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<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='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#BigPicture'>Big picture</a></li> <li><a href='#explicit_definition'>Explicit definition</a></li> <ul> <li><a href='#monad_distributing_over_monad'>Monad distributing over monad</a></li> <li><a href='#monad_distributing_over_a_comonad'>Monad distributing over a comonad</a></li> <li><a href='#ComonadDistributingOverMonad'>Comonad distributing over monad</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#products_distributing_over_coproducts'>Products distributing over coproducts</a></li> <li><a href='#in_cat'>In Cat</a></li> <ul> <li><a href='#tensor_products_distributing_over_direct_sums'>Tensor products distributing over direct sums</a></li> </ul> <li><a href='#in_other_2categories'>In other 2-categories</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#Literature'>Literature</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Sometimes in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> one considers objects equipped with two different types of <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> which interact in a suitable way. For instance, a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> is a <a class="existingWikiWord" href="/nlab/show/set">set</a> equipped with both (1) the structure of an (additive) <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and (2) the structure of a (multiplicative) <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, which satisfy the distributive laws <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a\cdot (b+c) = a\cdot b + a\cdot c</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a\cdot 0 = 0</annotation></semantics></math>.</p> <p>Abstractly, there are two <a class="existingWikiWord" href="/nlab/show/monads">monads</a> on the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, one (call it <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>) whose <a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebras</a> are abelian groups, and one (call it <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>) whose algebras are monoids, and so we might ask “can we construct, from these two monads, a third monad whose algebras are rings?” Such a monad would assign to each 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> the <a class="existingWikiWord" href="/nlab/show/free+object">free</a> ring on that set, which consists of formal sums of formal products of 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>—in other words, it can be identified with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(S(X))</annotation></semantics></math>. Thus the question becomes “given two monads <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>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, what further structure is required to make the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S</annotation></semantics></math> into a monad?”</p> <p>It is easy to give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S</annotation></semantics></math> a unit, as the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mover><mo>→</mo><mrow><msup><mi>η</mi> <mi>S</mi></msup></mrow></mover><mi>S</mi><mover><mo>→</mo><mrow><msup><mi>η</mi> <mi>T</mi></msup><mi>S</mi></mrow></mover><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">Id \xrightarrow{\eta^S} S \xrightarrow{\eta^T S} T S</annotation></semantics></math>, but to give it a multiplication we need a transformation from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S T S</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S</annotation></semantics></math>. We naturally want to use the multiplications <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>T</mi></msup><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>T</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\mu^T\colon T T \to T</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>S</mi></msup><mo lspace="verythinmathspace">:</mo><mi>S</mi><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\mu^S\colon S S \to S</annotation></semantics></math>, but in order to do this we first need to switch the order of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. However, if we have a transformation <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>T</mi><mo>→</mo><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\lambda\colon S T \to T S</annotation></semantics></math>, then we can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mrow><mi>T</mi><mi>S</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mu^{T S}</annotation></semantics></math> to be the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi><mi>T</mi><mi>S</mi><mover><mo>→</mo><mi>λ</mi></mover><mi>T</mi><mi>T</mi><mi>S</mi><mi>S</mi><mover><mo>→</mo><mrow><msup><mi>μ</mi> <mi>T</mi></msup><msup><mi>μ</mi> <mi>S</mi></msup></mrow></mover><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S T S \xrightarrow{\lambda} T T S S \xrightarrow{\mu^T\mu^S} T S</annotation></semantics></math>.