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change of enriching category in nLab

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<span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/9393/#Item_12" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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"> <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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#generalizations_and_enhancements'>Generalizations and enhancements</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given two <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_1,\otimes_1, I_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_2, \otimes_2, I_2)</annotation></semantics></math> used for enrichment in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> (for instance two <a class="existingWikiWord" href="/nlab/show/B%C3%A9nabou+cosmoi">Bénabou cosmoi</a>), a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> between them</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>I</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; (V_1, \otimes_1, I_1) \longrightarrow (V_2, \otimes_2, I_2) </annotation></semantics></math></div> <p>canonically induces a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mn>1</mn></msub><mi>Cat</mi><mo>⟶</mo><msub><mi>V</mi> <mn>2</mn></msub><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> F_\ast \;\colon\; V_1 Cat \longrightarrow V_2 Cat </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/categories+of+V-enriched+categories">categories of V-enriched categories</a> sending a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">V_1</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">\mathcal{C}</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">V_2</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><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_\ast(\mathcal{C})</annotation></semantics></math> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_\ast(\mathcal{C})</annotation></semantics></math> has the same <a class="existingWikiWord" href="/nlab/show/objects">objects</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_\ast(\mathcal{C})</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/images">images</a> of the hom-objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>F</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> F_\ast(\mathcal{C})(x,y) \;\coloneqq\; F\big(\mathcal{C}(x,y)\big) </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, <a class="existingWikiWord" href="/nlab/show/unit">unit</a>, <a class="existingWikiWord" href="/nlab/show/associator">associator</a> and <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a> <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_\ast(\mathcal{C})</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/images">images</a> of those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> composed with the structure morphisms of the <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> </li> </ol> <p>(<a href="#EilenbergKelly65">Eilenberg-Kelly 65</a>)</p> <p>This operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">F_\ast</annotation></semantics></math> is sometimes called <strong>change of enriching category</strong> or <strong>change of enriching base</strong> or just <strong>change of base</strong> (e.g. <a href="#Crutwell14">Crutwell 14, chapter 4</a>, <a href="#Riehl14">Riehl 14, lemma 3.4.3</a>). But notice that, despite some vague similarity, this is different from <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> of <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a>.</p> <p>In the special case 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">\mathcal{C}</annotation></semantics></math> has a single object and is hence (if thought of as <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> by that object) equivalently a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">V_1</annotation></semantics></math>, this statement reduces to the statement that lax monoidal functors preserve <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> (<a href="monoidal+functor#MonoidsToMonoidsByLaxMonoidal">this Prop.</a>)</p> <h2 id="generalizations_and_enhancements">Generalizations and enhancements</h2> <ul> <li> <p>The operation of change of enriching category is functorial from <a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a> to <a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a>. In particular, any <a class="existingWikiWord" href="/nlab/show/monoidal+adjunction">monoidal adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>⇄</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">V_1\rightleftarrows V_2</annotation></semantics></math> gives rise to a <a class="existingWikiWord" href="/nlab/show/2-adjunction">2-adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mi>Cat</mi><mo>⇄</mo><msub><mi>V</mi> <mn>2</mn></msub><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V_1 Cat\rightleftarrows V_2 Cat</annotation></semantics></math> (and also for profunctors).</p> </li> <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>-enriched categories can be defined more generally when <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> is a <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, and any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F:V_1\to V_2</annotation></semantics></math> between multicategories induces a change of enrichment 2-functor. Note that a functor of multicategorise between the underlying multicategories of two monoidal categories corresponds to a <em>lax</em> monoidal functor, as in the original version above. The multicategorical version also includes change of enrichment between <a class="existingWikiWord" href="/nlab/show/closed+categories">closed categories</a>.</p> </li> <li> <p>A different generalization is when <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> is a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, yielding <a class="existingWikiWord" href="/nlab/show/bicategory-enriched+categories">bicategory-enriched categories</a>. Any lax functor of bicategories induces a similar functor between its 2-categories of enriched categories.