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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></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/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="yoneda_lemma">Yoneda lemma</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#StatementOfYonedaLemma'>Statement and proof</a></li> <ul> <li><a href='#classical'>Classical</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> </ul> <li><a href='#YonedaCorollaries'>Corollaries</a></li> <ul> <li><a href='#corollary_i_yoneda_embedding'>corollary I: Yoneda embedding</a></li> <li><a href='#corollary_ii_uniqueness_of_representing_objects'>corollary II: uniqueness of representing objects</a></li> <li><a href='#corollary_iii_universality_of_representing_objects'>corollary III: universality of representing objects</a></li> <li><a href='#interpretation'>Interpretation</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <li><a href='#necessity_of_naturality'>Necessity of naturality</a></li> <li><a href='#the_yoneda_lemma_in_semicategories'>The Yoneda lemma in semicategories</a></li> <li><a href='#applications'>Applications</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <strong>Yoneda lemma</strong> says that the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> from a <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(c)</annotation></semantics></math> into an arbitrary <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <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> is in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(c)</annotation></semantics></math> assigned by <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> to the representing <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <p>The Yoneda lemma is an elementary but deep and central result in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and in particular in <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a>. It is essential background behind the central concepts of <em><a class="existingWikiWord" href="/nlab/show/representable+functors">representable functors</a></em>, <em><a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a></em>, and <em><a class="existingWikiWord" href="/nlab/show/universal+elements">universal elements</a></em>.</p> <h2 id="StatementOfYonedaLemma">Statement and proof</h2> <h3 id="classical">Classical</h3> <div class="num_defn" id="FunctorUnderlyingTheYonedaEmbedding"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/functor">functor</a> underlying the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> we write</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>Set</mi><mo stretchy="false">]</mo><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op}, Set] \coloneqq Func(C^{op}, Set) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> out of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> 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">\mathcal{C}</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>This is also called the <em><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. Other notation used for it includes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{C^{op}}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(C^{op},Set))</annotation></semantics></math>.</p> <p>There is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mi>y</mi></mover></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mi>c</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C &\overset{y}{\longrightarrow}& [C^op,Set] \\ c &\mapsto& Hom_{\mathcal{C}}(-,c) } </annotation></semantics></math></div> <p>(called the <em><a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></em> for reasons explained below) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a>, which sends each <a class="existingWikiWord" href="/nlab/show/object">object</a> to the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> into that object, also called the <a class="existingWikiWord" href="/nlab/show/representable+presheaf">presheaf represented</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a>)</strong></p> <p>The Yoneda embedding functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]</annotation></semantics></math> from Def. <a class="maruku-ref" href="#FunctorUnderlyingTheYonedaEmbedding"></a> is equivalently the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒞</mi><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set </annotation></semantics></math></div> <p>under the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a>/<a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom(C^{op} \times C, Set) \stackrel{\simeq}{\to} Hom(C, [C^{op}, Set]) </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">symmetric monoidal category</a> of categories.</p> </div> <div class="num_prop" id="YonedaLemma"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a>, with <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> denoted <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>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}^{op},Set]</annotation></semantics></math>, according to Def. <a class="maruku-ref" href="#FunctorUnderlyingTheYonedaEmbedding"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \in [\mathcal{C}^{op}, Set]</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>, there is a canonical <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{[C^op,Set]}(y(c),X) \;\simeq\; X(c) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> from the <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(c)</annotation></semantics></math> to <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 the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> </div> <p>This