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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #62 to #63: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-objects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences in</a>/<a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>of</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>reflective localization</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/join+of+quasi-categories'>join of quasi-categories</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flat+%28infinity%2C1%29-functor'>exact (∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/fibration+of+quasi-categories'>fibrations</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inner+fibration'>inner fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>left/right fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>essentially small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+small+%28infinity%2C1%29-category'>locally small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+complete+%28infinity%2C1%29-category'>idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor+theorem'>adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivator'>derivator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>model structure for quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation to simplicial categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure for Kan complexes</a></li> </ul> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#CharacterizationInTermsOfHomEquivalences'>In terms of hom-equivalences</a></li><li><a href='#InTermsOfCographsHeteromorphisms'>In terms of cographs/heteromorphisms</a></li><li><a href='#InTheHomotopy2Category'>In the homotopy 2-category</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#UniquenessOfAdjoints'>Uniqueness of adjoints</a></li><li><a href='#uniqueness_of_unit_and_counit'>Uniqueness of unit and counit</a></li><li><a href='#PresOfLims'>Preservation of limits and colimits</a></li><li><a href='#OnHomotopyCat'>Adjunctions on homotopy categories</a></li><li><a href='#FullAndFaithfulAdjoints'>Full and faithful adjoints</a></li><li><a href='#OnSlices'>Slicing of adjoint functors</a></li><li><a href='#UniversalArrows'>In terms of universal arrows</a></li><li><a href='#preservation_by_exponentiation'>Preservation by exponentiation</a></li></ul></li><li><a href='#category_of_adjunctions'>Category of adjunctions</a></li><li><a href='#examples'>Examples</a><ul><li><a href='#quillen_adjunctions'>Quillen adjunctions</a></li><li><a href='#SimplicialAndDerived'>Simplicial and derived adjunctions</a></li><li><a href='#localizations'>Localizations</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The notion of adjunction between two <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a> generalizes the notion of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> from <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> to <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category theory</a>.</p> <p>There are many equivalent definitions of the ordinary notion of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a>. Some of them have more evident generalizations to some parts of <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a> than others.</p> <ul> <li> <p>One definition of ordinary adjoint functors says that a pair of functors <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding='application/x-tex'>C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> is an adjunction if there is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><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 stretchy='false'>)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</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 stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Hom_C(L(-),-) \simeq Hom_D(-,R(-)) \,. </annotation></semantics></math></div> <p>The analog of this definition makes sense very generally in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, where <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>Hom_C(-,-) : C^{op} \times C \to \infty Grpd</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categorical hom-object.</p> </li> <li> <p>One other characterization of adjoint functors in terms of their <a class='existingWikiWord' href='/nlab/show/diff/cograph+of+a+functor'>cographs</a>/<a class='existingWikiWord' href='/nlab/show/diff/heteromorphism'>heteromorphisms</a>: the <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibrations</a> to which the <a href='http://ncatlab.org/nlab/show/(infinity%2C1)-Grothendieck+construction#FibsOverInterval'>functor is associated</a>. At <a class='existingWikiWord' href='/nlab/show/diff/cograph+of+a+functor'>cograph of a functor</a> it is discussed how two functors <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>L : C \to D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>R : D \to C</annotation></semantics></math> are adjoint precisely if the cograph of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> coincides with the cograph of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> up to the obvious reversal of arrows</p> </li> </ul> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>⇔</mo><mo stretchy='false'>(</mo><mi>cograph</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>cograph</mi><mo stretchy='false'>(</mo><msup><mi>R</mi> <mi>op</mi></msup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (L \dashv R) \Leftrightarrow (cograph(L) \simeq cograph(R^{op})^{op}) \,. </annotation></semantics></math></div> <p>Using the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a> the notion of cograph of a functor has an evident generalization to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories.</p> <h2 id='definition'>Definition</h2> <h3 id='CharacterizationInTermsOfHomEquivalences'>In terms of hom-equivalences</h3> <div class='num_defn' id='InTermsOfHomEquivalences'> <h6 id='definition_2'>Definition</h6> <p><strong>(in terms of hom equivalence induced by unit map)</strong></p> <p>A pair of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding='application/x-tex'> C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D </annotation></semantics></math></div> <p>is an adjunction, if there exists a <em>unit transformation</em> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</mi><mo>:</mo><msub><mi>Id</mi> <mi>D</mi></msub><mo>→</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>\epsilon : Id_D \to R \circ L</annotation></semantics></math> – a morphism in the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Func</mi><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Func(D,D)</annotation></semantics></math> – such that for all <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> the induced morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>R</mi> <mrow><mi>L</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi></mrow></msub></mrow></mover><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>ϵ</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></mover><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom_C(L(d),c) \stackrel{R_{L(d), c}}{\to} Hom_D(R(L(d)), R(c)) \stackrel{Hom_D(\epsilon, R(c))}{\to} Hom_D(d,R(c)) </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+infinity-groupoids'>equivalence of ∞-groupoids</a>.</p> </div> <p>In terms of the concrete incarnation of the notion of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category by the notion of <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a>, we have that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_C(L(d),c)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_D(d,R(c))</annotation></semantics></math> are incarnated as <a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-objects in quasi-categories</a>, which are <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complexes</a>, and the above equivalence is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a> of Kan complexes.</p> <p>In this form is due to <a href='#Lurie09'>Lurie 09, Def. 5.2.2.7</a>.