</p> <p>Such a transformation <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>T</mi><mo>→</mo><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\lambda\colon S T \to T S</annotation></semantics></math>, satisfying suitable axioms to make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">T S</annotation></semantics></math> into a monad, is called a <em>distributive law</em>, because of the motivating example relating addition to multiplication in a ring. In that case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">S T X</annotation></semantics></math> is a formal product of formal sums such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>5</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1 + x_2 + x_3)\cdot (x_4 + x_5)</annotation></semantics></math>, and the distributive law <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> is given by multiplying out such an expression formally, resulting in a formal sum of formal products such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>5</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>5</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>3</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>3</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">x_1\cdot x_4 + x_1 \cdot x_5 + x_2 \cdot x_4 + x_2 \cdot x_5 + x_3\cdot x_4 + x_3 \cdot x_5</annotation></semantics></math>.</p> <p>Given two monads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S, T</annotation></semantics></math> 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>, a distributive law <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∘</mo><mi>S</mi><mo>⟶</mo><mi>S</mi><mo>∘</mo><mi>T</mi></mrow><annotation encoding="application/x-tex"> T \circ S \longrightarrow S \circ T </annotation></semantics></math> gives a way of lifting the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to a monad on the category 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>-algebras, namely the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>T</mi></msup></mrow><annotation encoding="application/x-tex">C^T</annotation></semantics></math>. In the example above, the distributive law gives a way to lift the monad for monoids (which is a monad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C = Set</annotation></semantics></math>) to the monad for rings (which is a monad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>T</mi></msup><mo>=</mo><mi>AbGp</mi></mrow><annotation encoding="application/x-tex">C^T = AbGp</annotation></semantics></math>). This is a way of making rigorous the intuition that “rings are to abelian groups as monoids are to sets”.</p> <p> <div class='num_remark' id='TerminologyWhatDistributesOverWhat'> <h6>Remark</h6> <p><strong>(terminology – what distributes over what)</strong> <br /> The eponymous example of distributivity in <a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>b</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mi>a</mi><mo>×</mo><msub><mi>b</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> a \times \sum_i b_i \;=\; \sum_i a \times b_i </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big( a \times (-) \big) \circ \big( \sum (-) \big) \;\; = \;\; \big( \sum (-) \big) \circ \big( a \times (-) \big) </annotation></semantics></math></div> <p>(where, of course, the <a class="existingWikiWord" href="/nlab/show/equality">equality</a> as such works in both directions, but the <em>distribution</em> of factors over summands is the step from left to right) suggests that a suitable transformation of (co)monads of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∘</mo><mi>T</mi><mo>⟶</mo><mi>T</mi><mo>∘</mo><mi>S</mi></mrow><annotation encoding="application/x-tex"> S \circ T \longrightarrow T \circ S </annotation></semantics></math></div> <p>should be referred to as <em><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> distributing over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></em> instead of the other way around.</p> <p>However, already the original reference <a href="#Beck69">Beck 1969 §1</a> uses the opposite terminology.</p> <p>Authors sticking to this original but arguably reverse terminological convention include <a href="#BrookesVanStone93">Brookes & Van Stone 1993</a>, while other authors tacitly switch to the other terminological convention (eg. <a href="#BarrWells85">Barr & Wells 1985 §9 2.1</a>, <a href="#PowerWatanabe02">Power & Watanabe 2002 p. 138</a>).</p> </div> </p> <h2 id="BigPicture">Big picture</h2> <p><a class="existingWikiWord" href="/nlab/show/monad">Monads</a> in any <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>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> make themselves a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Mnd</mi></mrow><annotation encoding="application/x-tex">\mathrm{Mnd}</annotation></semantics></math> in which <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are either lax or colax <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of monads (cf. <em><a class="existingWikiWord" href="/nlab/show/monad+transformations">monad transformations</a></em>). By <a class="existingWikiWord" href="/nlab/show/formal+duality">formal duality</a> the analogue is true for <a class="existingWikiWord" href="/nlab/show/comonads">comonads</a>.</p> <p>Distributivity laws may be understood as monads <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to this 2-category of monads.</p> <p>In particular, distributive laws themselves make a 2-category.