is the standard notation used mostly in pure <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>. In other parts of the literature it is customary to denote the presheaf represented by <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> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">h_c</annotation></semantics></math>. In that case the above is often written</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mi>c</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom(h_c, X) \simeq X(c) </annotation></semantics></math></div> <p>or</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Nat</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mi>c</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Nat(h_c, X) \simeq X(c) </annotation></semantics></math></div> <p>to emphasize that the morphisms of presheaves are <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> of the corresponding functors.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>The proof is by chasing the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>c</mi></msub><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Id_c \in C(c, c)</annotation></semantics></math> around both legs of a <a class="existingWikiWord" href="/nlab/show/naturality+square">naturality square</a> for a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\eta: C(-, c) \to X</annotation></semantics></math> (hence a homomorphism of presheaves):</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>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>Id</mi> <mi>c</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>c</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>=</mo><mi>def</mi></mover></mtd> <mtd><mi>ξ</mi></mtd></mtr> <mtr><mtd><msub><mo></mo><mpadded width="0" lspace="-100%width"><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mpadded></msub><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>η</mi> <mi>b</mi></msub></mrow></munder></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>f</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>η</mi> <mi>b</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C(c, c) & \stackrel{\eta_c}{\to} & X(c) & & & & Id_c & \mapsto & \eta_c(Id_c) & \stackrel{def}{=} & \xi \\ _\mathllap{C(f, c)} \downarrow & & \downarrow _\mathrlap{X(f)} & & & & \downarrow & & \downarrow _\mathrlap{X(f)} & & \\ C(b, c) & \underset{\eta_b}{\to} & X(b) & & & & f & \mapsto & \eta_b(f) & & } </annotation></semantics></math></div> <p>What this diagram shows is that the entire transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\eta: C(-, c) \to X</annotation></semantics></math> is completely determined from the single value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>≔</mo><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>c</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi \coloneqq \eta_c(Id_c) \in X(c)</annotation></semantics></math>, because for each object <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> 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>, the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>b</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_b: C(b, c) \to X(b)</annotation></semantics></math> must take an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C(b, c)</annotation></semantics></math> (i.e., a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: b \to c</annotation></semantics></math>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(f)(\xi)</annotation></semantics></math>, according to the commutativity of this diagram.</p> <p>The crucial point is that the naturality condition on any <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⇒</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\eta : C(-,c) \Rightarrow X</annotation></semantics></math> is sufficient to ensure that <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 already entirely fixed by the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>c</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c(Id_c) \in X(c)</annotation></semantics></math> of its component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c : C(c,c) \to X(c)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">Id_c</annotation></semantics></math>. And every such value extends to a natural transformation <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>.</p> <p>More in detail, the bijection is established by the map</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>Set</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>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo stretchy="false">|</mo> <mi>c</mi></msub></mrow></mover><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ev</mi> <mrow><msub><mi>Id</mi> <mi>c</mi></msub></mrow></msub></mrow></mover><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op}, Set](C(-,c),X) \stackrel{|_{c}}{\to} Set(C(c,c), X(c)) \stackrel{ev_{Id_c}}{\to} X(c) </annotation></semantics></math></div> <p>where the first step is taking the component of a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> and the second step is <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>c</mi></msub><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Id_c \in C(c,c)</annotation></semantics></math>.</p> <p>The inverse of this map takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi \in X(c)</annotation></semantics></math> to the natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mi>ξ</mi></msup></mrow><annotation encoding="application/x-tex">\eta^\xi</annotation></semantics></math> with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>η</mi> <mi>d</mi> <mi>ξ</mi></msubsup><mo>:</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta^\xi_d := X(-)(\xi) : C(d,c) \to X(d) \,. </annotation></semantics></math></div></div> <h3 id="in_homotopy_type_theory">In homotopy type theory</h3> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>.