</p> <p id='RiehlVerityOnAdjunctionsViaHomEquivalences'> Streamlined discussion is in <a href='#RiehlVerity15'>Riehl & Verity 15, 4.4.2-4.4.4</a> and <a href='#RiehlVerity20'>Riehl & Verity 20, 3.3.3-3.5.1</a> and <a href='#RVElements'>Riehl & Verity “Elements”, Prop. 4.1.1</a>.</p> <h3 id='InTermsOfCographsHeteromorphisms'>In terms of cographs/heteromorphisms</h3> <p>We discuss here the quasi-category theoretic analog of <em><a href='cograph+of+a+functor#AdjointFunctorsInTermsOfCographs'>Adjoint functors in terms of cographs</a></em> (<a class='existingWikiWord' href='/nlab/show/diff/heteromorphism'>heteromorphisms</a>).</p> <p>We make use here of the explicit realization of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a> in its incarnation for <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a>: here an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>L : D \to C</annotation></semantics></math> may be regarded as a map <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><msup><mo stretchy='false'>]</mo> <mi>op</mi></msup><mo>→</mo></mrow><annotation encoding='application/x-tex'>\Delta[1]^{op} \to </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Cat'>(∞,1)Cat</a>, which corresponds under the Grothendieck construction to a <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a> of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>coGraph</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>coGraph(L) \to \Delta[1]</annotation></semantics></math>.</p> <div class='num_defn' id='InTermsOfCoCartesianFibrations'> <h6 id='definition_3'>Definition</h6> <p><strong>(in terms of Cartesian/coCartesian fibrations)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a>. An <strong>adjunction</strong> between <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is</p> <ul> <li> <p>a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>K \to \Delta[1]</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a>, which is both a <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a> as well as a coCartesian fibration.</p> </li> <li> <p>together with <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of quasi-categories</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>K</mi> <mrow><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>C \stackrel{\simeq}{\to} K_{\{0\}}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>K</mi> <mrow><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>D \stackrel{\simeq}{\to} K_{\{1\}}</annotation></semantics></math>.</p> </li> </ul> <p>Two <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>L : C \to D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>R : D \to C</annotation></semantics></math> are called <strong>adjoint</strong> – with <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> <em>left adjoint</em> to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> <em>right adjoint</em> to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> if</p> <ul> <li> <p>there exists an adjunction <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>K \to I</annotation></semantics></math> in the above sense</p> </li> <li> <p>and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> are the <a href='http://ncatlab.org/nlab/show/(infinity%2C1)-Grothendieck+construction#FibsOverInterval'>associated functors to</a> the Cartesian fibation <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>p \colon K \to \Delta[1]</annotation></semantics></math> and the Cartesian fibration <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>p</mi> <mi>op</mi></msup><mo>:</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><msup><mo stretchy='false'>]</mo> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>p^{op} : K^{op} \to \Delta[1]^{op}</annotation></semantics></math>, respectively.</p> </li> </ul> </div> <h3 id='InTheHomotopy2Category'>In the homotopy 2-category</h3> <div class='num_defn' id='InTermsOfTheHomotopy2Category'> <h6 id='definition_4'>Definition</h6> <p><strong>(in terms of the homotopy 2-category)</strong></p> <p>Say that a 2-categorical pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint (∞,1)-functors</a> is an <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-category+of+%28%E2%88%9E%2C1%29-categories'>homotopy 2-category of (∞,1)-categories</a>.</p> </div> <p>This concept, in the spirit of <a class='existingWikiWord' href='/nlab/show/diff/formal+%28infinity%2C1%29-category+theory'>formal $\infty$-category theory</a>, was mentioned, briefly, in <a href='#Joyal08'>Joyal 2008, p. 159 (11 of 348)</a> and then expanded on in <a href='#RiehlVerity15'>Riehl-Verity 15, Def. 4.0.1</a>.</p> <p>Such a 2-categorical adjunctions (Def. <a class='maruku-ref' href='#InTermsOfTheHomotopy2Category'>3</a>) determines an adjoint pair of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-functors in the sense of <a href='#Lurie'>Lurie 2009</a> (<a href='#RiehlVerity15'>Riehl-Verity 15, Rem. 4.4.5</a>):</p> <p>\begin{prop}\label{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho} An anti-parallal pair of morphisms in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Cat'>$Cat_\infty$</a> is a pair of adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-functors in the sense of <a href='#Lurie'>Lurie 2009, Sec. 5.2</a> if and only its image in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-category'>homotopy 2-category</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-category+of+%28%E2%88%9E%2C1%29-categories'>$Ho_2\big(Cat_\infty\big)$</a> forms an <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> in the classical sense of <a class='existingWikiWord' href='/nlab/show/diff/2-category+theory'>2-category theory</a> (Def. <a class='maruku-ref' href='#InTermsOfTheHomotopy2Category'>3</a>). \end{prop} (<a href='#RVElements'>Riehl & Verity 2022, Sec. F.5, Prop. F.5.6</a>)</p> <p>The conceptual content of Prop. \ref{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho} may be made manifest as follows: \begin{proposition} Every 2-categorical pair of adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-functors in the sense of Def. <a class='maruku-ref' href='#InTermsOfTheHomotopy2Category'>3</a> extends to a “homotopy coherent adjunction” in an essentially unique way. \end{proposition}</p> <p>(<a href='#RiehlVerity16'>Riehl & Verity 2016, Thm. 4.3.11, 4.4.11</a>)</p> <h2 id='properties'>Properties</h2> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a>, the two definitions of adjunction,</p> <ol> <li> <p>in terms of Hom-equivalence induced by unit maps (Def. <a class='maruku-ref' href='#InTermsOfHomEquivalences'>1</a>)</p> </li> <li> <p>in terms of Cartesian/coCartesian fibrations (Def. <a class='maruku-ref' href='#InTermsOfCoCartesianFibrations'>2</a>)</p> </li> </ol> <p>are equivalent.</p> </div> <p>This is <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop 5.2.2.8</a>.</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>First we discuss how to produce the unit for an adjunction from the data of a correspondence <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>K \to \Delta[1]</annotation></semantics></math> that encodes an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-adjunction <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>⊣</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f \dashv g)</annotation></semantics></math>.