</p> <p>There are other variants like distributive laws between a monad and an <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a>, <strong>mixed distributive laws</strong> between a monad and a comonad (the variants for algebras and coalgebras called <a class="existingWikiWord" href="/nlab/show/entwining+structures">entwining structures</a>), distributive laws between actions of two different monoidal categories on the same category, for <a class="existingWikiWord" href="/nlab/show/PROP">PROP</a>s and so on.</p> <p>Having a distributive law <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math> from one monad to another enables to define the composite monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>T</mi></mstyle><msub><mo>∘</mo> <mi>l</mi></msub><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf T\circ_l\mathbf P</annotation></semantics></math>. This correspondence extends to a 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">comp</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi mathvariant="normal">Mnd</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Mnd</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Mnd</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{comp} \,\colon\, \mathrm{Mnd}(\mathrm{Mnd}(C))\to\mathrm{Mnd}(C)</annotation></semantics></math>. An analogue of this 2-functor in the mixed setup is a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> from the bicategory of entwinings to a bicategory of <a class="existingWikiWord" href="/nlab/show/corings">corings</a>.</p> <h2 id="explicit_definition">Explicit definition</h2> <h3 id="monad_distributing_over_monad">Monad distributing over monad</h3> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <strong>distributive law</strong> for a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>T</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msup><mi>μ</mi> <mi>T</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>T</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{T} = (T, \mu^T, \eta^T)</annotation></semantics></math> in <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> over an endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is 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>l</mi><mo>:</mo><mi>T</mi><mi>P</mi><mo>⇒</mo><mi>P</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">l : T P \Rightarrow P T</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∘</mo><mo stretchy="false">(</mo><msup><mi>η</mi> <mi>T</mi></msup><msub><mo stretchy="false">)</mo> <mi>P</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>η</mi> <mi>T</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l \circ (\eta^T)_P = P(\eta^T)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∘</mo><mo stretchy="false">(</mo><msup><mi>μ</mi> <mi>T</mi></msup><msub><mo stretchy="false">)</mo> <mi>P</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>μ</mi> <mi>T</mi></msup><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>l</mi> <mi>T</mi></msub><mo>∘</mo><mi>T</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l \circ (\mu^T)_P = P(\mu^T) \circ l_T \circ T(l)</annotation></semantics></math>. 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-2.485006 2.868781 C -2.031881 1.146125 -1.020163 0.333625 -0.00063125 0.00159375 C -1.020163 -0.334344 -2.031881 -1.146844 -2.485006 -2.8695 " transform="matrix(1, 0, 0, -1, 208.2936, 76.943)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Y1AWf1W1ZzKlg2wOOH3b6tCh3ts=-glyph-1-4" x="122.874" y="73.427"></use> </g> </svg> <p></p> </div> </p> <p>Distributive laws for the monad <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> over the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> are in a canonical bijection with lifts of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> to 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><msup><mi>P</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">P^{\mathbf T}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">A^{\mathbf T}</annotation></semantics></math>, satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup><msup><mi>P</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup><mo>=</mo><mi>P</mi><msup><mi>U</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">U^{\mathbf T} P^{\mathbf T} = P U^{\mathbf T}</annotation></semantics></math>. Indeed, the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">P^{\mathbf T}</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>P</mi><mi>M</mi><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>l</mi> <mi>M</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M,\nu) \mapsto (P M,P(\nu)\circ l_M)</annotation></semantics></math>.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <strong>distributive law</strong> for a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>T</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msup><mi>μ</mi> <mi>T</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>T</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{T} = (T, \mu^T, \eta^T)</annotation></semantics></math> over a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>P</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msup><mi>μ</mi> <mi>P</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>P</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{P} = (P, \mu^P, \eta^P)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a distributive law for <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> over the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>, compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>P</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>P</mi></msup></mrow><annotation