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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> to <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> in the generality of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. For the different concept of <em><a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></em> of <a class="existingWikiWord" href="/nlab/show/comodules">comodules</a> see there.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </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 and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/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/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#in_1category_theory'>In 1-category theory</a></li> <li><a href='#PoweringOfInfinityToposesOverInfinityGroupoids'>Powering of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><msup><mi>V</mi> <mi>op</mi></msup><mo>×</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">[-,-] : V^{op} \times V \to V</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> \big[v_1,[v_2,v_3]\big] \;\simeq\; \big[v_2,[v_1,v_3]\big] </annotation></semantics></math></div> <p>for all <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_i \in V</annotation></semantics></math> (<a href="closed+monoidal+category#TensorHomIsoInternalizes">prop.</a>). If we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo lspace="thickmathspace" rspace="thickmathspace">:=</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">V(v_1,v_2) \mathrel{:=} [v_1,v_2]</annotation></semantics></math> and this reads</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V\big(v_1,V(v_2,v_3)\big) \;\simeq\; V\big(v_2,V(v_1,v_3)\big) \,. </annotation></semantics></math></div> <p>If we now pass more generally to any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> then we still have the enriched <a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C(-,-) : C^{op} \times C \to V</annotation></semantics></math>. One says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <em>powered</em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> if it is in addition equipped also with a mixed operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo>:</mo><msup><mi>V</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\pitchfork : V^{op} \times C \to C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pitchfork(v,c)</annotation></semantics></math> behaves as if it were a hom of the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in V</annotation></semantics></math> into the object <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> in that it comes with natural isomorphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mo>⋔</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>v</mi><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C\big(c_1, \pitchfork(v,c_2)\big) \;\simeq\; V\big(v, C(c_1,c_2)\big) \,. </annotation></semantics></math></div> <h2 id="definition">Definition</h2> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>. In a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the <strong>power</strong> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">y\in C</annotation></semantics></math> by an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v\in V</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\pitchfork(v,y) \in C</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo>⋔</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>V</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(x, \pitchfork(v,y)) \cong V(v, C(x,y)) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(-,-)</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued hom 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><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">V(-,-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <strong>powered</strong> or <strong>cotensored</strong> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> if all such power objects exist.</p> </div> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>Powers are frequently called <em>cotensors</em> and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-category having all powers is called <em>cotensored</em>, while the word “power” is reserved for the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">V=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a <a class="existingWikiWord" href="/nlab/show/copower">copower</a>) is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>, while a cotensor (= power) is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a>.</p> </div> <h2 id="properties">Properties</h2> <ul> <li>Powers are a special sort of <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a>: in particular, where the domain is the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-category. Conversely, all weighted limits can be constructed from powers together with <a class="existingWikiWord" href="/nlab/show/conical+limit">conical limit</a>s. The dual colimit notion of a power is a <a class="existingWikiWord" href="/nlab/show/copower">copower</a>.</li> </ul> <h2 id="examples">Examples</h2> <h3 id="in_1category_theory">In 1-category theory</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> itself is always powered over itself, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo lspace="thickmathspace" rspace="thickmathspace">:=</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\pitchfork(v_1,v_2) \mathrel{:=} [v_1,v_2]</annotation></semantics></math>.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mi>Set</mi><mo>,</mo><mo>×</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = (Set,\times)</annotation></semantics></math> ) with all <a class="existingWikiWord" href="/nlab/show/product">product</a>s is powered over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>: the powering operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo lspace="thickmathspace" rspace="thickmathspace">:=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>c</mi></mrow><annotation encoding="application/x-tex"> \pitchfork(S,c) \mathrel{:=} \prod_{s\in S} c </annotation></semantics></math></div> <p>of an object <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> by a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> forms the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|S|</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</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">c</annotation></semantics></math> with itself, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|S|</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>The defining natural isomorphism</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><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mo>⋔</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2)) </annotation></semantics></math></div> <p>is effectively the definition of the product (see <a class="existingWikiWord" href="/nlab/show/limit">limit</a>).