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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='#definition'>Definition</a></li> <ul> <li><a href='#equivalent_conditions'>Equivalent conditions</a></li> </ul> <li><a href='#algebras'>Algebras</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</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>A <strong>lax-idempotent 2-monad</strong>, also called a <strong>Kock–Zöberlein</strong> or <strong>KZ 2-monad</strong>, encodes a certain kind of <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">property-like structure</a> that a <a class="existingWikiWord" href="/nlab/show/category">category</a>, or more generally an <a class="existingWikiWord" href="/nlab/show/object">object</a> of a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, can carry.</p> <p>Lax-idempotent 2-monads have occasionally also been called <strong>KZ monads</strong> in the literature, but this terminology may be confusing, as it is inconsistent with terminology of <strong>lax-idempotent <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>-monads</strong>: a 1-monad may be viewed as a 2-monad on a <a class="existingWikiWord" href="/nlab/show/locally-discrete+2-category">locally-discrete 2-category</a>, in which case lax-idempotence is equivalent to <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotence</a>.</p> <p>The archetypal examples are given by <a class="existingWikiWord" href="/nlab/show/2-monads">2-monads</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> that take a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">T C</annotation></semantics></math> 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> under a given class of <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> – then an <a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T C \to C</annotation></semantics></math> is 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> with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, a functor <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 C \to D</annotation></semantics></math>, and a diagram <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> 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>, there is a unique arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>colim</mi><mi>T</mi><mi>F</mi><mi>S</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>colim</mi><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">colim T F S \to F(colim S)</annotation></semantics></math> given by the universal property of the colimit. It is this property that lax-idempotence generalizes.</p> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> on 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> is called <strong>lax-idempotent</strong> if given any two (strict) <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \colon T B \to B</annotation></semantics></math> and a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics></math>, there exists a unique 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>∘</mo><mi>T</mi><mi>f</mi><mo>⇒</mo><mi>f</mi><mo>∘</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\bar f \colon b \circ T f \Rightarrow f \circ a</annotation></semantics></math> making <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,\bar f)</annotation></semantics></math> a lax morphism 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:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>T</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>T</mi><mi>B</mi></mtd></mtr> <mtr><mtd><mi>a</mi><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mtd> <mtd><mo stretchy="false">↓</mo><mi>b</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>→</mo><mi>f</mi></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \swArrow \bar f & \downarrow b \\ A & \underset{f}{\to} & B } </annotation></semantics></math></div> <p>Dually, a 2-monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is called <strong>colax-idempotent</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics></math> gives rise to a colax <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>-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,\tilde f)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>T</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>T</mi><mi>B</mi></mtd></mtr> <mtr><mtd><mi>a</mi><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇗</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mo stretchy="false">↓</mo><mi>b</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>→</mo><mi>f</mi></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \neArrow \tilde f & \downarrow b \\ A & \underset{f}{\to} & B } </annotation></semantics></math></div> <h3 id="equivalent_conditions">Equivalent conditions</h3> <p>A 2-monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T, \mu, \eta)</annotation></semantics></math> is lax idempotent if and only if, for every 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>, there is an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub><mo>⊣</mo><msub><mi>μ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">T \eta_A \dashv \mu_A</annotation></semantics></math> with invertible counit or an adjunction <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><msub><mi>η</mi> <mrow><mi>T</mi><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_A \dashv \eta_{T A}</annotation></semantics></math> with invertible counit (with either adjunction implying the other).</p> <p>An extensive list of equivalent conditions is given on the page of <a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-adjunction">lax-idempotent 2-adjunction</a>s.