</p> <p>For that, define a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>′</mo><mo>:</mo><mi>Λ</mi><mo stretchy='false'>[</mo><mn>2</mn><msub><mo stretchy='false'>]</mo> <mn>2</mn></msub><mo>×</mo><mi>C</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>F' : \Lambda[2]_2 \times C \to K</annotation></semantics></math> as follows:</p> <ul> <li> <p>on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{0,2\}</annotation></semantics></math> it is the morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>F : C \times \Delta[1] \to K</annotation></semantics></math> that exhibits <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> as associated to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, being <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>Id_C</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>C \times \{0\}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>C \times \{2\}</annotation></semantics></math>;</p> </li> <li> <p>on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{1,2\}</annotation></semantics></math> it is the morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mover><mo>→</mo><mrow><mi>f</mi><mo>×</mo><mi>Id</mi></mrow></mover><mi>D</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mover><mo>→</mo><mi>G</mi></mover><mi>K</mi></mrow><annotation encoding='application/x-tex'>C \times \Delta[1] \stackrel{f \times Id}{\to} D \times \Delta[1] \stackrel{G}{\to} K</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is the morphism that exhibits <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> as associated to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>;</p> </li> </ul> <p>Now observe that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>F'</annotation></semantics></math> in particular sends <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{1,2\}</annotation></semantics></math> to <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a>s in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> (by definition of functor associated to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>). By one of the equivalent characterizations of <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a>s, this means that the lift in the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>Λ</mi><mo stretchy='false'>[</mo><mn>2</mn><msub><mo stretchy='false'>]</mo> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>K</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><msup><mrow /> <mrow><mi>F</mi><mo>″</mo></mrow></msup><mo>↗</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>×</mo><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \Lambda[2]_2 &\stackrel{F'}{\to}& K \\ \downarrow &{}^{F''}\nearrow& \downarrow \\ \Delta[2] \times C &\to & \Delta[1] } </annotation></semantics></math></div> <p>exists. This defines a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo>→</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>C \times \{0,1\} \to K</annotation></semantics></math> whose components may be regarded as forming a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>:</mo><msub><mi>d</mi> <mi>C</mi></msub><mo>→</mo><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>u : d_C \to g \circ f</annotation></semantics></math>.</p> <p>To show that this is indeed a unit transformation, we need to show that the maps of <a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-object in a quasi-category</a> for all <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom_D(f(f), d) \to Hom_C(g(f(c)), g(d)) \to Hom_C(c, g(d)) </annotation></semantics></math></div> <p>is an equivalence, hence an isomorphism in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a>. Once checks that this fits into a commuting diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>K</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mo>=</mo></mtd> <mtd /> <mtd><msub><mi>Hom</mi> <mi>K</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ Hom_D(f(c), d) &\to& Hom_C(g(f(c)), g(d)) &\to& Hom_C(c, g(d)) \\ \downarrow &&&& \downarrow \\ Hom_K(C,D) &&=&& Hom_K(C,D) } \,. </annotation></semantics></math></div> <p>For illustration, chasing a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>f(c) \to d</annotation></semantics></math> through this diagram yields</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>c</mi><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mi>c</mi><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mo>=</mo></mtd> <mtd /> <mtd><mo stretchy='false'>(</mo><mi>c</mi><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \array{ (f(c) \to d) &\mapsto& (g(f(c)) \to g(d)) &\mapsto& (c \to g(f(c)) \to g(d)) \\ \downarrow && && \downarrow \\ (c \to g(f(c)) \to f(c) \to d) &&=&& (c \to g(f(c)) \to g(d) \to d) } \,, </annotation></semantics></math></div> <p>where on the left we precomposed with the Cartesian morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>g</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mo>≃</mo></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>c</mi></mtd> <mtd /> <mtd><mo>→</mo></mtd> <mtd /> <mtd><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && g(f(c)) \\ & \nearrow &\Downarrow^{\simeq}& \searrow \\ c &&\to&& f(c) } </annotation></semantics></math></div> <p>given by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>″</mo><msub><mo stretchy='false'>|</mo> <mi>c</mi></msub><mo>:</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>F''|_{c} : \Delta[2] \to K</annotation></semantics></math>, by …</p> </div> <h3 id='UniquenessOfAdjoints'>Uniqueness of adjoints</h3> <p>The adjoint of a functor is, if it exists, essentially unique:</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>If the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-functor between quasi-categories <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>L : D \to C</annotation></semantics></math> admits a right adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>R : C \to D</annotation></semantics></math>, then this is unique up to homotopy.</p> <p>Moreover, even the choice of homotopy is unique, up to ever higher homotopy, i.e. the collection of all right adjoints to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> forms a <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible</a> <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a>, in the following sense:</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>L</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>,</mo><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>Func</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Func^L(C,D), Func^R(C,D) \subset Func(C,D)</annotation></semantics></math> be the full sub-quasi-categories on the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a> between <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> on those functors that are left adjoint and those that are right adjoints, respectively. Then there is a canonical <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of quasi-categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>L</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>C</mi><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'> Func^L(C,D) \stackrel{\simeq}{\to} Func^R(D,C)^{op} </annotation></semantics></math></div> <p>(to the <a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite quasi-category</a>), which takes every left adjoint functor to a corresponding right adjoint.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>This is <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop 5.2.1.3</a> (also remark 5.2.2.2), and <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 5.2.6.2</a>.</p> <p>The idea is to construct the category of right adjoints as an intersection of full subcategories</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>C</mi> <mi>D</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msup><mi>D</mi> <mi>C</mi></msup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>∞</mn><msup><mi>Gpd</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>D</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ Func^R(C,D) &\to& C^D \\ \downarrow & & \downarrow \\ (D^C)^{op} &\to& \infty Gpd^{C^{op} \times D} } </annotation></semantics></math></div> <p>where the inclusions are given by the yoneda embedding. An element of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Func^R(C,D)</annotation></semantics></math> corresponds to a functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>D</mi><mo>→</mo><mn>∞</mn><mi>Gpd</mi></mrow><annotation encoding='application/x-tex'>p : C^{op} \times D \to \infty Gpd</annotation></semantics></math> for which there exists a pair of functors <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>g : D \to C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>f : C \to D</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>≃</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>≃</mo><mi>C</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>g</mi><mo>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p \simeq D(f-,-) \simeq C(-,g-)</annotation></semantics></math>.