</p> </li> <li> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">V_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">V_2</annotation></semantics></math> are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Prof">Prof</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math>-categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> exist, change of enrichment also induces a <a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mi>Prof</mi><mo>→</mo><msub><mi>V</mi> <mn>2</mn></msub><mi>Prof</mi></mrow><annotation encoding="application/x-tex">V_1 Prof \to V_2 Prof</annotation></semantics></math>. To make this version functorial on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MonCat</mi> <mi>cocomplete</mi></msub></mrow><annotation encoding="application/x-tex">MonCat_{cocomplete}</annotation></semantics></math>, we need to consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub><mi>Prof</mi></mrow><annotation encoding="application/x-tex">V_i Prof</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a>, yielding a functor from <a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a> to <span class="newWikiWord">DblCat<a href="/nlab/new/DblCat">?</a></span>.</p> </li> <li> <p>Finally, a generalization subsuming all of these is that for any <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double 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> we can construct another virtual double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Prof</mi></mrow><annotation encoding="application/x-tex">V Prof</annotation></semantics></math>, and this construction is functorial.</p> </li> </ul> <h2 id="examples">Examples</h2> <div class="num_example" id="Underlying"> <h6 id="example">Example</h6> <p>For any a monoidal category <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 functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>V</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V(I,-): V \to Set</annotation></semantics></math> is lax monoidal, hence induces a 2-functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V Cat</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>. This assigns 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>-enriched category, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, its <a class="existingWikiWord" href="/nlab/show/underlying+ordinary+category">underlying ordinary category</a>, usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_0</annotation></semantics></math>, defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>hom</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">\mathcal{C}_0(x,y) = V(I, hom(x,y))</annotation></semantics></math>.</p> </div> <div class="num_example" id="FreeCats"> <h6 id="example_2">Example</h6> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a> monoidal category, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(I,-)</annotation></semantics></math> above has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> that takes a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/copower">copower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. The resulting 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><mo>→</mo><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat \to V Cat</annotation></semantics></math> takes an ordinary category to the “free” <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 it generates. Such <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>-categories are used, for instance, in subsuming “conical” <a class="existingWikiWord" href="/nlab/show/limits">limits</a> under enriched <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a>.</p> <p>By functoriality, the adjunction between these two functors gives rise to a 2-adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><mo>⇄</mo><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat \rightleftarrows V Cat</annotation></semantics></math>.</p> </div> <div class="num_example" id="GenUnderlying"> <h6 id="example_3">Example</h6> <p>More generally, if an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⇄</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>:</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">F:V_1 \rightleftarrows V_2:U</annotation></semantics></math> can be regarded as a <a class="existingWikiWord" href="/nlab/show/free-forgetful+adjunction">free-forgetful adjunction</a>, then the corresponding adjunction between enriched categories can also be so regarded. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>=</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">V_1 = Top</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub><mo>=</mo><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">V_2 = G Top</annotation></semantics></math> for some <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, then the right adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub><mi>Cat</mi><mo>→</mo><msub><mi>V</mi> <mn>1</mn></msub><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V_2 Cat \to V_1 Cat</annotation></semantics></math> takes the “underlying topologically enriched category” of a category enriched in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-spaces (its morphisms are the fixed-point spaces of the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-space-enriched category; we think of these as the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant maps” while the original spaces consisted of not-necessarily-equivariant maps acted on by conjugation). Similarly, any category enriched in <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> has an underlying category enriched in spaces (whose hom-spaces are the 0-spaces of the original hom-spectra), any <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> has an underlying <a class="existingWikiWord" href="/nlab/show/Ab-category">Ab-category</a> (whose morphisms are the degree-0 ones in the original category), and so on.</p> </div> <div class="num_example" id="RealSing"> <h6 id="example_4">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> and <span class="newWikiWord">total singular complex<a href="/nlab/new/total+singular+complex">?