encoding="application/x-tex">\mu^P,\eta^P</annotation></semantics></math> in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∘</mo><mi>T</mi><mo stretchy="false">(</mo><msup><mi>η</mi> <mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>η</mi> <mi>P</mi></msup><msub><mo stretchy="false">)</mo> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">l \circ T(\eta^P) = (\eta^P)_T</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∘</mo><mi>T</mi><mo stretchy="false">(</mo><msup><mi>μ</mi> <mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>μ</mi> <mi>P</mi></msup><msub><mo stretchy="false">)</mo> <mi>T</mi></msub><mo>∘</mo><mi>P</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>l</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">l \circ T(\mu^P) = (\mu^P)_T \circ P(l) \circ l_P</annotation></semantics></math>. In diagrams:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="205.736" height="84.441" viewBox="0 0 205.736 84.441"> <defs> <g> <g id="eOxReKCFQrMKKy-xhyo5AOHeo7I=-glyph-0-0"> </g> <g id="eOxReKCFQrMKKy-xhyo5AOHeo7I=-glyph-0-1"> <path d="M 6.234375 -9.109375 C 6.3125 -9.46875 6.34375 -9.609375 6.578125 -9.671875 C 6.6875 -9.6875 7.1875 -9.6875 7.5 -9.6875 C 9 -9.6875 9.6875 -9.640625 9.6875 -8.46875 C 9.6875 -8.25 9.640625 -7.671875 9.546875 -7.125 L 9.53125 -6.953125 C 9.53125 -6.890625 9.59375 -6.796875 9.671875 -6.796875 C 9.828125 -6.796875 9.828125 -6.875 9.875 -7.109375 L 10.3125 -9.75 C 10.34375 -9.890625 10.34375 -9.921875 10.34375 -9.96875 C 10.34375 -10.125 10.25 -10.125 9.953125 -10.125 L 1.78125 -10.125 C 1.4375 -10.125 1.421875 -10.109375 1.328125 -9.84375 L 0.421875 -7.15625 C 0.40625 -7.125 0.359375 -6.953125 0.359375 -6.953125 C 0.359375 -6.875 0.421875 -6.796875 0.515625 -6.796875 C 0.625 -6.796875 0.65625 -6.859375 0.71875 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fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#lIqpFp2fS2V4Xvqh6euvCdP074M=-glyph-1-3" x="124.157" y="73.427"></use> </g> </svg> <p></p> </div> (due to <a href="#Beck69">Beck 1969</a>, review includes <a href="#BarrWells85">Barr & Wells 1985 §9 2.1</a>)</p> <p>The correspondence between distributive laws and <em>endofunctor</em> liftings extends to a correspondence between distributive laws and <em>monad</em> liftings. That is, distributive laws <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>P</mi><mo>⇒</mo><mi>P</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">l \colon T P \Rightarrow P T</annotation></semantics></math> from the monad <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> to the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{P}</annotation></semantics></math> are in a canonical bijection with lifts of the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{P}</annotation></semantics></math> to a monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathbf{P}^{\mathbf T}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">A^{\mathbf T}</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup><mo lspace="verythinmathspace">:</mo><msup><mi>A</mi> <mstyle mathvariant="bold"><mi>T</mi></mstyle></msup><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U^{\mathbf T} \colon A^{\mathbf T}\to A</annotation></semantics></math> preserves the monad structure.</p> <p>Thus all together a distributive law for a monad over a monad is a 2-cell for which 2 triangles and 2 pentagons commute. In the entwining case, Brzeziński and Majid combined the 4 diagrams into one picture which they call the <em>bow-tie diagram</em>.</p> <p> <div class='num_remark' id='DistributivityAsMonadsInMonads'> <h6>Remark</h6> <p><strong>(distributivity as monads in monads)</strong> <br /> As mentioned earlier, one can understand a distributivity law of 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>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,s)</annotation></semantics></math> over another 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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,t)</annotation></semantics></math> as displaying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">s \colon a \to a</annotation></semantics></math> as a monad on <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></mrow><annotation encoding="application/x-tex">(a,t)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> 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><mstyle mathvariant="bold"><mi>Mnd</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Mnd}(A)</annotation></semantics></math> of monads in a 2-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>.