</p> </div> <h3 id='uniqueness_of_unit_and_counit'>Uniqueness of unit and counit</h3> <p>Given functors <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>f : C \to D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>g : D \to C</annotation></semantics></math>, we can use the <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-end'>(∞,1)-end</a> to determine compute a chain of equivalences</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msup><mi>C</mi> <mi>C</mi></msup><mo stretchy='false'>(</mo><mi>id</mi><mo>,</mo><mi>gf</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>gf</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mn>∞</mn><msup><mi>Gpd</mi> <mi>D</mi></msup><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>g</mi><mo>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mi>Gpd</mi> <mrow><msup><mi>C</mi> <mo lspace='0em' rspace='thinmathspace'>op</mo></msup><mo>×</mo><mi>D</mi></mrow></msup><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>g</mi><mo>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} C^C(id, gf) &\simeq \int_{c \in C} C(c, gf(c)) \\ &\simeq \int_{c \in C} \infty Gpd^D(D(f(c), -), C(c, g-)) \\ &\simeq Gpd^{C^{\op} \times D}(D(f-, -), C(-, g-)) \end{aligned} </annotation></semantics></math></div> <p>dually, we can identify the space of counits as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mi>D</mi></msup><mo stretchy='false'>(</mo><mi>fg</mi><mo>,</mo><mi>id</mi><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>Gpd</mi> <mrow><msup><mi>C</mi> <mo lspace='0em' rspace='thinmathspace'>op</mo></msup><mo>×</mo><mi>D</mi></mrow></msup><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>g</mi><mo>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> D^D(fg, id) \simeq Gpd^{C^{\op} \times D}(C(-, g-), D(f-, -)) </annotation></semantics></math></div> <p>So each half of the equivalence <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>≃</mo><mi>C</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>g</mi><mo>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(f-,-) \simeq C(-,g-)</annotation></semantics></math> corresponds essentially uniquely to a choice of unit and counit transformation.</p> <h3 id='PresOfLims'>Preservation of limits and colimits</h3> <p>Recall that for <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(L \dashv R)</annotation></semantics></math> an ordinary pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a>s, the fact that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a>s (and that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>s) is a formal consequence of</p> <ol> <li> <p>the hom-isomorphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><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 stretchy='false'>)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</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 stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_C(L(-),-) \simeq Hom_D(-,R(-))</annotation></semantics></math>;</p> </li> <li> <p>the fact that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Hom_C(-,-) : C^{op} \times C \to Set</annotation></semantics></math> preserves all limits in both arguments;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a>, which says that two objects are isomorphic if all homs out of (into them) are.</p> </li> </ol> <p>Using this one computes for all <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> and diagram <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d : I \to D</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><munder><mi>lim</mi> <mo>→</mo></munder><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><munder><mi>lim</mi> <mo>→</mo></munder><msub><mi>d</mi> <mi>i</mi></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><munder><mi>lim</mi> <mo>←</mo></munder><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy='false'>(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><munder><mi>lim</mi> <mo>←</mo></munder><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><munder><mi>lim</mi> <mo>→</mo></munder><mi>L</mi><mo stretchy='false'>(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} Hom_C(L(\lim_{\to} d_i), c) & \simeq Hom_D(\lim_\to d_i, R(c)) \\ & \simeq \lim_{\leftarrow} Hom_D(d_i, R(c)) \\ & \simeq \lim_{\leftarrow} Hom_C(L(d_i), c) \\ & \simeq Hom_C(\lim_{\to} L(d_i), c) \,, \end{aligned} </annotation></semantics></math></div> <p>which implies that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo stretchy='false'>(</mo><msub><mi>lim</mi> <mo>→</mo></msub><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>lim</mi> <mo>→</mo></msub><mi>L</mi><mo stretchy='false'>(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>L(\lim_\to d_i) \simeq \lim_\to L(d_i)</annotation></semantics></math>.</p> <p>Now to see this in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category theory (…) HTT Proposition 5.2.3.5</p> <h3 id='OnHomotopyCat'>Adjunctions on homotopy categories</h3> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mover><mo>→</mo><mo>←</mo></mover><mi>D</mi></mrow><annotation encoding='application/x-tex'>(L \dashv R) : C \stackrel{\leftarrow}{\to} D</annotation></semantics></math> an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-adjunction, its image under decategorifying to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy categories</a> is a pair of ordinary <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a>s</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Ho</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>⊣</mo><mi>Ho</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>Ho</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>←</mo></mover><mi>Ho</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (Ho(L) \dashv Ho(R)) : Ho(C) \stackrel{\leftarrow}{\to} Ho(D) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>This is <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop 5.2.2.9</a>.</p> <p>This follows from that fact that for <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</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'>\epsilon : Id_C \to R \circ L</annotation></semantics></math> a unit of the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-adjunction, its image <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>ϵ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(\epsilon)</annotation></semantics></math> is a unit for an ordinary adjunction.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>The converse statement is in general false. A near converse is given by <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop 5.2.2.12</a> if one instead considers <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi></mrow><annotation encoding='application/x-tex'>Ho</annotation></semantics></math>-enriched homotopy categories: if <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(L)</annotation></semantics></math> has a right adjoint, then so does <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>.</p> <p>It is important to consider the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi></mrow><annotation encoding='application/x-tex'>Ho</annotation></semantics></math>-enriched homotopy category rather than the ordinary one. For a counterexample, when <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi></mrow><annotation encoding='application/x-tex'>Ho</annotation></semantics></math> is considered as an ordinary category, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>:</mo><mi>Ho</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\pi_0 : Ho \to Set</annotation></semantics></math> is both left and right adjoint to the inclusion <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi><mo>⊆</mo><mi>Ho</mi></mrow><annotation encoding='application/x-tex'>Set \subseteq Ho</annotation></semantics></math>. However, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>:</mo><mn>∞</mn><mi>Gpd</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\pi_0 : \infty Gpd \to Set</annotation></semantics></math> does not have a left adjoint.