</p> <div class="num_theorem" id="AlgebraAdjoint"> <h6 id="theorem">Theorem</h6> <p>A 2-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 above, with unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\eta: 1 \to T</annotation></semantics></math>, is lax-idempotent if and only if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math> there is a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>⇒</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\theta_a \colon 1 \Rightarrow \eta_A \circ a</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>θ</mi> <mi>a</mi></msub><mo>,</mo><msub><mn>1</mn> <mrow><msub><mn>1</mn> <mi>A</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\theta_a ,1_{1_A})</annotation></semantics></math> are the unit and counit of an <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><mi>a</mi><mo>⊣</mo><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">a \dashv \eta_A</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p><strong>(Adapted from Kelly–Lack)</strong>. 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 lspace="verythinmathspace">:</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 \colon T^2 A \to T A</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">T A</annotation></semantics></math>, and <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><mi>T</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta_A \colon A \to T A</annotation></semantics></math> is a morphism from the underlying object 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 that of <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>. So there is a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>∘</mo><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub><mo>=</mo><msub><mn>1</mn> <mrow><mi>T</mi><mi>A</mi></mrow></msub><mo>⇒</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\bar\eta_A \colon \mu_A \circ T \eta_A = 1_{T A} \Rightarrow \eta_A \circ a</annotation></semantics></math> making <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\eta_A</annotation></semantics></math> into a lax <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>-morphism. Set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub><mo>=</mo><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\theta_a = \bar\eta_A</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/adjunction">triangle equalities</a> then require that:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>⇒</mo><mi>a</mi><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a</annotation></semantics></math> is equal to <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>. The composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∘</mo><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">a \circ \bar\eta_A</annotation></semantics></math> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">a \circ \eta_A</annotation></semantics></math> a lax <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>-morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> (paste <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\bar\eta_A</annotation></semantics></math> with the identity square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∘</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>=</mo><mi>a</mi><mo>∘</mo><mi>T</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">a \circ \mu_A = a \circ T a</annotation></semantics></math>). But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">a \circ \eta_A = 1_A</annotation></semantics></math>, and <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> also makes this into a lax <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>-morphism, so by uniqueness <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">a \bar\eta_A = 1_a</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub><msub><mi>η</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>⇒</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>=</mo><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A</annotation></semantics></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><msub><mi>η</mi> <mi>A</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">1_{\eta_A}</annotation></semantics></math>. But this follows directly from the unit coherence condition for the lax <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>-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>η</mi><mo stretchy="false">¯</mo></mover> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\bar\eta_A</annotation></semantics></math>.</p> </li> </ol> <p>Conversely, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\theta_a</annotation></semantics></math>, algebras <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,b</annotation></semantics></math> on <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,B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics></math> are given. Take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/mate">mate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>f</mi></msub><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>∘</mo><mi>T</mi><mi>f</mi><mo>∘</mo><mi>η</mi><mi>A</mi><mo>=</mo><mi>f</mi><mo>⇒</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">1_f \colon b \circ T f \circ \eta A = f \Rightarrow f</annotation></semantics></math> with respect to the adjunctions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊣</mo><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">a \dashv \eta_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⊣</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \dashv 1</annotation></semantics></math>, which is given in this case by pasting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\theta_a</annotation></semantics></math>, so we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>=</mo><mi>b</mi><mo>∘</mo><mi>T</mi><mi>f</mi><mo>∘</mo><msub><mi>θ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\bar f = b \circ T f \circ \theta_a</annotation></semantics></math>. The mate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> in turn is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\bar f \circ \eta_A</annotation></semantics></math>, which because mates correspond bijectively is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">1_f</annotation></semantics></math>. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> satisfies the unit condition.