</p> <p>Specifically, a monad in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Mnd</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Mnd}(A)</annotation></semantics></math> over <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></mrow><annotation encoding="application/x-tex">(a,t)</annotation></semantics></math> (which is a monad in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>!) is comprised of the following data:</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>s</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">s \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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, together with an intertwiner <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo lspace="verythinmathspace">:</mo><mi>t</mi><mi>s</mi><mo>→</mo><mi>s</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\lambda \colon t s \to s t</annotation></semantics></math> satisfying the equations <a class="existingWikiWord" href="/nlab/show/monad#BicategoryOfMonads">here</a></p> </li> <li> <p>Two <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Mnd</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Mnd}(A)</annotation></semantics></math>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>⇒</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma \colon 1 \Rightarrow (s,\lambda)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nu \colon (s,\lambda)(s,\lambda) \Rightarrow (s,\lambda)</annotation></semantics></math> which correspond to two 2-morpisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>⇒</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\sigma \colon 1 \Rightarrow s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo lspace="verythinmathspace">:</mo><mi>s</mi><mi>s</mi><mo>⇒</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\nu \colon s s \Rightarrow s</annotation></semantics></math> commuting with the intertwiners 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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ss</mi></mrow><annotation encoding="application/x-tex">ss</annotation></semantics></math>.</p> </li> </ol> <p>Then <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> is the distributive law sought, and the laws <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math> satisfy correspond to those of a distributive law.</p> </div> </p> <h3 id="monad_distributing_over_a_comonad">Monad distributing over a comonad</h3> <p><a href="#vanOsdol73">van Osdol 1973 p. 456</a></p> <p>(…)</p> <h3 id="ComonadDistributingOverMonad">Comonad distributing over monad</h3> <p>The distributivity law of</p> <ul> <li> <p>a comonad <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> over</p> </li> <li> <p>a 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></p> </li> </ul> <p>on the same <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></p> <p>is as follows (<a href="#BrookesVanStone93">Brookes & Van Stone 1993 Def. 3</a>, <a href="#PowerWatanabe02">Power & Watanabe 2002</a>):</p> <p>A <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>distr</mi> <mrow><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo></mrow> <mrow><mi>𝒞</mi><mo>,</mo><mi>ℰ</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℰ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⟶</mo><mi>ℰ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> distr^{\mathcal{C}, \mathcal{E}}_{(\text{-})} \;\;\colon\;\; \mathcal{C} \big( \mathcal{E}(-) \big) \longrightarrow \mathcal{E} \big( \mathcal{C}(-) \big) </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a> for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>:</p> <div style="margin: -20px 0px 20px 0px"> <img src="/nlab/files/MixedCoMonadDistributivity-230929.jpg" width="540px" /> </div> <p id="TwoSidedKleisliComposition"> Given this distributivity structure, there is a two-sided (“double”) <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a> (<a href="#BrookesVanStone93">Brookes & Van Stone 1993 Thm. 2</a>, <a href="#PowerWatanabe02">Power & Watanabe 2002, Prop. 7.4</a>) whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are those of <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>, and whose morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>D</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_1 \to D_2</annotation></semantics></math> are morphisms in <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> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>prog</mi> <mn>12</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>ℰ</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> prog_{12} \;\colon\; \mathcal{C}(D_1) \longrightarrow \mathcal{E}(D_2) </annotation></semantics></math></div> <p>with two-sided Kleisli composition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>prog</mi> <mn>12</mn></msub><mtext>>=></mtext><msub><mi>prog</mi> <mn>23</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>ℰ</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> prog_{12} \text{>=>} prog_{23} \;\; \colon \;\; \mathcal{C}(D_1) \longrightarrow \mathcal{E}(D_3) </annotation></semantics></math></div> <p>given by the (co-)<a class="existingWikiWord" href="/nlab/show/Kleisli+extension">bind-operation</a> on the factors connected by the distributivity transformation:</p> <div style="margin: -20px 0px 20px 0px"> <img src="/nlab/files/CoMonKleisliComposition-230930b.pdf" width="800px" /> </div> <h2 id="examples">Examples</h2> <h3 id="products_distributing_over_coproducts">Products distributing over coproducts</h3> <p>In a <em><a class="existingWikiWord" href="/nlab/show/distributive+category">distributive category</a></em> <a class="existingWikiWord" href="/nlab/show/products">products</a> distribute over <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>.</p> <h3 id="in_cat">In Cat</h3> <ul> <li>There is a distributive law of the monad (on <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) for <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> over the monad for <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, whose composite is the monad for <a class="existingWikiWord" href="/nlab/show/rings">rings</a>. This is the canonical example which gives the name to the whole concept.