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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> </li> </ul> <p><strong>Incarnations</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a></p> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></p> </li> </ul> <p><strong>Universal aspects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+element">universal element</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>, <a class="existingWikiWord" href="/nlab/show/classifying+stack">classifying stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>, <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+moduli+space">derived moduli space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+morphism">classifying morphism</a></p> </li> </ul> <p><strong>Induced theorems</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></li> </ul> <p>…</p> <p><strong>In higher category theory</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+higher+categories">Yoneda lemma for higher categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+%28%E2%88%9E%2C1%29-categories">Yoneda lemma for (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+tricategories">Yoneda lemma for tricategories</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/Yoneda+lemma+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="2category_theory">2-category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#statement'>Statement</a></li> <li><a href='#implications'>Implications</a></li> <li><a href='#proof'>Proof</a></li> <ul> <li><a href='#explicit_proof'>Explicit proof</a></li> <li><a href='#hightechnology_proof'>High-technology proof</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="statement">Statement</h2> <p>The <strong>Yoneda lemma for bicategories</strong> is a version of the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> that applies to <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>, the most common algebraic sort of weak <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>. It says that for any bicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x\in C</annotation></semantics></math>, and any <a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F\colon C^{op}\to Cat</annotation></semantics></math>, there is an equivalence of categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op},Cat](C(-,x), F) \simeq F(x) </annotation></semantics></math></div> <p>which is <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformation">pseudonatural</a> in <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, and which is given by evaluation at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">1_x</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\alpha\colon C(-,x)\to F</annotation></semantics></math> maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_x(1_x)</annotation></semantics></math>.</p> <p>For bicategories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,B]</annotation></semantics></math> denotes the bicategory of <a class="existingWikiWord" href="/nlab/show/pseudofunctors">pseudofunctors</a>, <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformations">pseudonatural transformations</a>, and <a class="existingWikiWord" href="/nlab/show/modifications">modifications</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. Note that it is a strict 2-category as soon as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is.</p> <h2 id="implications">Implications</h2> <p>In particular, the Yoneda lemma for bicategories implies that there is a <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> for bicategories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C\to [C^{op},Cat]</annotation></semantics></math> which is <span class="newWikiWord">2-fully-faithful<a href="/nlab/new/2-fully-faithful+functor">?</a></span>, i.e. an equivalence on hom-categories. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/equivalence">equivalent</a> to a sub-bicategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op},Cat]</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalent to a strict 2-category, which is one form of the <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+bicategories">coherence theorem for bicategories</a>. (Conversely, another form of the coherence theorem can be used to prove the Yoneda lemma; see below.)</p> <h2 id="proof">Proof</h2> <p>A detailed proof of the bicategorical Yoneda lemma is given in (<a href="#JohnsonYau">Johnson & Yau 20, Chap. 8</a>).</p> <h3 id="explicit_proof">Explicit proof</h3> <p>An explicit proof, involving many diagrams, has been written up by <a class="existingWikiWord" href="/nlab/show/Igor+Bakovi%C4%87">Igor Baković</a> and can be found <a href="http://www.irb.hr/korisnici/ibakovic/yoneda.pdf">here</a>.</p> <h3 id="hightechnology_proof">High-technology proof</h3> <p>We will take it for granted that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op},Cat]</annotation></semantics></math> is a well-defined bicategory; this is a basic fact having nothing to do with the Yoneda lemma. We also take it as given that “evaluation at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">1_x</annotation></semantics></math>” functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op},Cat](C(-,x), F) \to F(x) </annotation></semantics></math></div> <p>is well-defined and pseudonatural in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <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>; our goal is to prove that it is an equivalence. (Granted, these basic facts require a fair amount of verification as well.)</p> <p>We will use part of the <span class="newWikiWord">coherence theorem for pseudoalgebras<a href="/nlab/new/coherence+theorem+for+pseudoalgebras">?</a></span>, which says that for a suitably well-behaved strict <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>, the inclusion <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mi>strict</mi></msub><mo>↪</mo><mi>Ps</mi></mrow><annotation encoding="application/x-tex">Alg_{strict} \hookrightarrow Ps</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi></mrow><annotation encoding="application/x-tex">Alg</annotation></semantics></math> of the 2-category of 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 and 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>-morphisms into the 2-category of 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 and 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>-morphisms has a left adjoint, usually written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">(-)'</annotation></semantics></math>. Moreover, for any 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A\to A'</annotation></semantics></math> is an equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ps</mi></mrow><annotation encoding="application/x-tex">Ps</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi></mrow><annotation encoding="application/x-tex">Alg</annotation></semantics></math>.</p> <p>First, there is 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> such that 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 are strict 2-categories, 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>-morphisms are strict 2-functors, 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 are <a class="existingWikiWord" href="/nlab/show/biased">unbiased bicategories</a>, and 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>-morphisms are <a class="existingWikiWord" href="/nlab/show/pseudofunctors">pseudofunctors</a>. By Mac Lane’s coherence theorem for bicategories, any ordinary bicategory can equally well be considered as an unbiased one. Thus, since <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> is a strict 2-category, for any bicategory <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 strict 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> such that pseudofunctors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C\to Cat</annotation></semantics></math> are in bijection with strict 2-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C'\to Cat</annotation></semantics></math>.