</p> <p>Note: the <a class="existingWikiWord" href="/nlab/show/HoTT+book">HoTT book</a> calls a <a class="existingWikiWord" href="/nlab/show/internal+category+in+HoTT">internal category in HoTT</a> a “precategory” and a <a class="existingWikiWord" href="/nlab/show/univalent+category">univalent category</a> a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.</p> <p>By Lemma 9.5.3 in the HoTT book (see <a class="existingWikiWord" href="/nlab/show/product+category">product category</a>), we have an induced functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>y</mi></mstyle><mo>:</mo><mi>A</mi><mo>→</mo><msup><mstyle mathvariant="italic"><mi>Set</mi></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{y} : A \to \mathit{Set}^{A^{op}}</annotation></semantics></math> which we call the <strong>yoneda embedding</strong>.</p> <p><strong>Theorem 9.5.4 (The Yoneda Lemma)</strong> For any <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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math>, and any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mstyle mathvariant="italic"><mi>Set</mi></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">F: \mathit{Set}^{A^{op}}</annotation></semantics></math>, we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mrow><msup><mstyle mathvariant="italic"><mi>Set</mi></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mi>a</mi><mspace width="2em"></mspace><mspace width="2em"></mspace><mo stretchy="false">(</mo><mn>9.5</mn><mn>.5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,F) \cong F a \qquad \qquad(9.5.5)</annotation></semantics></math></div> <p>Moreover this is natural in both <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>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> <p><strong>Proof.</strong> Given a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\alpha : \mathbf{y}a \to F</annotation></semantics></math>, we can consider the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>a</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\alpha_a : \mathbf{y}a(a) \to F a</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{y} a(a)\equiv hom_A(a,a)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>a</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1_a:\mathbf{y}a(a)</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>a</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>F</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\alpha_a(1_a):F a</annotation></semantics></math>. This gives a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>↦</mo><msub><mi>α</mi> <mi>a</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \mapsto \alpha_a(1_a)</annotation></semantics></math> from left to right in (9.5.5).</p> <p>In the other direction, given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>F</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">x: F a</annotation></semantics></math>, we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\alpha : \mathbf{y} a \to F</annotation></semantics></math> by</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> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_{a'}(f) \equiv F_{a,a'}(f)(x)</annotation></semantics></math></div> <p>Naturality is easy to check, so this gives a function from right to left in (9.5.5).</p> <p>To show that these are inverses, first suppose given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>F</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">x: F a</annotation></semantics></math>. Then with <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> defined as above, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\alpha : \mathbf{y}a \to F</annotation></semantics></math> and define <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> as above, then for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:hom_A(a',a)</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>α</mi> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>α</mi> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><msub><mi>a</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mrow><mi>a</mi><mo>′</mo></mrow></msub><mo>∘</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><msub><mi>a</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>α</mi> <mi>a</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mo stretchy="false">(</mo></msub><msub><mi>α</mi> <mi>a</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>a</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \alpha_{a'}(f) &= \alpha_{a'}(\mathbf{y} a_{a,a'}(f)(1_a))\\ &= (\alpha_{a'} \circ \mathbf{y}a_{a,a'}(f))(1_a)\\ &= (F_{a,a'}(f) \circ \alpha_a)(1_a)\\ &= F_{a,a'}(f_(\alpha_a(1_a))\\ &= F_{a,a'}(f)(x). \end{aligned} </annotation></semantics></math></div> <p>Thus, both composites are equal to identities. The proof of naturality follows from this. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>□</mo></mrow><annotation encoding="application/x-tex">\square</annotation></semantics></math></p> <p><strong>Corollary 9.5.6</strong> The Yoneda embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>y</mi></mstyle><mo>:</mo><mi>A</mi><mo>→</mo><msup><mstyle mathvariant="italic"><mi>Set</mi></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{y} : A \to \mathit{Set}^{A^{op}}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a>.</p> <p><strong>Proof.