</p> <p>Consider the diagrams expressing the multiplication condition: because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∘</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>=</mo><mi>a</mi><mo>∘</mo><mi>T</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">a \circ \mu_A = a \circ T a</annotation></semantics></math> (and the same for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>), their boundaries are equal, so we have 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>∘</mo><mi>T</mi><mi>b</mi><mo>∘</mo><msup><mi>T</mi> <mn>2</mn></msup><mi>f</mi><mo>⇒</mo><mi>f</mi><mo>∘</mo><mi>a</mi><mo>∘</mo><mi>T</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\alpha, \beta \colon b \circ T b \circ T^2 f \Rightarrow f \circ a \circ T a</annotation></semantics></math>. Their mates under the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><msub><mi>θ</mi> <mi>a</mi></msub><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>a</mi><mo>⊣</mo><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">(T\theta_a, 1) \colon T a \dashv T\eta_A</annotation></semantics></math> are given by pasting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">T \eta_A</annotation></semantics></math>. One is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> pasted with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>∘</mo><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msub><mi>η</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><msub><mn>1</mn> <mi>f</mi></msub><mo>=</mo><msub><mn>1</mn> <mrow><mi>T</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T \bar f \circ T \eta_A = T(f \circ \eta_A) = T 1_f = 1_{T f}</annotation></semantics></math>, and the other is given by composing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">T \eta_A</annotation></semantics></math> with the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>B</mi></msub><mo>∘</mo><msup><mi>T</mi> <mn>2</mn></msup><mi>f</mi><mo>=</mo><mi>T</mi><mi>f</mi><mo>∘</mo><msub><mi>μ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\mu_B \circ T^2 f = T f \circ \mu_A</annotation></semantics></math> (and then pasting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math>), but because <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><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub><mo>=</mo><msub><mn>1</mn> <mrow><mi>T</mi><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_A \circ T \eta_A = 1_{T A}</annotation></semantics></math> this is also equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><mi>T</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">1_{T f}</annotation></semantics></math>. The two original 2-cells are hence equal, because their mates are equal, and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> is indeed a lax <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>-morphism.</p> </div> <p>Since <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>‘s multiplication <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> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> itself into a (generalized) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra, the above implies (and in fact is implied by) the requirement that there exist a <a class="existingWikiWord" href="/nlab/show/modification">modification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo lspace="verythinmathspace">:</mo><msub><mn>1</mn> <mrow><msup><mi>T</mi> <mn>2</mn></msup></mrow></msub><mo>→</mo><mi>η</mi><mi>T</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\ell \colon 1_{T^2} \to \eta T \circ \mu</annotation></semantics></math> making <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><mn>1</mn><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>μ</mi><mo>⊣</mo><mi>η</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">(\ell,1) \colon \mu \dashv \eta T</annotation></semantics></math>. Conversely, given an algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math>, the 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\theta_a</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>a</mi><mo>∘</mo><msub><mi>ℓ</mi> <mi>A</mi></msub><mo>∘</mo><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">T a \circ \ell_A \circ T \eta_A</annotation></semantics></math>.</p> <p>A different but equivalent condition is that there be a modification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>η</mi><mo>→</mo><mi>η</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">d \colon T \eta \to \eta T</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>η</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d \eta = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mi>d</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mu d = 1</annotation></semantics></math>; and given <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">\ell</annotation></semantics></math> as above, <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> is given by <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>η</mi></mrow><annotation encoding="application/x-tex">\ell \circ T \eta</annotation></semantics></math>.</p> <p>These various conditions can also be regarded as ways to say that the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore adjunction</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-adjunction">lax-idempotent 2-adjunction</a>. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a lax-idempotent 2-monad exactly when this 2-adjunction is lax-idempotent, and therefore also just when it is the 2-monad induced by <em>some</em> lax-idempotent 2-adjunction.</p> <p>Dually, for <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> to be colax-idempotent, it is necessary and sufficient that any of the following hold.</p> <ul> <li> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math> there is a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mi>a</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi><mo>⇒</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\zeta_a \colon \eta_A \circ a \Rightarrow 1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>ζ</mi> <mi>a</mi></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>⊣</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">(1,\zeta_a) \colon \eta_A \dashv a</annotation></semantics></math>.