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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#section_For_categories'>For categories</a></li> <li><a href='#section_For_enriched_categories'>For enriched categories</a></li> <li><a href='#section_In_a_2-category'>In a 2-category</a></li> <li><a href='#section_For_preorders_and_posets'>For preorders and posets</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#InHomotopyTypeTheory'>In homotopy type theory</a></li> </ul> <li><a href='#section_Examples'>Examples</a></li> <li><a href='#related_entries'>Related entries</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The left part of a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s is one of two best approximations to a <a class="existingWikiWord" href="/nlab/show/weak+inverse">weak inverse</a> of the other functor of the pair. (The other best approximation is the functor’s <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, if it exists.) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>.</p> <p>A left adjoint to a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> is called a <a class="existingWikiWord" href="/nlab/show/free+functor">free functor</a>. Many left adjoints can be constructed as quotients of free functors.</p> <p>The concept generalises immediately to <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched categories</a> and in <a class="existingWikiWord" href="/nlab/show/2-category">2-categories</a>.</p> <h2 id="definitions">Definitions</h2> <p><h3 id='section_For_categories'>For categories</h3></p> <p> <div class='num_defn' id='DefinitionLeftAdjointForCategories'> <h6>Definition</h6> <p>Given categories <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R: \mathcal{D} \to \mathcal{C}</annotation></semantics></math>, a <em>left adjoint</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">L: \mathcal{C} \to \mathcal{D}</annotation></semantics></math> together with <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>:</mo><msub><mi>id</mi> <mi>𝒞</mi></msub><mo>→</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\iota: id_\mathcal{C} \to R \circ L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo>→</mo><msub><mi>id</mi> <mi>𝒟</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon: L \circ R \to id_\mathcal{D} </annotation></semantics></math> such that the following diagrams (known as the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a>) commute, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> of a functor with a natural transformation.</p> <div 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rspace="0em">−</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Set</mi></mstyle><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_\mathcal{C}\left(L(-),-\right), Hom_\mathcal{D}\left(-,R(-)\right): D^{op} \times C \to \mathsf{Set}. </annotation></semantics></math></div> <p>Depending upon one’s interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Set</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Set}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a>, one may strictly speaking need to restrict to <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> categories for this equivalence to parse.</p> </div> </p> <p><h3 id='section_For_enriched_categories'>For enriched categories</h3></p> <p>The equivalent formulation of Definition <a class="maruku-ref" href="#DefinitionLeftAdjointForCategories"></a> given in Remark <a class="maruku-ref" href="#RemarkEquivalentDefinitionLeftAdjointForCategories"></a> generalises immediately to the setting of <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched categories</a>.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>Given <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">\mathbb{V}</annotation></semantics></math>-enriched categories <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> 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">\mathbb{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R: \mathcal{D} \to \mathcal{C}</annotation></semantics></math>, a <em>left adjoint</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is 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">\mathbb{V}</annotation></semantics></math>-enriched functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">L: \mathcal{C} \to \mathcal{D}</annotation></semantics></math> together with 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">\mathbb{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural isomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/hom-functor">Hom functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>)</mo></mrow><mo>,</mo><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>𝕍</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_\mathcal{C}\left((L(-),-\right), Hom_\mathcal{D}\left(-,R(-)\right): D^{op} \times C \to \mathbb{V}. </annotation></semantics></math></div> <p></p> </div> </p> <p><h3 id='section_In_a_2-category'>In a 2-category</h3></p> <p>Definition <a class="maruku-ref" href="#DefinitionLeftAdjointForCategories"></a> generalises immediately from <a class="existingWikiWord" href="/nlab/show/CAT">CAT</a>, the 2-category of (large) categories, to any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>.