</p> <p>Now note that a pseudonatural transformation between two pseudofunctors (resp. strict 2-functors) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C\to D</annotation></semantics></math> is the same as a single pseudofunctor (resp. strict 2-functor) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C\to Cyl(D)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(D)</annotation></semantics></math> is the bicategory whose objects are the 1-cells of <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>, whose 1-cells are squares in <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> commuting up to isomorphism, and whose 2-cells are “cylinders” in <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>. Likewise, a modification between two such transformations is the same as a single functor (of whichever sort) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mn>2</mn><mi>Cyl</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C\to 2Cyl(D)</annotation></semantics></math>, where the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>Cyl</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2Cyl(D)</annotation></semantics></math> are the 2-cells of <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>, and so on. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> classifies not only pseudofunctors out 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>, but transformations and modifications between them; thus we have an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo><mo>≅</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op},Cat] \cong [(C')^{op},Cat]_{strict,pseudo}</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[A,B]_{strict,pseudo}</annotation></semantics></math> denotes the 2-category of strict 2-functors, pseudonatural transformations, and modifications between two strict 2-categories. Thus we can equally well analyze the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub><mo stretchy="false">(</mo><mover><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [(C')^{op},Cat]_{strict,pseudo}(\overline{C(-,x)}, \overline{F}) \to \overline{F}(x) = F(x) </annotation></semantics></math></div> <p>given by evaluation at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">1_x</annotation></semantics></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{C(-,x)}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{F}</annotation></semantics></math> denote the strict 2-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(C')^{op}\to Cat</annotation></semantics></math> corresponding to the pseudofunctors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(-,x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> under the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">(-)'</annotation></semantics></math> adjunction. However, we also have a strict 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C'(-,x)</annotation></semantics></math>, and the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C\simeq C'</annotation></semantics></math> induces an equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">C'(-,x)\simeq \overline{C(-,x)}</annotation></semantics></math>. Therefore, it suffices to analyze the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). </annotation></semantics></math></div> <p>Now for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, we have an inclusion functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>strict</mi></mrow></msub><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[A,B]_{strict,strict} \to [A,B]_{strict,pseudo}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>strict</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[A,B]_{strict,strict}</annotation></semantics></math> denotes the 2-category of strict 2-functors, strict 2-natural transformations, and modifications. This functor is <a class="existingWikiWord" href="/nlab/show/bijective+on+objects+functor">bijective on objects</a> and <a class="existingWikiWord" href="/nlab/show/locally+fully+faithful+2-functor">locally fully faithful</a>. Moreover, the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>strict</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). </annotation></semantics></math></div> <p>is an <em>isomorphism</em>, by the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a>, in the special case of <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>-enrichment. Since</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>strict</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mrow><mi>strict</mi><mo>,</mo><mi>pseudo</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F})</annotation></semantics></math></div> <p>is fully faithful, if we can show that it is essentially surjective, then the <a class="existingWikiWord" href="/nlab/show/2-out-of-3+property">2-out-of-3 property</a> for equivalences of categories will imply that the desired functor is an equivalence.</p> <p>Here we at last descend to something concrete. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mover><mi>F</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\alpha\colon C'(-,x)\to \overline{F}</annotation></semantics></math>, we have an obvious choice for a strict transformation for it to be equivalent to, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> whose components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>y</mi></msub><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta_y\colon C'(y,x)\to \overline{F}(y)</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \mapsto \overline{F}(f)(a)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><msub><mi>α</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a = \alpha_x(1_x)\in \overline{F}(x)</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is pseudonatural, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">f\colon y\to x</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> we have an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>y</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>y</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msub><mn>1</mn> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>≅</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>β</mi> <mi>y</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\alpha_y(f) = \alpha_y(f\circ 1_x) \cong \overline{F}(f)(\alpha_x(1_x)) = \overline{F}(f)(a) = \beta_y(f).</annotation></semantics></math></div> <p>We then simply verify that these isomorphisms are the components of an (invertible) modification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>≅</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha\cong \beta</annotation></semantics></math>. This completes the proof.</p> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+tricategories">Yoneda lemma for tricategories</a></li> </ul> <h2 id="references">References</h2> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Angelo+Vistoli">Angelo Vistoli</a>, §3.6.2 in: <em>Grothendieck topologies, fibered categories and descent theory</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Fundamental+algebraic+geometry+--+Grothendieck%27s+FGA+explained">Fundamental algebraic geometry – Grothendieck's FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) 1-104 [<a href="https://bookstore.ams.org/surv-123-s">ISBN:978-0-8218-4245-4</a>, <a href="http://arxiv.org/abs/math/0412512">math.AG/0412512</a>]</li> </ul> <p>See also:</p> <ul> <li id="JohnsonYau"><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, <a class="existingWikiWord" href="/nlab/show/Donald+Yau">Donald Yau</a>, Chap. 8: <em>2-Dimensional Categories</em>, (<a href="https://arxiv.org/abs/2002.06055">arXiv:2002.06055</a>)</li> </ul> <p>An account of <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a> as a corollary of the Yoneda lemma for bicategories is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, <em>Morita Theory For Derived Categories: A Bicategorical Perspective</em> (<a href="http://arxiv.org/abs/0805.3673">arXiv:0805.3673</a>)</li> </ul> <p>The stricter case of 2-categories is detailed in</p> <ul> <li>Jonas Hedman, <em>2-Categories and Yoneda lemma</em> (<a href="http://www.diva-portal.org/smash/get/diva2:1064822/FULLTEXT01.pdf">pdf</a>)</li> <li>Max Kelly, <em>Basic Concepts of Enriched Category Theory</em> (<a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html">TAC</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 7, 2023 at 07:02:33. 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