</p> </li> <li> <p>There is a modification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>μ</mi><mo>∘</mo><mi>η</mi><mi>T</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">m \colon \mu \circ \eta T \to 1</annotation></semantics></math> making <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>m</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>η</mi><mi>T</mi><mo>⊣</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">(1,m) \colon \eta T \dashv \mu</annotation></semantics></math>.</p> </li> <li> <p>There is a modification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo lspace="verythinmathspace">:</mo><mi>η</mi><mi>T</mi><mo>→</mo><mi>T</mi><mi>η</mi></mrow><annotation encoding="application/x-tex">e \colon \eta T \to T\eta</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mi>η</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">e\eta = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mi>e</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mu e = 1</annotation></semantics></math>.</p> </li> </ul> <h2 id="algebras">Algebras</h2> <p>Theorem <a class="maruku-ref" href="#AlgebraAdjoint"></a> gives a necessary condition for 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> to admit a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra structure, namely that <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> admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.</p> <div class="num_theorem" id="PseudoAlgebras"> <h6 id="theorem_2">Theorem</h6> <p>To give a pseudo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra structure 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> is equivalently to give a left adjoint to <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> with invertible counit.</p> </div> <p>In particular, an object admits at most one pseudo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra structure, up to unique isomorphism. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra structure is <a class="existingWikiWord" href="/nlab/show/property-like+structure">property-like structure</a>.</p> <p>In many cases it is interesting to consider the pseudo <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 for which the algebra structure <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>A</mi></mrow><annotation encoding="application/x-tex">T A \to A</annotation></semantics></math> has a further left adjoint, forming an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a>. Algebras of this sort are sometimes called <a class="existingWikiWord" href="/nlab/show/continuous+algebras">continuous algebras</a>.</p> <h2 id="examples">Examples</h2> <p>As mentioned above, the standard examples of lax-idempotent 2-monads are those on <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> whose algebras are categories with all colimits of a specified class. In this case, the 2-monad is a <a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a> operation. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class. A converse is given by (<a href="#PowerCattaniWinskel">PowerCattaniWinskel</a>), who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit <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> is dense (plus a coherence condition).</p> <p>Another important example of a colax-idempotent 2-monad is the monad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Cat/B</annotation></semantics></math> that takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p \colon E \to B</annotation></semantics></math> to the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">/</mo><mi>p</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B/p \to B</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>. The algebras for this monad are <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibrations">Grothendieck fibrations</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>; see also <a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a>. The monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>↦</mo><mi>p</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p \mapsto p/B</annotation></semantics></math> is lax-idempotent, and its algebras are opfibrations.</p> <p>This latter is actually a special case of a general situation. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a (2-)monad relative to which one can define <a class="existingWikiWord" href="/nlab/show/generalized+multicategories">generalized multicategories</a>, then often it induces a lax-idempotent 2-monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>T</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{T}</annotation></semantics></math> on the 2-category of such generalized multicategories (aka “virtual <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”), such that (pseudo) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>T</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{T}</annotation></semantics></math>-algebras are equivalent to (pseudo) <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. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the 2-monad whose algebras are strict 2-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">B\to Cat</annotation></semantics></math> and whose pseudo algebras are pseudofunctors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">B\to Cat</annotation></semantics></math>, then a virtual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebra is a category over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, and it is a pseudo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>T</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{T}</annotation></semantics></math>-algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a> whose pseudo algebras are <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>, and so on.