</p> <p> <div class='num_defn' id='DefinitionLeftAdjointInA2Category'> <h6>Definition</h6> <p>Let <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">\mathcal{A}</annotation></semantics></math> be a 2-category. Given objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and a 1-arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R: \mathcal{D} \to \mathcal{C}</annotation></semantics></math> 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">\mathcal{A}</annotation></semantics></math>, a <em>left adjoint</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a 1-arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">L: \mathcal{C} \to \mathcal{D}</annotation></semantics></math> together with 2-arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>:</mo><msub><mi>id</mi> <mi>𝒞</mi></msub><mo>→</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\iota: id_\mathcal{C} \to R \circ L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo>→</mo><msub><mi>id</mi> <mi>𝒟</mi></msub></mrow><annotation 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stroke-miterlimit="10" d="M -2.484965 2.868084 C -2.031796 1.149309 -1.020041 0.332906 -0.000482527 0.000888758 C -1.020027 -0.335076 -2.03175 -1.147612 -2.48485 -2.870312 " transform="matrix(-0.00002, 0.99998, 0.99998, 0.00002, 117.10458, 64.39892)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#bjE8U6L_4s9O6W56nVURPf5MLn4=-glyph-2-2" x="120.621" y="47.22"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#bjE8U6L_4s9O6W56nVURPf5MLn4=-glyph-3-1" x="128.65475" y="47.22"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#bjE8U6L_4s9O6W56nVURPf5MLn4=-glyph-2-5" x="131.59475" y="47.22"></use> </g> </svg> </div> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>If one assumes that one’s ambient 2-category has more structure, bringing it closer to being a <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a>, for example a <a class="existingWikiWord" href="/nlab/show/Yoneda+structure">Yoneda structure</a>, one should be able to give an equivalent formulation of Definition <a class="maruku-ref" href="#DefinitionLeftAdjointInA2Category"></a> akin to that of Remark <a class="maruku-ref" href="#RemarkEquivalentDefinitionLeftAdjointForCategories"></a>.</p> </div> </p> <p><h3 id='section_For_preorders_and_posets'>For preorders and posets</h3></p> <p>Restricted to <a class="existingWikiWord" href="/nlab/show/preorder">preorders</a> or <a class="existingWikiWord" href="/nlab/show/poset">posets</a>, Definition <a class="maruku-ref" href="#DefinitionLeftAdjointForCategories"></a> in its equivalent formulation of Remark <a class="maruku-ref" href="#RemarkEquivalentDefinitionLeftAdjointForCategories"></a> can be expressed in the following terminology.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>Given <a class="existingWikiWord" href="/nlab/show/partial+order">posets</a> or <a class="existingWikiWord" href="/nlab/show/preorder">preorders</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R: \mathcal{D} \to \mathcal{C}</annotation></semantics></math>, a <em>left adjoint</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a monotone function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">L: \mathcal{C} \to \mathcal{D}</annotation></semantics></math> such that, for all <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> in <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>, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">L(x) \leq y</annotation></semantics></math> holds if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≤</mo><mi>R</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \leq R(y) </annotation></semantics></math> holds.</p> </div> </p> <h2 id="properties">Properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">left adjoints preserve colimits</a></p> </li> <li> <p>left adjoints preserve <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a>.</p> </li> </ul> <h3 id="InHomotopyTypeTheory">In homotopy type theory</h3> <p>Note: the <a class="existingWikiWord" href="/nlab/show/HoTT+book">HoTT book</a> calls a <a class="existingWikiWord" href="/nlab/show/internal+category+in+HoTT">internal category in HoTT</a> a “precategory” and a <a class="existingWikiWord" href="/nlab/show/univalent+category">univalent category</a> a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.</p> <p> <div class='num_lemma' id='Lemma932'> <h6>Lemma</h6> <p><strong>(Lemma 9.3.2 in the <a class="existingWikiWord" href="/nlab/show/HoTT+book">HoTT book</a>)</strong> <br /> If <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 an <a class="existingWikiWord" href="/nlab/show/univalent+category">univalent category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> then the type “<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> is a left adjoint” is a <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>Suppose we are given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G, \eta, \epsilon)</annotation></semantics></math> with the triangle identities and also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mo>,</mo><mi>η</mi><mo>′</mo><mo>,</mo><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G', \eta', \epsilon')</annotation></semantics></math>. Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\gamma: G \to G'</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mi>G</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G' \epsilon )(\eta G')</annotation></semantics></math>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>δ</mi><mi>γ</mi></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mi>G</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mo>′</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mi>G</mi><mo>′</mo><msub><mi>ϵ</mi> <mo stretchy="false">)</mo></msub><mo stretchy="false">)</mo><mi>η</mi><mi>G</mi><mo>′</mo><mi>F</mi><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mo>′</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mi>ϵ</mi><mo>′</mo><mi>F</mi><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mi>η</mi><mo>′</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mn>1</mn> <mi>G</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \delta \gamma &= (G \epsilon')(\eta G')(G' \epsilon) (\eta' G)\\ &= (G \epsilon')(G F G' \epsilon_))\eta G' F G)(\eta' G)\\ &= (G \epsilon ')(G \epsilon' F G)(G F \eta' G)(\eta G)\\ &= (G \epsilon)(\eta G)\\ &= 1_G \end{aligned} </annotation></semantics></math></div> <p>using Lemma 9.2.8 (see <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) and the triangle identities. Similarly, we show <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mi>δ</mi><mo>=</mo><msub><mn>1</mn> <mrow><mi>G</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma \delta=1_{G'}</annotation></semantics></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>≅</mo><mi>G</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">G \cong G'</annotation></semantics></math>. By Theorem 9.2.5 (see <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>), we have an <a class="existingWikiWord" href="/nlab/show/identity">identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>G</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">G=G'</annotation></semantics></math>.</p> <p>Now we need to know that when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> are <span class="newWikiWord">transported<a href="/nlab/new/transported">?</a></span> along this <a class="existingWikiWord" href="/nlab/show/identity">identity</a>, they become equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\eta'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\epsilon '</annotation></semantics></math>. By Lemma 9.1.9,</p> <div class="query"> <p>Lemma 9.1.9 needs to be included. For now as transports are not yet written up I didn’t bother including a reference to the page <a class="existingWikiWord" href="/nlab/show/category">category</a>. -Ali</p> </div> <p>this <a class="existingWikiWord" href="/nlab/show/transport">transport</a> is given by composing with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> as appropriate. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>, this yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mi>ϵ</mi><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>η</mi><mo>′</mo><mi>G</mi><mi>F</mi><mo stretchy="false">)</mo><mi>η</mi><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mi>ϵ</mi><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mo>′</mo><mi>F</mi><mi>η</mi><mo stretchy="false">)</mo><mi>η</mi><mo>′</mo><mo>=</mo><mi>η</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">(G' \epsilon F)(\eta' G F)\eta = (G' \epsilon F)(G' F \eta)\eta'=\eta'</annotation></semantics></math></div> <p>using Lemma 9.2.8 (see <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) and the traingle identity. The case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is similar. FInally, the triangle identities transport correctly automatically, since hom-sets are sets.</p> </div> </p> <p><h2 id='section_Examples'>Examples</h2></p> <ul> <li>The left adjoint of the <a class="existingWikiWord" href="/nlab/show/nerve+functor">nerve functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><mstyle mathvariant="sans-serif"><mi>Grpd</mi></mstyle><mo>→</mo><msup><mstyle mathvariant="sans-serif"><mi>Set</mi></mstyle> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">N: \mathsf{Grpd} \to \mathsf{Set}^{\Delta^{op}}</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/category+of+groupoids">category of groupoids</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+sets">category of simplicial sets</a> is the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> functor.</li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> for more on left adjoints of monotone functions.</li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> for more on left adjoints of functors.</li> <li><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> for more on left adjoints in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories.</li> <li><a class="existingWikiWord" href="/nlab/show/examples+of+adjoint+functors">examples of adjoint functors</a> for examples.</li> <li><a class="existingWikiWord" href="/nlab/show/pro-left+adjoint">pro-left adjoint</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-adjunction">2-adjunction</a> for a categorified notion of adjunction.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 9, 2022 at 08:21:49. 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