</strong> By the Yoneda lemma, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mrow><msup><mstyle mathvariant="italic"><mi>Set</mi></mstyle> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>a</mi><mo>,</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>b</mi><mo stretchy="false">)</mo><mo>≅</mo><mstyle mathvariant="bold"><mi>y</mi></mstyle><mi>b</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,\mathbf{y}b) \cong \mathbf{y} b(a) \equiv hom_A(a,b)</annotation></semantics></math></div> <p>It is easy to check that this isomorphism is in fact the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>y</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math> on hom-sets. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>□</mo></mrow><annotation encoding="application/x-tex">\square</annotation></semantics></math></p> <h2 id="YonedaCorollaries">Corollaries</h2> <p>The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.</p> <h3 id="corollary_i_yoneda_embedding">corollary I: Yoneda embedding</h3> <p>The Yoneda lemma implies that the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">y \colon C \to [C^op,Set]</annotation></semantics></math> really is an <em>embedding</em> in that it is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>, because for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c,d \in C</annotation></semantics></math> it naturally induces the isomorphism of Hom-sets.</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>Set</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>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op},Set](C(-,c),C(-,d)) \simeq (C(-,d))(c) = C(c,d) </annotation></semantics></math></div> <h3 id="corollary_ii_uniqueness_of_representing_objects">corollary II: uniqueness of representing objects</h3> <p>Since the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>, an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(c) \simeq y(d)</annotation></semantics></math> must come from an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of the representing objects <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 \simeq d</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>c</mi><mo>≃</mo><mi>d</mi></mrow><annotation encoding="application/x-tex"> y(c) \simeq y(d) \;\; \Leftrightarrow \;\; c \simeq d </annotation></semantics></math></div> <h3 id="corollary_iii_universality_of_representing_objects">corollary III: universality of representing objects</h3> <p>A <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X \colon C^{op} \to Set</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> precisely if the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mi>const</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y,const_X)</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>. If a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>y</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d, g : y(d) \to X) \simeq (d, g \in X(d))</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><mi>y</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \simeq y(d)</annotation></semantics></math>.</p> <p>This follows from unwrapping the definition of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mi>const</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y,const_X)</annotation></semantics></math> and applying the Yoneda lemma to find</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><mi>y</mi><mo>,</mo><msub><mi>const</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">{</mo><mi>u</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (y,const_X)((c,f \in X(c)), (d, g \in X(d))) \simeq \{ u \in C(c,d) : X(u)(g) = f \} \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mi>const</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">(y,const_X)((c,f \in X(c)), (d, g \in X(d))) \simeq pt</annotation></semantics></math> says precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(-)(f) \colon C(c,d) \to X(c)</annotation></semantics></math> is a bijection.</p> <h3 id="interpretation">Interpretation</h3> <p>For emphasis, here is the interpretation of these three corollaries in words:</p> <ul> <li> <p><strong>corollary I</strong> says that the interpretation of presheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> as generalized objects probeable by objects <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> 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> is consistent: the probes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by <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> are indeed the maps of generalized objects from <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> into <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>;</p> </li> <li> <p><strong>corollary II</strong> says that probes by objects 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> are sufficient to distinguish objects 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>: two objects 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> are the same if they have the same probes by other objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </li> <li> <p><strong>corollary III</strong> characterizes <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functors</a> by a <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> and is hence the bridge between the notion of <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> and <a class="existingWikiWord" href="/nlab/show/universal+construction">universal constructions</a>.</p> </li> </ul> <h2 id="generalizations">Generalizations</h2> <p>The Yoneda lemma tends to carry over to all important generalizations of the context of <a class="existingWikiWord" href="/nlab/show/locally+small+category">categories</a>:</p> <ul> <li> <p>There is an analog of the Yoneda lemma in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>. See <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a>.