</p> <p>One way to find that an ordinary adjunction of homotopy categories lifts to an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-adjunction is to exhibit it as a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> between <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a>-structures. This is discussed in the Examples-section <a href='#SimplicialAndDerived'>Simplicial and derived adjunction</a> below.</p> </div> <h3 id='FullAndFaithfulAdjoints'>Full and faithful adjoints</h3> <p>As for ordinary <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> we have the following relations between full and faithful adjoints and idempotent monads.</p> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>Given an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-adjunction <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>(L \dashv R) : C \to D</annotation></semantics></math></p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/fully+faithful+%28infinity%2C1%29-functor'>full and faithful (∞,1)-functor</a> precisely is the counit <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi><mover><mo>→</mo><mrow /></mover><mi>Id</mi></mrow><annotation encoding='application/x-tex'>L R \stackrel{}{\to} Id</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a>.</p> <p>In this case <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective (∞,1)-subcategory</a> of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/fully+faithful+%28infinity%2C1%29-functor'>full and faithful (∞,1)-functor</a> precisely is the unit <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Id</mi><mo>→</mo><mi>R</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>Id \to R L</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a>.</p> </li> </ul> </div> <p><a href='#Lurie'>Lurie, prop. 5.2.7.4</a>, See also top of p. 308.</p> <h3 id='OnSlices'>Slicing of adjoint functors</h3> <p>\begin{proposition}\label{SliceAdjoints} <strong>(sliced adjoints)</strong> \linebreak Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>R</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>L</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'> \mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div> <p>be a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> (<a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint ∞-functors</a>), where the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> (<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>∞-category</a>) <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> has all <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullbacks</a> (<a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullbacks</a>).</p> <p>Then:</p> <ol> <li> <p>For every <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>b \in \mathcal{C}</annotation></semantics></math> there is induced a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> between the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice categories</a> (<a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>slice ∞-categories</a>) of the form</p> <div class='maruku-equation' id='eq:SlicedAdjointFunctorsOverLb'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mpadded width='0'><mrow><mspace width='thinmathspace' /><mo>,</mo></mrow></mpadded></mrow><annotation encoding='application/x-tex'> \mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{/b}</annotation></semantics></math> is the evident induced functor (applying <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> to the entire triangle <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagrams</a> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> which represent the morphisms in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_{/b}</annotation></semantics></math>);</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>R_{/b}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composite</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mrow><mi>L</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>R</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mrow><mo stretchy='false'>(</mo><mi>R</mi><mo>∘</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>η</mi> <mi>b</mi></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>;</p> </li> <li> <p>the (<a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy</a>) <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> along the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(L \dashv R)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>unit</a> at <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> (i.e. the <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> along <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>η</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>\eta_b</annotation></semantics></math>).</p> </li> </ol> </li> </ul> </li> <li> <p>For every <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>b \in \mathcal{D}</annotation></semantics></math> there is induced a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> between the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice categories</a> of the form</p> <div class='maruku-equation' id='eq:SlicedAdjointFunctorsOverRb'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></msub><mpadded width='0'><mrow><mspace width='thinmathspace' /><mo>,</mo></mrow></mpadded></mrow><annotation encoding='application/x-tex'> \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>R_{/b}</annotation></semantics></math> is the evident induced functor (applying <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> to the entire triangle <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagrams</a> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{D}</annotation></semantics></math> which represent the morphisms in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{D}_{/b}</annotation></semantics></math>);</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{/b}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composite</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>L</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>ϵ</mi> <mi>b</mi></msub><msub><mo stretchy='false'>)</mo> <mo>!</mo></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> with the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(L \dashv R)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>counit</a> at <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> (i.e. the left <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> along <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϵ</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>\epsilon_b</annotation></semantics></math>).</p> </li> </ol> </li> </ul> </li> </ol> <p>\end{proposition} The first statement appears, in the generality of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, as <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 5.2.5.1</a>. For discussion in <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a> see at <em><a href='slice+model+structure#SlicedQuillenAdjunction'>sliced Quillen adjunctions</a></em>. \begin{proof} (in <a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-</a><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>)</p> <p>Recall that (<a href='adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit'>this Prop.</a>) the hom-isomorphism that defines an adjunction of functors (<a href='adjoint+functor#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets'>this Def.