</p> </li> <li> <p>In a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">\mathcal{K}</annotation></semantics></math> (seen as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Cat}</annotation></semantics></math>-enriched category), powers by the walking arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math> are ways to internalize ‘generalized arrows’ of a given object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">A:\mathcal{K}</annotation></semantics></math>. Specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><msup><mi>A</mi> <mo stretchy="false">↓</mo></msup></mrow><mo>:</mo><mo>=</mo><mrow><mo stretchy="false">↓</mo><mo>⋔</mo><mi>A</mi></mrow></mrow><annotation encoding="application/x-tex">{A^\downarrow} := {\downarrow \pitchfork A}</annotation></semantics></math>, called the <strong>object of arrows</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is, when it exists, an object such that:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>A</mi> <mo stretchy="false">↓</mo></msup><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>Cat</mi></mstyle><mo stretchy="false">(</mo><mo stretchy="false">↓</mo><mo>,</mo><mi>𝒦</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>𝒦</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><msup><mo stretchy="false">)</mo> <mo stretchy="false">↓</mo></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{K}(X, A^\downarrow) \simeq \mathbf{Cat}(\downarrow, \mathcal{K}(X,A)) = \mathcal{K}(X,A)^\downarrow. </annotation></semantics></math></div> <p>Thus <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo stretchy="false">↓</mo></msup></mrow><annotation encoding="application/x-tex">A^\downarrow</annotation></semantics></math> correspond to 2-cells between generalized elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, explaining why <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo stretchy="false">↓</mo></msup></mrow><annotation encoding="application/x-tex">A^\downarrow</annotation></semantics></math> can be considered a ‘view from the inside’ of the internal structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </li> </ul> <div> <h3 id="PoweringOfInfinityToposesOverInfinityGroupoids">Powering of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</h3> <p>We discuss how the <a class="existingWikiWord" href="/nlab/show/powering">powering</a> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-toposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-toposes</a> over <a class="existingWikiWord" href="/nlab/show/Infinity-Grpd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Grpd</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">Grpd_\infty</annotation> </semantics> </math></a> is given by forming <a class="existingWikiWord" href="/nlab/show/mapping+stacks">mapping stacks</a> out of <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stacks">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>. All of the following formulas and their proofs hold verbatim also for <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a>, as they just use general abstract properties.</p> <p><br /></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-topos</a></p> <ul> <li> <p>with terminal <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-geometric+morphism">geometric morphism</a> denoted</p> <div class="maruku-equation" id="eq:TerminalGeometricMorphismAdjunction"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></munderover><msub><mi>Grp</mi> <mn>∞</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grp_\infty \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> constructs <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stacks">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>,</p> </li> <li> <p>and with its <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (<a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a>) <a class="existingWikiWord" href="/nlab/show/adjoint+%28infinity%2C1%29-functor">adjunction</a> denoted</p> <div class="maruku-equation" id="eq:MappingStackAdjunction"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></mover></munderover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \,\in\, \mathbf{H}</annotation></semantics></math>.</p> <p>Notice that this construction is also <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-functorial</a> in the first argument: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Maps\big( X \xrightarrow{f} Y ,\, A \big)</annotation></semantics></math> is the morphism which under the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-Yoneda+lemma"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-Yoneda lemma</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply) corresponds to</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>f</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Y</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \;\simeq\; \mathbf{H} \big( (-) \times X ,\, A \big) \xrightarrow{ \mathbf{H} \big( (-) \times f ,\, A \big) } \mathbf{H} \big( (-) \times Y ,\, A \big) \;\simeq\; \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \,. </annotation></semantics></math></div> <p>By definition, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">S \in Grpd_\infty</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/powering">powering</a>] is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> over the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> constant on <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>K</mi></msup><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> X^K \,=\, {\lim_\leftarrow}_K X </annotation></semantics></math></div> <p>while the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> over the diagram constant on <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⋅</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K \cdot X \,=\, {\lim_{\to}}_K X \,. </annotation></semantics></math></div> <p> <div class="num_remark"> <h6>Remark</h6> <p>Under <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a>, the powering operations on homotopy types <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> corresponds to higher order <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of suitable algebras of functions on <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 discussed there.