</p> </li> <li> <p>In the context of <a class="existingWikiWord" href="/nlab/show/module">modules</a> (see also <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>) the Yoneda lemma becomes the important statement of <a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a>, which identifies the bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom_C(-, -)</annotation></semantics></math> as a unit bimodule.</p> </li> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a>.</p> </li> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+tricategories">Yoneda lemma for tricategories</a>.</p> </li> <li> <p>There is a <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>In <a class="existingWikiWord" href="/nlab/show/functional+programming">functional programming</a>, the Yoneda embedding is related to the <a class="existingWikiWord" href="/nlab/show/continuation+passing+style">continuation passing style</a> transform.</p> </li> <li> <p>Formulation of the lemma in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>: <a href="http://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/">A type theoretical Yoneda lemma</a> at <a href="http://homotopytypetheory.org">homotopytypetheory.org</a></p> </li> </ul> <h2 id="necessity_of_naturality">Necessity of naturality</h2> <p>The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects <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>, in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ℤ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Hom(A,A) = Hom(A,B) = Hom(B,B) = \mathbb{Z}_{\geq 0}</annotation></semantics></math>, the set of integers greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ℤ</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Hom(B,A) = \mathbb{Z}_{\geq 1}</annotation></semantics></math>, the set of integers greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, and in which composition is addition. Here it is certainly the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(A,-)</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(B,-)</annotation></semantics></math> for any choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math>, but <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> are not isomorphic (composition with any arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B \rightarrow A</annotation></semantics></math> is greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, so cannot have an inverse, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> 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>).</p> <p>A finite counter-example is given by the category with two objects <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>, in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Hom(A,A) = Hom(A,B) = Hom(B,B) = \{0, 1\}</annotation></semantics></math>, in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Hom(B,A) = \{0, 2\}</annotation></semantics></math>, and composition is multiplication modulo 2. Here, again, it is certainly the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(A,-)</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(B,-)</annotation></semantics></math> for any choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math>, but <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> are not isomorphic (composition with any arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B \rightarrow A</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, so cannot have an inverse, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> 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>).</p> <p>On the other hand, there have been examples of locally finite categories where naturality is not necessary. For example, (<a href="#Lovasz">Lovász, Theorem 3.6 (iv)</a>) states precisely that finite relational structures <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> are isomorphic if, and only if, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(C,A) \cong Hom(C,B)</annotation></semantics></math> for every finite relational structure <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>. Later (<a href="#Pultr">Pultr, Theorem 2.2</a>) generalised the result to finitely well-powered, locally finite categories with (extremal epi, mono) <a class="existingWikiWord" href="/nlab/show/factorization+system">factorization system</a>.</p> <h2 id="the_yoneda_lemma_in_semicategories">The Yoneda lemma in semicategories</h2> <p>An interesting phenomenon arises in the case of <a class="existingWikiWord" href="/nlab/show/semicategory">semicategories</a> i.e. “categories” (possibly) lacking <a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a>:</p> <p>the Yoneda lemma fails in general, since its validity in a semicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> is in fact already a category because the Yoneda lemma permits to embed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PrSh</mi><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PrSh(\mathcal{G})</annotation></semantics></math> and the latter is always a category, the embedding then implying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> is itself a category!</p> <p>But for <a class="existingWikiWord" href="/nlab/show/regular+semicategories">regular semicategories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/unity+of+opposites">unity of opposites</a> in the category of all <a class="existingWikiWord" href="/nlab/show/semipresheaves">semipresheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> between the so called regular presheaves that are <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representables</a> and presheaves satisfying the Yoneda lemma, whence <em>the Yoneda lemma holds dialectically for regular presheaves!