</a>) is equivalently given in terms of <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> with</p> <ul> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>adjunction unit</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>η</mi> <mi>c</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>c</mi><mover><mo>→</mo><mspace width='thickmathspace' /></mover><mi>R</mi><mo>∘</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)</annotation></semantics></math></p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>adjunction counit</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>ϵ</mi> <mi>d</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mspace width='thickmathspace' /></mover><mi>d</mi></mrow><annotation encoding='application/x-tex'>\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d</annotation></semantics></math></p> </li> </ul> <p>as follows:</p> <p>\begin{tikzcd} L(c) \ar[rr, f] && d &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}& c \ar[rr, \eta_c] \ar[rrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8.5pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rr, R(f)] && R(d) \end{tikzcd}</p> <p>\begin{tikzcd} c \ar[rr, \widetilde f] && R(d) &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}& L(c) \ar[rr, L(\widetilde{f})] \ar[rrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8.5pt\tikztotarget.north) (\tikztotarget.north)}] && L \circ R(d) \ar[rr, \epsilon_d] && d \end{tikzcd}</p> <p>Using this, consider the following transformations of morphisms in slice categories, for the <strong>first case</strong>:</p> <p><strong>(1a)</strong></p> <p>\begin{tikzcd} L(c) \ar[rr, f, dashed] \ar[dr] && d \ar[dl, p] \ & L(b) \end{tikzcd}</p> <p><strong>(2a)</strong></p> <p>\begin{tikzcd} c \ar[rr, {\eta_c}] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+30pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+27pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rr, {R(f)}, dashed] \ar[dr] && R(d) \ar[dl, R(p)] \ & b \ar[rr, \eta_b{below}] & & R \circ L(b) \end{tikzcd}</p> <p><strong>(2b)</strong></p> <p>\begin{tikzcd} c \ar[rr, dashed] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && \eta_b^\ast\big(R(d)\big) \ar[rr] \ar[dl] \ar[dr, phantom, \mbox{\tiny\rmfamily(pb)}] && R(d) \ar[dl, R(p)] \ & b \ar[rr, \eta_b{below}] & & R \circ L(b) \end{tikzcd}</p> <p><strong>(1b)</strong></p> <p>\begin{tikzcd} L(c) \ar[dr] \ar[rrrr, L(\widetilde{f})] \ar[rrrrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && && L \circ R(d) \ar[rr, {\epsilon_d}] \ar[dl] && d \ar[dl, p] \ & L(b) \ar[rr, L(\eta_b){below}] \ar[rrrr, rounded corners, to path={ (yshift=-8pt\tikztostart.south) nodebelow{ \scalebox{} } (yshift=-8pt\tikztotarget.south) (\tikztotarget.south)}] & & L \circ R \circ L(b) \ar[rr, \epsilon_{L(b)}{below}] && L(b) \end{tikzcd}</p> <p>Here:</p> <ul> <li> <p>(1a) and (1b) are equivalent expressions of the same morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{D}_{/L(b)}</annotation></semantics></math>, by (at the top of the diagrams) the above expression of <a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjuncts</a> between <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{D}</annotation></semantics></math> and (at the bottom) by the <a class='existingWikiWord' href='/nlab/show/diff/triangle+identities'>triangle identity</a>.</p> </li> <li> <p>(2a) and (2b) are equivalent expression of the same morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_{/b}</annotation></semantics></math>, by the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>.</p> </li> </ul> <p>Hence:</p> <ul> <li> <p>starting with a morphism as in (1a) and transforming it to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2)</annotation></semantics></math> and then to (1b) is the identity operation;</p> </li> <li> <p>starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.</p> </li> </ul> <p>In conclusion, the transformations (1) <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↔</mo></mrow><annotation encoding='application/x-tex'>\leftrightarrow</annotation></semantics></math> (2) consitute a <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> that witnesses an adjunction of the first claimed form <a class='maruku-eqref' href='#eq:SlicedAdjointFunctorsOverLb'>(1)</a>.</p> <p>\linebreak</p> <p>The <strong>second case</strong> follows analogously, but a little more directly since no pullback is involved:</p> <p><strong>(1a)</strong></p> <p>\begin{tikzcd} c \ar[rr, dashed, f] \ar[dr] && R(d) \ar[dl] \ & R(b) \end{tikzcd}</p> <p><strong>(2)</strong></p> <p>\begin{tikzcd} L(c) \ar[rr, dashed, L(f)] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && L \circ R(d) \ar[rr, \epsilon_d{above}] \ar[dl] && d \ar[dl] \ & L \circ R(b) \ar[rr, \epsilon_b{below}] && b \end{tikzcd}</p> <p><strong>(1b)</strong></p> <p>\begin{tikzcd} c \ar[rr, \eta_c{above}] \ar[dr] \ar[rrrrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rrrr,R(\widetilde{f})] \ar[dr] && %R \circ L \circ R(d) %\ar[rr, R(\epsilon_d){above}] %\ar[dl] && R(d) \ar[dl] \ & R(b) \ar[rr, \eta_{R(b)}{below}] \ar[rrrr, rounded corners, to path={ (yshift=-8pt\tikztostart.south) nodebelow{ \scalebox{} } (yshift=-8pt\tikztotarget.south) (\tikztotarget.south)}] && R \circ L \circ R(b) \ar[rr, R(\epsilon_b){below}] && R(b) \end{tikzcd}</p> <p>In conclusion, the transformations (1) <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↔</mo></mrow><annotation encoding='application/x-tex'>\leftrightarrow</annotation></semantics></math> (2) consitute a <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-isomorphism</a> that witnesses an adjunction of the second claimed form <a class='maruku-eqref' href='#eq:SlicedAdjointFunctorsOverRb'>(2)</a>. \end{proof}</p> <p>\begin{remark} \label{LeftAdjointOfSlicedAdjunctionFormsAdjuncts} <strong>(left adjoint of sliced adjunction forms adjuncts)</strong> \linebreak The sliced adjunction (Prop. \ref{SliceAdjoints}) in the second form <a class='maruku-eqref' href='#eq:SlicedAdjointFunctorsOverRb'>(2)</a> is such that the sliced <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> sends slicing morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi></mrow><annotation encoding='application/x-tex'>\tau</annotation></semantics></math> to their <a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjuncts</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>τ</mi><mo>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\widetilde{\tau}</annotation></semantics></math>, in that (again by <a href='adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit'>this Prop.</a>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub><mspace width='thinmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mi>τ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>R</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mover><mi>τ</mi><mo>˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∈</mo><mspace width='thickmathspace' /><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b} </annotation></semantics></math></div> <p>\end{remark}</p> <p>The two adjunctions in \ref{SliceAdjoints} admit the following joint generalisation, which is proven <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, lem. 5.2.5.2</a>. (Note that the statement there is even more general and here we only use the case where <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>=</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow><annotation encoding='application/x-tex'>K = \Delta^0</annotation></semantics></math>.)</p> <p>\begin{proposition}\label{SliceAdjointsGeneralized} <strong>(sliced adjoints)</strong> \linebreak Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>R</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>L</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'> \mathcal{C} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D} </annotation></semantics></math></div> <p>be a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint ∞-functors</a>, where the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>∞-category</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> has all <a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullbacks</a>. Suppose further we are given objects <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>c \in \mathcal{C}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>d \in \mathcal{D}</annotation></semantics></math> together with a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha: c \to R(d)</annotation></semantics></math> and its adjunct <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mo>:</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>\beta:L(c) \to d</annotation></semantics></math>.