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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #15 to #16: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='yoneda_lemma'>Yoneda lemma</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a></p> </li> </ul> <p><strong>Incarnations</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+Yoneda+lemma'>enriched Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/co-Yoneda+lemma'>co-Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+reduction'>Yoneda reduction</a></p> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletion</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></p> </li> </ul> <p><strong>Universal aspects</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+element'>universal element</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a>, <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack'>classifying stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/moduli+space'>moduli space</a>, <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack'>moduli stack</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+moduli+space'>derived moduli space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+topos'>classifying topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+principal+bundle'>universal principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+morphism'>classifying morphism</a></p> </li> </ul> <p><strong>Induced theorems</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></li> </ul> <p>…</p> <p><strong>In higher category theory</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+higher+categories'>Yoneda lemma for higher categories</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>Yoneda lemma for (∞,1)-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+bicategories'>Yoneda lemma for bicategories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h4 id='limits_and_colimits'>Limits and colimits</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits and colimits</a></strong></p> <h2 id='1categorical'>1-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit and colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limits+and+colimits+by+example'>limits and colimits by example</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+colimit'>directed colimit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sequential+limit'>sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+pullback'>wide pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a>, <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a>, <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobase+change'>cobase change</a>, <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+product'>direct product</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end and coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibered limit</a></p> </li> </ul> <h2 id='2categorical'>2-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isoinserter'>isoinserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PIE-limit'>PIE-limit</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> </ul> <h2 id='1categorical_2'>(∞,1)-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id='modelcategorical'>Model-categorical</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+product'>homotopy product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equalizer'>homotopy equalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+totalization'>homotopy totalization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy end</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coproduct'>homotopy coproduct</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coequalizer'>homotopy coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+realization'>homotopy realization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/infinity-limits+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#remarks'>Remarks</a></li><li><a href='#formula'>Formula</a></li></ul></li><li><a href='#properties'>Properties</a></li><li><a href='#generalizations'>Generalizations</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <em>Yoneda extension</em> of a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F : C \to D</annotation></semantics></math> is a universal <a class='existingWikiWord' href='/nlab/show/diff/extension'>extension</a> (<a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a>) along the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_2' xmlns='http://www.w3.org/1998/Math/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> of its <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a> category to a functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>D</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \tilde F \colon [C^{op}, Set] \to D \,. </annotation></semantics></math></div> <p>The Yoneda extension exhibits the <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a> category <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PSh(C)</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletion</a> of <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <h2 id='definition'>Definition</h2> <p>For <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a> and <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F \colon C \to D</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a>, its <strong>Yoneda extension</strong></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'> \tilde F \;\colon\; [C^{op},Set] \to D </annotation></semantics></math></div> <p>is the left <a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a> <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Lan</mi> <mi>Y</mi></msub><mi>F</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>Lan_Y F \;\colon\; [C^{op}, Set] \to D</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> along the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msub><mi>Lan</mi> <mi>Y</mi></msub><mi>F</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \tilde F \;\coloneqq\; Lan_Y F \,. </annotation></semantics></math></div> <h3 id='remarks'>Remarks</h3> <p>Often it is of interest to Yoneda extend not <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F \;\colon\; C \to D</annotation></semantics></math> itself, but the composition <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>∘</mo><mi>F</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>C</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Y \circ F \;\colon\; C \to [D^{op}, Set]</annotation></semantics></math> to get a functor entirely between presheaf categories</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>^</mo></mover><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mover><mrow><mi>Y</mi><mo>∘</mo><mi>F</mi></mrow><mo stretchy='false'>˜</mo></mover><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \hat F \;\coloneqq\; \tilde{Y \circ F} \;\colon\; [C^{op},Set] \to [D^{op}, Set] \,. </annotation></semantics></math></div> <p>This is in fact a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> to the restriction functor <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><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'>F^\ast \;\colon \; [D^{op}, Set] \to [C^{op}, Set]</annotation></semantics></math> which maps <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mo>↦</mo><mi>H</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>H \mapsto H \circ F</annotation></semantics></math>. This is relevant, for instance, to <a class='existingWikiWord' href='/nlab/show/diff/restriction+and+extension+of+sheaves'>restriction and extension of sheaves</a>.</p> <h3 id='formula'>Formula</h3> <p>Recalling the general formula for the left <a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a> of a functor <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F \;\colon\; C \to D</annotation></semantics></math> through a functor <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>p : C \to C'</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>≃</mo><msub><mo lspace='0em' rspace='thinmathspace'>colim</mo> <mrow><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>∈</mo><mo stretchy='false'>(</mo><mi>p</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (Lan_p F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c) </annotation></semantics></math></div> <p>one finds for the Yoneda extension the formula</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd> <mtd><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>Lan</mi> <mi>Y</mi></msub><mi>F</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mo lspace='0em' rspace='thinmathspace'>colim</mo> <mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} \tilde F (A) & \;\coloneqq\; (Lan_Y F)(A) \\ & \simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U) \end{aligned} \,. </annotation></semantics></math></div> <p>(Recall the notation for the <a class='existingWikiWord' href='/nlab/show/diff/comma+category'>comma category</a> <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>const</mi> <mi>A</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Y,A) \;\coloneqq\; (Y, const_A)</annotation></semantics></math> whose objects are pairs <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>∈</mo><mi>C</mi><mo>,</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo><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'>(U \in C, (Y(U) \to A) \in [C^{op}, Set] )</annotation></semantics></math>.</p> <p>For the full extension <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>^</mo></mover><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo><span><del class='diffmod'> .</del><ins class='diffmod'> ,</ins></span></mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'><span> \hat F : [C^{op}, Set] \to<del class='diffmod'> [D^{op}.</del><ins class='diffmod'> [D^{op},</ins> Set]</span></annotation></semantics></math> this yields</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_273ffc57c928bd174e088187848522311ea57083_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mover><mi>F</mi><mo stretchy='false'>^</mo></mover><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mo stretchy='false'>(</mo><msub><mo lspace='0em' rspace='thinmathspace'>colim</mo> <mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mo lspace='0em' rspace='thinmathspace'>colim</mo> <mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mo lspace='0em' rspace='thinmathspace'>colim</mo> <mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} \hat F(A)(V) &= (\colim_{(Y(U) \to A) \in (Y,A) } F(U))(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U)(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } Hom_{D}(V,F(U)) \end{aligned} \,. </annotation></semantics></math></div> <p>Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a>.</p> <h2 id='properties'>Properties</h2> <ul> <li> <p>The restriction of the Yoneda extension to <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> coincides with the original functor: <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo>∘</mo><mi>Y</mi><mo>≃</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'> \tilde F \circ Y \simeq F </annotation></semantics></math>.</p> </li> <li> <p>The Yoneda extension commutes with small colimits in <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, i.e. for <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>\alpha : A \to C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a>, we have <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mi>colim</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>∘</mo><mi>α</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≃</mo><mi>colim</mi><mi>F</mi><mo>∘</mo><mi>α</mi></mrow><annotation encoding='application/x-tex'>\tilde F (colim (Y \circ \alpha)) \simeq colim F \circ \alpha</annotation></semantics></math> .</p> </li> <li> <p>Moreover, <math class='maruku-mathml' display='inline' id='mathml_273ffc57c928bd174e088187848522311ea57083_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde F</annotation></semantics></math> is defined up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> by these two properties.</p> </li> </ul> <h2 id='generalizations'>Generalizations</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Yoneda+extension'>(infinity,1)-Yoneda extension</a></li> </ul> </div> <div class="revisedby"> <p> Last revised on September 11, 2024 at 09:30:18. 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