</em></p> <p>For some of the details see at <em><a class="existingWikiWord" href="/nlab/show/regular+semicategory">regular semicategory</a></em> and the references therein.</p> <h2 id="applications">Applications</h2> <ul> <li> <p>The Yoneda lemma is the or a central ingredient in various <a class="existingWikiWord" href="/nlab/show/reconstruction+theorem">reconstruction theorems</a>, such as those of <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a>. See there for a detailed account.</p> </li> <li> <p>In its incarnations as <a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a> the Yoneda lemma governs the algebra of <a class="existingWikiWord" href="/nlab/show/end">ends</a> and <a class="existingWikiWord" href="/nlab/show/coend">coends</a> and hence that of <a class="existingWikiWord" href="/nlab/show/bimodule">bimodules</a> and <a class="existingWikiWord" href="/nlab/show/profunctor">profunctors</a>.</p> </li> <li> <p>The Yoneda lemma is effectively the reason that <a class="existingWikiWord" href="/nlab/show/Isbell+conjugation">Isbell conjugation</a> exists. This is a fundamental duality that relates <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> and <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> in large part of mathematics.</p> </li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</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+structure">Yoneda structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuation-passing+style">continuation-passing style</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+semicategory">regular semicategory</a></p> </li> <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> </ul> <p><br /></p> <h2 id="References">References</h2> <p>For general references see any text on <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, as listed in the references <a href="category+theory#References">there</a>.</p> <p>The term <em>Yoneda lemma</em> originated in an interview of <a class="existingWikiWord" href="/nlab/show/Nobuo+Yoneda">Nobuo Yoneda</a> by <a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a> at Paris Gare du Nord:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoshiki+Kinoshita">Yoshiki Kinoshita</a>, <em>Nobuo Yoneda (1930-1996)</em></p> <p>&</p> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>The Yoneda Lemma</em>,</p> <p>Math. Japonica, <strong>47</strong>, No. 1 (1998) 155-156</p> <p>(<a href="https://dmitripavlov.org/scans/yoneda.pdf">pdf</a> <a class="existingWikiWord" href="/nlab/files/MathJaponica_YonedaObituary.pdf" title="pdf">pdf</a>, also: <a href="http://www.mta.ca/~cat-dist/catlist/1999/yoneda">posting to catlist in 1996</a>)</p> </li> </ul> <p>In <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em> MacLane writes that this happened in 1954.</p> <p><img src="http://ncatlab.org/nlab/files/YonedaObituary.jpg" /></p> <p>Review and exposition:</p> <ul> <li> <p><a href="https://www.cs.bham.ac.uk/~mhe/yoneda/yoneda.html">Using Yoneda rather than J to present the identity type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, Section A.1 of: <em>Technique de descente et théorèmes d’existence en géométrie algébriques. II. Le théorème d’existence en théorie formelle des modules</em>, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. (<a href="http://www.numdam.org/item/?id=SB_1958-1960__5__369_0">numdam:SB_1958-1960__5__369_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §III.2 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em><a class="existingWikiWord" href="/nlab/files/LeinsterYoneda.ps" title="The Yoneda Lemma: What's it all about?">The Yoneda Lemma: What's it all about?</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>Category Theory in Context. Chapter 2. Universal Properties, Representability, and the Yoneda Lemma</em> <a href="http://www.math.jhu.edu/~eriehl/context.pdf">pdf</a></p> </li> <li id="RRZ2004"> <p>Marie La Palme Reyes, Gonzalo E. Reyes, and Houman Zolfaghari, <em>Generic figures and their glueings: A constructive approach to functor categories</em>, Polimetrica sas, 2004 (<a href="https://reyes-reyes.com/2004/06/01/generic-figures-and-their-glueings-a-constructive-approach-to-functor-categories/">author page</a>,<a href="https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf">pdf</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a>, <em>Notes on Category Theory with examples from basic mathematics</em>, Chapter 2. (<a href="http://arxiv.org/abs/1912.10642">arXiv</a>)</p> </li> </ul> <p>A discussion of the Yoneda lemma from the point of view of <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vaughan+Pratt">Vaughan Pratt</a>, <em>The Yoneda lemma without category theory: algebra and applications</em> (<a href="http://boole.stanford.edu/pub/yon.pdf">pdf</a>).</li> </ul> <p>A treatment of the Yoneda lemma for <a class="existingWikiWord" href="/nlab/show/internal+category+in+an+%28infinity%2C1%29-category">categories internal to an (∞,1)-topos</a> is in</p> <ul> <li>Louis Martini, <em>Yoneda’s lemma for internal higher categories</em>, (<a href="https://arxiv.org/abs/2103.17141">arXiv:2103.17141</a>)</li> </ul> <p>Early Lovász-Type results include</p> <ul> <li id="Lovasz">László Lovász, <em>Operations with structures</em>, Acta Mathematica Academiae Scientiarum Hungarica 18.3-4 (1967): 321-328.</li> <li id="Pultr">Aleš Pultr. <em>Isomorphism types of objects in categories determined by numbers of morphisms</em>, Acta Scientiarum Mathematicarum, 35:155–160, 1973.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 1, 2024 at 19:36:24. 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