</p> <p>Then there is an induced a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint ∞-functors</a> between the <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>slice ∞-categories</a> of the form</p> <div class='maruku-equation' id='eq:SlicedAdjointFunctorsGeneral'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>c</mi></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>b</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub><mpadded width='0'><mrow><mspace width='thinmathspace' /><mo>,</mo></mrow></mpadded></mrow><annotation encoding='application/x-tex'> \mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>c</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{/c}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composite</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy='false'>/</mo><mi>c</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>c</mi></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>L</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mrow><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>β</mi> <mo>!</mo></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> with <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mo>:</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>\beta:L(c) \to d</annotation></semantics></math> (i.e. the left <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> along <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math>).</p> </li> </ol> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>R_{/d}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composite</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>𝒟</mi> <mrow><mo stretchy='false'>/</mo><mi>d</mi></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>R</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msup><mi>α</mi> <mo>*</mo></msup><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>c</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy</a> along <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha:c \to R(d)</annotation></semantics></math> (i.e. the <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> along <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>).</p> </li> </ol> </li> </ul> <p>\end{proposition}</p> <h3 id='UniversalArrows'>In terms of universal arrows</h3> <div class='num_prop' id='UnivArr'> <h6 id='proposition_5'>Proposition</h6> <p>An <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>G:D\to C</annotation></semantics></math> admits a left adjoint if and only if for each <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>X\in C</annotation></semantics></math>, the <span class='newWikiWord'>comma (infinity,1)-category<a href='/nlab/new/comma+%28infinity%2C1%29-category'>?</a></span> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>↓</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X \downarrow G)</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>initial object</a>, i.e. every object <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>X\in C</annotation></semantics></math> admits a <a class='existingWikiWord' href='/nlab/show/diff/reflection+along+a+functor'>universal arrow</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mi>F</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\to G F X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> </div> <p>This is stated explicitly as <a href='#RVElements'>Riehl-Verity, Corollary 16.2.7</a>, and can be extracted with some work from <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, Proposition 5.2.4.2</a>.</p> <h3 id='preservation_by_exponentiation'>Preservation by exponentiation</h3> <div class='num_prop'> <h6 id='proposition_6'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>f : C \to D</annotation></semantics></math> be left adjoint to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>g : D \to C</annotation></semantics></math>. Then for any <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mi>A</mi></msup></mrow><annotation encoding='application/x-tex'>f^A</annotation></semantics></math> is left adjoint to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>g</mi> <mi>A</mi></msup></mrow><annotation encoding='application/x-tex'>g^A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>g</mi></msup></mrow><annotation encoding='application/x-tex'>A^g</annotation></semantics></math> is left adjoint to <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>f</mi></msup></mrow><annotation encoding='application/x-tex'>A^f</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>:</mo><msub><mi>id</mi> <mi>C</mi></msub><mo>⇒</mo><mi>gf</mi></mrow><annotation encoding='application/x-tex'>\eta : id_C \Rightarrow gf</annotation></semantics></math> be a unit transformation. The property of being a unit transformation can be detected at the level of enriched homotopy categories, so <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>η</mi></msup><mo>:</mo><msub><mi>id</mi> <mrow><msup><mi>A</mi> <mi>C</mi></msup></mrow></msub><mo>⇒</mo><msup><mi>A</mi> <mi>f</mi></msup><msup><mi>A</mi> <mi>g</mi></msup></mrow><annotation encoding='application/x-tex'>A^\eta: id_{A^C} \Rightarrow A^f A^g</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>η</mi> <mi>A</mi></msup><mo>:</mo><msub><mi>id</mi> <mrow><msup><mi>C</mi> <mi>A</mi></msup></mrow></msub><mo>⇒</mo><msup><mi>g</mi> <mi>A</mi></msup><msup><mi>f</mi> <mi>A</mi></msup></mrow><annotation encoding='application/x-tex'>\eta^A : id_{C^A} \Rightarrow g^A f^A</annotation></semantics></math> are also unit transformations.</p> </div> <h2 id='category_of_adjunctions'>Category of adjunctions</h2> <p>The functorality of adjunctions can be organized into the existence of two wide subcategories <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LAdj</mi><mo>⊆</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>LAdj \subseteq (\infty,1)Cat</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>RAdj</mi><mo>⊆</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>RAdj \subseteq (\infty,1)Cat</annotation></semantics></math> whose functors are the left adjoints and the right adjoints respectively.</p> <p>We can then define the functor categories</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>L</mi></msup><mo>:</mo><msup><mi>LAdj</mi> <mi>op</mi></msup><mo>×</mo><mi>LAdj</mi><mo>→</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Func^L : LAdj^{op} \times LAdj \to (\infty,1)Cat</annotation></semantics></math> is defined by taking <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>L</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>⊆</mo><mi>Func</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Func^L(C,D) \subseteq Func(C, D)</annotation></semantics></math> to be the full subcategory spanned by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LAdj</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>LAdj(C, D)</annotation></semantics></math>.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>R</mi></msup><mo>:</mo><msup><mi>RAdj</mi> <mi>op</mi></msup><mo>×</mo><mi>RAdj</mi><mo>→</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Func^R : RAdj^{op} \times RAdj \to (\infty,1)Cat</annotation></semantics></math> is defined by taking <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>⊆</mo><mi>Func</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Func^R(C,D) \subseteq Func(C, D)</annotation></semantics></math> to be the full subcategory spanned by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>RAdj</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>RAdj(C, D)</annotation></semantics></math>.