</li> </ul> <h4 id="tensor_products_distributing_over_direct_sums">Tensor products distributing over direct sums</h4> <p>For many standard choices of <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> in the presence of <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> the former distribute over the latter. See at <em><a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a></em> and <em><a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a></em>.</p> <h3 id="in_other_2categories">In other 2-categories</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+factorization+systems">strict factorization systems</a> can be identified with distributive laws between categories regarded as monads in <a class="existingWikiWord" href="/nlab/show/span">Span(Set)</a>.</p> </li> <li> <p>More generally, <a class="existingWikiWord" href="/nlab/show/factorization+systems+over+a+subcategory">factorization systems over a subcategory</a> can be identified with distributive laws in <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a>. Ordinary <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+systems">orthogonal factorization systems</a> are a special case. The latter can also be obtained by other weakenings; see for instance <a href="http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html">this discussion</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+distributive+law">weak distributive law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/iterated+distributive+law">iterated distributive law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wreath">wreath</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributivity+for+monoidal+structures">distributivity for monoidal structures</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad+transformer">monad transformer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-distributive+law">pseudo-distributive law</a></p> </li> </ul> <h2 id="Literature">Literature</h2> <ul> <li id="Beck69"> <p><a class="existingWikiWord" href="/nlab/show/Jon+Beck">Jon Beck</a>, <em>Distributive Laws</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>, ETH 1966/67, Lecture Notes in Mathemativs, Springer (1969), 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#page=95">pdf</a>]</p> </li> <li id="vanOsdol73"> <p><a class="existingWikiWord" href="/nlab/show/Donovan+van+Osdol">Donovan van Osdol</a>, <em>Bicohomology Theory</em>, Transactions of the American Mathematical Society <strong>183</strong> (1973) 449-476 [<a href="https://www.jstor.org/stable/1996479">jstor:1996479</a>]</p> </li> <li id="Street72"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, §6 of: <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 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>, <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em>, Springer (1985) republished in: <a class="existingWikiWord" href="/nlab/show/TAC+reprints+series">Reprints in Theory and Applications of Categories</a>, <strong>12</strong> (2005) 1-287 [<a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">tac:tr12</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">tac:tr12</a>]</p> </li> <li id="BrookesVanStone93"> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Brookes">Stephen Brookes</a>, <a class="existingWikiWord" href="/nlab/show/Kathryn+Van+Stone">Kathryn Van Stone</a>, <em>Monads and Comonads in Intensional Semantics</em> (1993) [<a href="https://apps.dtic.mil/sti/citations/ADA266522">dtic:ADA266522</a>, <a href="https://www.cs.cmu.edu/~brookes/papers/MonadsComonads.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BrookesVanStone-CoMonads.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, <em>Distributive laws and Koszulness</em>, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323 (<a href="http://www.numdam.org/numdam-bin/fitem?id=AIF_1996__46_2_307_0">numdam</a>)</p> </li> <li> <p>T. F. Fox, <a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, <em>Distributive laws, bialgebras, and cohomology</em>, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math. <strong>202</strong> AMS (1997) 167-205</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, <a class="existingWikiWord" href="/nlab/show/S.+Majid">S. Majid</a>, <em>Coalgebra bundles</em>, Comm. Math. Phys. <strong>191</strong> 2 (1998) 467–492 [<a href="http://arxiv.org/abs/q-alg/9602022">arXiv:q-alg/9602022</a>]</p> </li> </ul> <p>For a study of distributive laws between monads and (pointed) endofunctors, see:</p> <ul> <li>Marina Lenisa, John Power, and Hiroshi Watanabe, <em>Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads</em>, Electronic Notes in Theoretical Computer Science 33 (2000): 230-260.</li> </ul> <p>For a thorough study of mixed distributive laws, see:</p> <ul> <li id="PowerWatanabe99"> <p><a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <a class="existingWikiWord" href="/nlab/show/Hiroshi+Watanabe">Hiroshi Watanabe</a>, <em>Distributivity for a monad and a comonad</em>, Electronic Notes in Theoretical Computer Science <strong>19</strong> (1999) 102 [<a href="https://doi.org/10.1016/S1571-0661(05)80271-3">doi:10.1016/S1571-0661(05)80271-3</a>, <a class="existingWikiWord" href="/nlab/files/PowerWatanabe-Distributivity.