [<a href="https://eudml.org/doc/161964">eudml:161964</a>, <a class="existingWikiWord" href="/nlab/files/Pumpluen-Monaden.pdf" title="pdf">pdf</a>]</p> </li> <li id="Coppey70"> <p>Laurent Coppey, <em>Morphismes et comorphismes de cotriples</em>, <a class="existingWikiWord" href="/nlab/show/Comptes+rendus+hebdomadaires+des+s%C3%A9ances+de+l%27Acad%C3%A9mie+des+sciences">C. R. Acad. Sc. Paris</a> <strong>271</strong> (1970), [<a href="https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f27">ark:12148/bpt6k5619186c/f27</a>]</p> </li> <li id="MacLane71"> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Ch. VI of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, <em>Les Triples</em>, Séminaire de mathématiquepure pure <strong>26</strong>, Université de Louvain (1972) [<a class="existingWikiWord" href="/nlab/files/Benabou-LesTriples.pdf" title="pdf">pdf</a>]</p> </li> <li id="Street72"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>The formal theory of monads</em>, Journal of Pure and Applied Algebra <strong>2</strong> 2 (1972) 149-168 [<a href="https://doi.org/10.1016/0022-4049(72)90019-9">doi:10.1016/0022-4049(72)90019-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>The formal theory of monads II</em>, Journal of Pure and Applied Algebra</p> <p><strong>175</strong> 1–3 (2002) 243-265 [<a href="https://doi.org/10.1016/S0022-4049(02)00137-8">doi:10.1016/S0022-4049(02)00137-8</a>]</p> </li> <li id="BarrWells85"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, Chapter 3 of: <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em>, Grundlehren der math. Wissenschaften <strong>278</strong>, Springer (1985), Reprints in Theory and Applications of Categories <strong>12</strong> (2005) 1-287 [<a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">tac:tr12</a>]</p> </li> </ul> <p>Further textbook accounts:</p> <ul> <li id="Borceux94"><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Ch. 4 “Monads” in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> vol. 2, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994)</li> </ul> <p>Expositions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/The+Catsters">The Catsters</a>, <em>Monads</em>, short video lectures (2007) [1:<a href="https://www.youtube.com/watch?v=9fohXBj2UEI&list=PL0E91279846EC843E&index=1">YT</a>, 2:<a href="https://www.youtube.com/watch?v=Si6_oG7ZdK4&list=PL0E91279846EC843E&index=2">YT</a>, 3:<a href="https://www.youtube.com/watch?v=eBQnysX7oLI&list=PL0E91279846EC843E&index=3">YT</a>, 3A:<a href="https://www.youtube.com/watch?v=uYY5c1kkoIo&list=PL0E91279846EC843E&index=4">YT</a>, 4:<a href="https://www.youtube.com/watch?v=Cm-O_ZWEIGY&list=PL0E91279846EC843E&index=5">YT</a>, 5/6:<a href="https://www.youtube.com/watch?v=g-SCYArh5RY&list=PL0E91279846EC843E&index=6">YT</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em><a href="http://math.ucr.edu/home/baez/universal/universal_hyper.pdf">Universal Algebra and Diagrammatic Reasoning</a></em> (Introductory slides).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, p. 154 of: <em><a class="existingWikiWord" href="/nlab/show/Category+Theory+in+Context">Category Theory in Context</a></em> (2017)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a>, <em>Notes on Category Theory with examples from basic mathematics</em>, Chapter 5. (<a href="http://arxiv.org/abs/1912.10642">arXiv</a>)</p> </li> </ul> <p>More on the relation to <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Hyland">Martin Hyland</a> and <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>The category theoretic understanding of universal algebra: Lawvere theories and monads</em> (<a href="http://www.dpmms.cam.ac.uk/~jmeh1/Research/Publications/2007/hp07.pdf">pdf</a>).</p> </li> <li id="Voutas12"> <p><a class="existingWikiWord" href="/nlab/show/Anthony+Voutas">Anthony Voutas</a>, <em>The basic theory of monads and their connection to universal algebra</em>, (2012) [<a href="https://voutasaur.us/monad-algebra.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Voutas-Monads.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>An elementary proof of the equivalence between infinitary <a class="existingWikiWord" href="/nlab/show/Lawvere+theories">Lawvere theories</a> and <a class="existingWikiWord" href="/nlab/show/monads">monads</a> on the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a> is given in Appendix A of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Brandenburg">Martin Brandenburg</a>, <em>Large limit sketches and topological space objects</em> [<a href="https://arxiv.org/abs/2106.11115">arXiv:2106.11115</a>]</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+M.+Fiore">T. M. Fiore</a>, <a class="existingWikiWord" href="/nlab/show/N.+Gambino">N. Gambino</a>, <a class="existingWikiWord" href="/nlab/show/J.+Kock">J. Kock</a>, <em>Monads in double categories</em>, <a href="http://arxiv.org/abs/1006.0797">arxiv/1006.0797</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriella+B%C3%B6hm">Gabriella Böhm</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Weak bimonads</em>, <a href="http://arxiv.org/abs/1002.4493">arxiv/1002.4493</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 9, 2025 at 16:35:55. 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