</a></span> adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Real</mi><mo>:</mo><mi>SSet</mi><mo>⇄</mo><mi>Top</mi><mo>:</mo><mi>Sing</mi></mrow><annotation encoding="application/x-tex">Real:SSet \rightleftarrows Top : Sing</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> and <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> induces another adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SSet</mi><mi>Cat</mi><mo>⇄</mo><mi>Top</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">SSet Cat \rightleftarrows Top Cat</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> and <a class="existingWikiWord" href="/nlab/show/topologically+enriched+categories">topologically enriched categories</a>.</p> </div> <div class="num_example" id="HomotopyCat"> <h6 id="example_5">Example</h6> <p>If <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/SSet">SSet</a> or <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, then the set of connected components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>:</mo><mi>V</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi_0:V\to Set</annotation></semantics></math> is lax monoidal. The resulting change of enrichment functor takes 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to its “naive homotopy category” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</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">h C(x,y) = \pi_0(C(x,y))</annotation></semantics></math> obtained by “identifying homotopic morphisms”.</p> <p>Similarly, the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo>:</mo><mi>V</mi><mo>→</mo><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">\Pi_1:V\to Gpd</annotation></semantics></math> is lax monoidal, so 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>-category has an underlying <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a>.</p> </div> <div class="num_example" id="EnrichedHomotopyCat"> <h6 id="example_6">Example</h6> <p>An enhancement of the last example is that if <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> is any <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>, then its <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(V)</annotation></semantics></math> comes with a lax monoidal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma : V \to Ho(V)</annotation></semantics></math>. Thus 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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has an underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(V)</annotation></semantics></math>-enriched “homotopy category” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">h C</annotation></semantics></math>. Usually the underlying ordinary category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">h C</annotation></semantics></math> is the naive homotopy category from the previous example.</p> </div> <div class="num_example" id="Tau1"> <h6 id="example_7">Example</h6> <p>The nerve functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><mi>Cat</mi><mo>→</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex">N:Cat \to SSet</annotation></semantics></math> preserves products and has a left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tau_1</annotation></semantics></math> that also preserves products. Thus, we have a change of enrichment adjunction in which any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> can be regarded as a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a>, and similarly any simplicially enriched category has a “homotopy 2-category”. The latter plays an important role in the theory of <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>.</p> </div> <div class="num_example" id="PolyMorphisms"> <h6 id="example_8">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/poly-morphisms">poly-morphisms</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F = P \;\colon\; Set \to Set</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a>, the change of base functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Cat</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">P_\ast \;\colon\; Cat \to Cat</annotation></semantics></math> sends plain <a class="existingWikiWord" href="/nlab/show/categories">categories</a> to plain categories. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> any category, the morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>poly</mi></msup><mo>≔</mo><mi>P</mi><mo>*</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^{poly} \coloneqq P\ast(C)</annotation></semantics></math> are called the <em><a class="existingWikiWord" href="/nlab/show/poly-morphisms">poly-morphisms</a></em> 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> in <a href="poly-morphism#Mochizuki12">Mochizuki 12, section 0</a>.</p> </div> <h2 id="references">References</h2> <ul> <li id="EilenbergKelly65"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <em>Closed categories</em>, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer (1966) &lbrack;<a href="https://doi.org/10.1007/978-3-642-99902-4_22">doi:10.1007/978-3-642-99902-4_22</a>&rbrack;</p> </li> <li id="Crutwell14"> <p><a class="existingWikiWord" href="/nlab/show/Geoff+Cruttwell">Geoff Cruttwell</a>, chapter 4 of: <em>Normed spaces and the Change of Base for Enriched Categories</em>, 2014 (<a href="https://www.reluctantm.com/gcruttw/publications/thesis4.pdf">pdf</a>)</p> </li> <li id="Riehl14"> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, lemma 3.4.3 inL <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press (2014)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 14, 2024 at 16:31:14. See the <a href="/nlab/history/change+of+enriching+category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/change+of+enriching+category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9393/#Item_12">Discuss</a><span class="backintime"><a href="/nlab/revision/change+of+enriching+category/7" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/change+of+enriching+category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/change+of+enriching+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (7 revisions)</a> <a href="/nlab/show/change+of+enriching+category/cite" style="color: black">Cite</a> <a href="/nlab/print/change+of+enriching+category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/change+of+enriching+category" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>

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