</p> </div> </p> <p> <div class="num_prop"> <h6>Proposition</h6> <p>The <em>powering</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/Infinity-Grpd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Grpd</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">Grpd_\infty</annotation> </semantics> </math></a> is given by the <a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a> out of the <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stack">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</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><msubsup><mi>Grpd</mi> <mn>∞</mn> <mi>op</mi></msubsup><mo>×</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>LConst</mi> <mi>op</mi></msup><mo>×</mo><mi mathvariant="normal">id</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>op</mi></msup><mo>×</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Grpd_\infty^{op} \times \mathbf{H} & \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow} & \mathbf{H}^{op} \times \mathbf{H} & \overset{Maps(-,-)}{\longrightarrow} & \mathbf{H} } </annotation></semantics></math></div> <p>in that this operation has the following properties:</p> <ol> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X,\,A \,\in\, \mathbf{H}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">S \,\in\, Grpd_\infty</annotation></semantics></math> we have a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) \;\; \simeq \;\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) </annotation></semantics></math></div></li> <li> <p>In its first argument the operation</p> <ol> <li> <p>sends the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> (the <a class="existingWikiWord" href="/nlab/show/point">point</a>) to the identity:</p> <div class="maruku-equation" id="eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>X</mi></mrow><annotation encoding="application/x-tex"> Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X </annotation></semantics></math></div></li> <li> <p>sends <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimits"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-colimits</a> to <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-limits"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-limits</a>:</p> <div class="maruku-equation" id="eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>LConst</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Maps \Big( \underset{ \longrightarrow }{\lim} \, LConst\big(S_\bullet\big) ,\, X \Big) \;\; \simeq \;\; \underset{ \longleftarrow }{\lim} \, Maps \Big( LConst\big(S_\bullet\big) ,\, X \Big) \,, </annotation></semantics></math></div></li> </ol> <p>where all <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalences</a> shown are <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural</a>.</p> </li> </ol> <p></p> </div> </p> <p> <div class="proof"> <h6>Proof</h6> <p></p> <p>For the first statement to be proven, consider the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left left"><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Γ</mi><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:TerminalGeometricMorphismAdjunction">(1)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="2.4em" minsize="2.4em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mo maxsize="2.4em" minsize="2.4em">)</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>×</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A \big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:TerminalGeometricMorphismAdjunction">(1)</a> } \\ & \;\simeq\; Grpd_\infty \bigg( S ,\, \mathbf{H} \Big( \ast_{\mathbf{H}} ,\, Maps \big( X ,\, A \big) \Big) \bigg) & \text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( \ast_{\mathbf{H}} \times X ,\, A \big) \Big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array} </annotation></semantics></math></div> <p>For the second statement, recall that <a class="existingWikiWord" href="/nlab/show/hom-functors+preserve+limits">hom-functors preserve limits</a> in that there are <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural</a> <a class="existingWikiWord" href="/nlab/show/equivalences+in+an+%28infinity%2C1%29-category">equivalences</a> of the form</p> <div class="maruku-equation" id="eq:HomFunctorPreservesLimits"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>i</mi></munder></munder><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \Big( \underset{\underset{i}{\longrightarrow}}{\lim} \,, X_i ,\, \underset{\underset{j}{\longleftarrow}}{\lim} \,, A_j \Big) \;\; \simeq \;\; \underset{\underset{i}{\longleftarrow}}{\lim} \, \underset{\underset{j}{\longleftarrow}}{\lim} \, \mathbf{H} \Big( X_i ,\, A_j \Big) \,, </annotation></semantics></math></div> <p>and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes have <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a>, in particular that the <a class="existingWikiWord" href="/nlab/show/product">product</a> operation is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> and <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">hence preserves colimits</a>:</p> <div class="maruku-equation" id="eq:ProductsPreserveColimits"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>×</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>×</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-) \,\times\, \underset{{\longrightarrow}}{\lim} \, S_\bullet \;\; \simeq \;\; \underset{{\longrightarrow}}{\lim} \, \big( (-) \,\times\, S_\bullet \big) \,. </annotation></semantics></math></div> <p>With this, we get the following sequences of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left left"><mtr><mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="2.4em" minsize="2.4em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mo maxsize="2.4em" minsize="2.4em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:ProductsPreserveColimits">(6)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> </mtext><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ <a class="maruku-eqref" href="#eq:ProductsPreserveColimits">(6)</a> } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) & \text{ <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> } \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> } \,. \end{array} </annotation></semantics></math></div> <p>This implies <a class="maruku-eqref" href="#eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument">(4)</a> by the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-Yoneda+lemma"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-Yoneda lemma</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply).</p> <p>Finally <a class="maruku-eqref" href="#eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg">(3)</a> is immediate from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> preserves the terminal object, by definition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Maps \big( LConst(\ast) ,\, X \big) \;\simeq\; Maps \big( \ast_{\mathbf{H}} ,\, X \big) \;\simeq\; X \,. </annotation></semantics></math></div> <p></p> </div> </p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensored+and+cotensored+category">tensored and cotensored category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/copower">copower</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-copower">(∞,1)-copower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback-power">pullback-power</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, section 3.7 of: <em>Basic concepts of enriched category theory</em> (<a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html">tac</a> ,<a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf">pdf</a>)</p> </li> <li id="Borceux94"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Vol 2, Section 6.5 of <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Cambridge University Press (1994)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, §3.7 in: <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [<a href="https://doi.org/10.1017/CBO9781107261457">doi:10.1017/CBO9781107261457</a>, <a href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">pdf</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 10, 2023 at 16:35:10. 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