</p> <h2 id="properties">Properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-distributive+laws">pseudo-distributive laws</a> involving lax-idempotent 2-monads have an especially nice form; see <a href="#MarmolejoDL">(Marmolejo)</a> and <a href="#WalkerDL">(Walker)</a>.</p> </li> <li> <p>For ordinary 1-monads there exists a presentation due to Manes as “Kleisli triples” with primary data a family of unit morphisms and lifts avoiding the iteration of the endofunctor. A similar presentation exists for lax-idempotent 2-monads as shown in Marmolejo-Wood (<a href="#MW12">2012</a>). It is shown then in Walker <a href="#WalkerYS">(2017)</a> that provided the units of this presentation are fully faithful (a reflection of the fully-faithfulness of the Yoneda embedding) (almost) all the axioms of a <a class="existingWikiWord" href="/nlab/show/Yoneda+structure">Yoneda structure</a> are satisfied. In cases where size plays no role like e.g. the ideal completion of posets the two concepts coincide. For further details see at <a class="existingWikiWord" href="/nlab/show/Yoneda+structure">Yoneda structure</a> or Walker <a href="#WalkerYS">(2017)</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></li> <li><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">stuff, structure, property</a></li> <li><a class="existingWikiWord" href="/nlab/show/property-like+structure">property-like structure</a></li> <li><a class="existingWikiWord" href="/nlab/show/continuous+algebra">continuous algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-adjunction">lax-idempotent 2-adjunction</a></li> </ul> <h2 id="references">References</h2> <p>Classical references are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>On property-like structures</em>, TAC 3(9), 1997. (<a href="http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html">abstract</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Monads for which structures are adjoint to units</em> , Aarhus Preprint 1972/73 No. 35. (<a href="http://home.imf.au.dk/kock/msau1.PDF">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Monads for which structures are adjoint to units</em>, JPAA 104:41–59, 1995.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Fibrations and Yoneda’s lemma in a 2-category</em>, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 104–133. [<a href="https://doi.org/10.1007/BFb0063102">doi:10.1007/BFb0063102</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Fibrations in Bicategories</em> , Cah. Top. Géom. Diff. <strong>XXI</strong> no.2 (1980). (<a href="http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0">numdam</a>)</p> </li> <li> <p>Volker Zöberlein, <em>Doctrines on 2-categories</em> , Math. Zeitschrift <strong>148</strong> (1976) pp.267-279. (<a href="https://gdz.sub.uni-goettingen.de/id/PPN266833020_0148?tify={%22pages%22:[273],%22view%22:%22info%22}">gdz</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francisco+Marmolejo">Francisco Marmolejo</a>, <em>Doctrines whose structure forms a fully faithful adjoint string</em>, Theory and Applications of Categories 3 (1997), 23–44. (<a href="http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html">TAC</a>)</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Sketches of an elephant vol.1</em> , Oxford UP 2004. (B1.1.11, pp.250-54)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Jonathon+Funk">Jonathon Funk</a>, <em>Singular coverings of Toposes</em> , Springer Heidelberg 2006. (pp.79ff)</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <em>Tightly Bounded Completions</em> , TAC <strong>28</strong> no.8 (2013) pp.213-240. (<a href="http://tac.mta.ca/tac/volumes/28/8/28-8abs.html">abstract</a>)</p> </li> <li id="BF99"> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Jonathon+Funk">Jonathon Funk</a>, <em>On a bicomma object condition for KZ-doctrines</em>, JPAA <strong>143</strong> (1999) 69-105 [<a href="https://doi.org/10.1016/S0022-4049(98)00108-X">doi:10.1016/S0022-4049(98)00108-X</a>]</p> </li> <li> <p>A. J. Power, G. L. Cattani, G. Winskel, <em>A representation result for free cocompletions</em>, JPAA 151:273–286, 2000 <a href=" http://dx.doi.org/10.1016/S0022-4049(99)00063-8">doi</a></p> </li> </ul> <p>Their distributive laws come into focus in</p> <ul id="WalkerDL"> <li id="MarmolejoDL"> <p><a class="existingWikiWord" href="/nlab/show/Francisco+Marmolejo">Francisco Marmolejo</a>, <em>Distributive laws for pseudomonads</em>, Theory and Applications of Categories, <strong>5</strong> 5 (1999) 81-147 [<a href="http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html">tac</a>]</p> </li> <li id="MW12"> <p><a class="existingWikiWord" href="/nlab/show/Francisco+Marmolejo">Francisco Marmolejo</a>, <a class="existingWikiWord" href="/nlab/show/Richard+J.+Wood">Richard J. Wood</a>, <em>Kan extensions and lax idempotent pseudomonads</em> , TAC <strong>26</strong> no.1 (2012) pp.1-19. (<a href="http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html">abstract</a>)</p> </li> <li> <p>Charles Walker, <em>Distributive Laws via Admissibility</em>, <a href="https://arxiv.org/abs/1706.09575">arXiv</a></p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/Yoneda+structures">Yoneda structures</a>:</p> <ul id="WalkerYS"> <li>Charles Walker, <em>Yoneda Structures and KZ Doctrines</em>, <a href="https://arxiv.org/abs/1703.08693">arxiv</a></li> </ul> <p>The logical-syntactical side:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jiri+Adamek">Jiri Adamek</a>, Lurdes Sousa, <em>KZ-monadic categories and their logic</em>, <a href="http://tac.mta.ca/tac/volumes/32/10/32-10abs.html">tac</a></li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>:</p> <ul> <li id="ALL23"><a class="existingWikiWord" href="/nlab/show/Nathanael+Arkor">Nathanael Arkor</a>, <a class="existingWikiWord" href="/nlab/show/Ivan+Di+Liberti">Ivan Di Liberti</a>, <a class="existingWikiWord" href="/nlab/show/Fosco+Loregian">Fosco Loregian</a>, <em>Adjoint functor theorems for lax-idempotent pseudomonads</em>, [<a href="https://arxiv.org/abs/2306.10389">arXiv:2306.10389</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 17, 2024 at 08:40:02. 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