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>Composing PROPS</em>, <a href="http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html">Theory Appl. Categ.</a> 13 (2004), No. 9, 147–163.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>The formal theory of monads II</em>, Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra <strong>175</strong> 1-3 (2002) 243-265</p> </li> <li id="PowerWatanabe02"> <p><a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <a class="existingWikiWord" href="/nlab/show/Hiroshi+Watanabe">Hiroshi Watanabe</a>, <em>Combining a monad and a comonad</em>, Theoretical Computer Science <strong>280</strong> 1–2 (2002) 137-162 [<a href="https://doi.org/10.1016/S0304-3975(01)00024-X">doi:10.1016/S0304-3975(01)00024-X</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wisbauer">Robert Wisbauer</a>, <em>Corings and comodules</em>, London Math. Soc. Lec. Note Series 309, Cambridge (2003)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabi+B%C3%B6hm">Gabi Böhm</a>, <em>Internal bialgebroids, entwining structures and corings</em>, AMS Contemp. Math. <strong>376</strong> (2005) 207-226 [<a href="http://front.math.ucdavis.edu/math.QA/0311244">arXiv:math.QA/0311244</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ernie+Manes">Ernie Manes</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Mulry">Philip Mulry</a>: <em>Monad compositions I: general constructions and recursive distributive laws</em>, Theory and Applications of Categories <strong>18</strong> 7 (2007) 172-208 [<a href="http://www.tac.mta.ca/tac/volumes/18/7/18-07abs.html">tac:18-07</a>, <a href="http://www.tac.mta.ca/tac/volumes/18/7/18-07.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zoran+%C5%A0koda">Zoran Škoda</a>, <em>Distributive laws for monoidal categories</em> (<a href="http://front.math.ucdavis.edu/math.CT/0406310">arXiv:0406310</a>); <em>Equivariant monads and equivariant lifts versus a 2-category of distributive laws</em> (<a href="http://front.math.ucdavis.edu/0707.1609">arXiv:0707.1609</a>); <em>Bicategory of entwinings</em> (<a href="http://arxiv.org/abs/0805.4611">arXiv:0805.4611</a>)</p> </li> <li> <p>R. Wisbauer, <em>Algebras versus coalgebras</em>, Appl. Categ. Structures <strong>16</strong> 1-2 (2008) 255–295 [<a href="https://doi.org/10.1007/s10485-007-9076-5">doi:10.1007/s10485-007-9076-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zoran+%C5%A0koda">Zoran Škoda</a>, <em>Some equivariant constructions in noncommutative geometry</em>, Georgian Math. J. 16 (2009) 1; 183–202 (<a href="http://front.math.ucdavis.edu/0811.4770">arXiv:0811.4770</a>)</p> </li> <li> <p>Bachuki Mesablishvili, <a class="existingWikiWord" href="/nlab/show/Robert+Wisbauer">Robert Wisbauer</a>, <em>Bimonads and Hopf monads on categories</em>, Journal of K-Theory <strong>7</strong> 2 (2011) 349-388 [<a href="https://arxiv.org/abs/0710.1163">arXiv:0710.1163</a>, <a href="https://doi.org/10.1017/is010001014jkt105">doi:10.1017/is010001014jkt105</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francisco+Marmolejo">Francisco Marmolejo</a>, Adrian Vazquez-Marquez, <em>No-iteration mixed distributive laws</em>, Mathematical Structures in Computer Science <strong>27</strong> 1 (2017) 1-16 [<a href="https://doi.org/10.1017/S0960129514000656">doi:10.1017/S0960129514000656</a>]</p> </li> <li> <p>Liang Ze Wong, <em>Distributive laws</em>, post at <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>-<a href="https://golem.ph.utexas.edu/category/2017/02/distributive_laws.html">cafe</a>, Feb 2017</p> </li> <li> <p>Enrique Ruiz Hernández, <em>Another characterization of no-iteration distributive laws</em>, <a href="https://arxiv.org/abs/1910.06531">arxiv</a></p> </li> <li> <p>Werner Struckmann and Dietmar Wätjen, <em>A note on the number of distributive laws</em>, Algebra universalis 21 (1985): 305-306.</p> </li> </ul> <p>On distributive laws for <a class="existingWikiWord" href="/nlab/show/relative+monads">relative monads</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gabriele+Lobbia">Gabriele Lobbia</a>, <em>Distributive Laws for Relative Monads</em>, Applied Categorical Structures <strong>31</strong> 19 (2023) [<a href="https://doi.org/10.1007/s10485-023-09716-1">doi:10.1007/s10485-023-09716-1</a>, <a href="https://arxiv.org/abs/2007.12982">arXiv:2007.12982</a>]</li> </ul> <p>Invertible distributive laws are considered in Lemma 4.12 of:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bob+Rosebrugh">Bob Rosebrugh</a>, <a class="existingWikiWord" href="/nlab/show/Richard+J.+Wood">Richard J. Wood</a>, <em>Distributive Adjoint Strings</em>, Theory and Applications of Categories, <strong>1</strong> 6 (1995) 119-145 [<a href="http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html">tac:1-06</a>]</p> </li> <li> <p>Stefano Kasangian, <a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, and <a class="existingWikiWord" href="/nlab/show/Enrico+Vitale">Enrico Vitale</a>. <em>Coalgebras, braidings, and distributive laws</em>, Theory and Applications of Categories 13.8 (2004): 129-146. (<a href="https://www.emis.de/journals/TAC/volumes/13/8/13-08abs.html">html</a>)</p> </li> <li> <p>Alain Bruguieres and Alexis Virelizier, <em>Quantum double of Hopf monads and categorical centers</em>, Transactions of the American Mathematical Society 364.3 (2012): 1225-1279.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 2, 2024 at 16:31:41. 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