</p> </li> </ul> <p>Lurie defines an adjunction to be a functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>X \to [1]</annotation></semantics></math> that is both a cartesian and a cocartesian fibration. We can generalize this to</p> <div class='num_defn' id='AdjunctFibration'> <h6 id='definition_5'>Definition</h6> <p>A functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>p : X \to S</annotation></semantics></math> is an <em>adjunct fibration</em> iff it is both a cartesian fibration and a cocartesian fibration</p> </div> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a> construction, adjunct fibrations over <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> correspond to category-valued functors on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> that send arrows of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> to adjoint pairs of categories.</p> <div class='num_lemma'> <h6 id='lemma'>Lemma</h6> <p>For a functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>p : X \to S</annotation></semantics></math> of (∞,1)-categories with small fibers.</p> <ul> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a cartesian fibration classified by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo>:</mo><msup><mi>S</mi> <mo lspace='0em' rspace='thinmathspace'>op</mo></msup><mo>→</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>\chi : S^\op \to (\infty,1)Cat</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi></mrow><annotation encoding='application/x-tex'>\chi</annotation></semantics></math> factors through <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>RAdj</mi></mrow><annotation encoding='application/x-tex'>RAdj</annotation></semantics></math> iff <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is an adjunct fibration</p> </li> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a cocartesian fibration classified by <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo>:</mo><mi>S</mi><mo>→</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>\chi : S \to (\infty,1)Cat</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi></mrow><annotation encoding='application/x-tex'>\chi</annotation></semantics></math> factors through <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LAdj</mi></mrow><annotation encoding='application/x-tex'>LAdj</annotation></semantics></math> iff <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is an adjunct fibration</p> </li> </ul> </div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>This is a restatement of <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, corr. 5.2.2.5</a>.</p> </div> <p>\begin{lemma}\label{LadjAndRadj} There are anti-equivalences <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ladj</mi><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><msup><mi>RAdj</mi> <mi>op</mi></msup><mo>→</mo><mi>LAdj</mi></mrow><annotation encoding='application/x-tex'>ladj \,\colon\, RAdj^{op} \to LAdj</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>radj</mi><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><msup><mi>LAdj</mi> <mi>op</mi></msup><mo>→</mo><mi>RAdj</mi></mrow><annotation encoding='application/x-tex'>radj \,\colon\, LAdj^{op} \to RAdj</annotation></semantics></math> that are the identity on objects and the action on homspaces <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LAdj</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>RAdj</mi><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>LAdj(C, D) \simeq RAdj(D,C)</annotation></semantics></math> is the equivalence sending a functor to its adjoint. \end{lemma}</p> <div class='proof'> <h6 id='proof_6'>Proof</h6> <p>By the covariant Grothendieck construction, for any (∞,1)-category C, <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Map</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>LAdj</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Map(C, LAdj)</annotation></semantics></math> can be identified with the ∞-groupoid of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><msub><mover><mi>Cat</mi><mo>^</mo></mover> <mrow><mo stretchy='false'>/</mo><mi>C</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>(\infty,1)\widehat{Cat}_{/C}</annotation></semantics></math> spanned by adjunct fibrations over <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with small fibers and all equivalences between them. The same is true of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Map</mi><mo stretchy='false'>(</mo><msup><mi>C</mi> <mo lspace='0em' rspace='thinmathspace'>op</mo></msup><mo>,</mo><mi>RAdj</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Map(C^{\op}, RAdj)</annotation></semantics></math>.</p> <p>Since the Grothendieck construction is natural in the base category, we obtain the asserted equivalence between <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LAdj</mi></mrow><annotation encoding='application/x-tex'>LAdj</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>RAdj</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>RAdj^{op}</annotation></semantics></math>. Taking <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>C = [1]</annotation></semantics></math>, this establishes the correspondence between an adjunction and its associated adjoint pair of functors.</p> </div> <p>As discussed at <a href='#UniquenessOfAdjoints'>Uniqueness of Adjoints</a>, this anti-equivalence extends to the (∞,2)-enrichment, in the sense they induce anti-equivalences <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>radj</mi><mo>:</mo><msup><mi>Func</mi> <mi>L</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>radj : Func^L(C, D)^{op} \to Func^R(D, C)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ladj</mi><mo>:</mo><msup><mi>Func</mi> <mi>R</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>Func</mi> <mi>L</mi></msup><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>ladj : Func^R(C, D)^{op} \to Func^L(D, C)</annotation></semantics></math>.</p> <p>The preservation of adjunctions by products and exponentials implies</p> <div class='num_lemma'> <h6 id='lemma_2'>Lemma</h6> <p>The product and exponential on <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>(\infty,1)Cat</annotation></semantics></math> restrict to functors</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>×</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>:</mo><mi>LAdj</mi><mo>×</mo><mi>LAdj</mi><mo>→</mo><mi>LAdj</mi></mrow><annotation encoding='application/x-tex'>- \times - : LAdj \times LAdj \to LAdj</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>×</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>:</mo><mi>RAdj</mi><mo>×</mo><mi>RAdj</mi><mo>→</mo><mi>RAdj</mi></mrow><annotation encoding='application/x-tex'>- \times - : RAdj \times RAdj \to RAdj</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Func</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>RAdj</mi> <mi>op</mi></msup><mo>×</mo><mi>LAdj</mi><mo>→</mo><mi>LAdj</mi></mrow><annotation encoding='application/x-tex'>Func(-,-) : RAdj^{op} \times LAdj \to LAdj</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Func</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>LAdj</mi> <mi>op</mi></msup><mo>×</mo><mi>RAdj</mi><mo>→</mo><mi>RAdj</mi></mrow><annotation encoding='application/x-tex'>Func(-,-) : LAdj^{op} \times RAdj \to RAdj</annotation></semantics></math></li> </ul> </div> <h2 id='examples'>Examples</h2> <p>A large class of examples of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-adjunctions arises from <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunctions</a> of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a>, or adjunctions in <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a>.</p> <h3 id='quillen_adjunctions'>Quillen adjunctions</h3> <p>Any <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> induces an adjunction of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(infinity,1)-categories</a> on the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>simplicial localizations</a>. See <a href='#Hinich14'>Hinich 14</a> or <a href='#MazelGee15'>Mazel-Gee 15</a>.</p> <h3 id='SimplicialAndDerived'>Simplicial and derived adjunctions</h3> <p>We want to produce Cartesian/coCartesian fibration <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>K \to \Delta[1]</annotation></semantics></math> from a given <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched</a> adjunction. For that first consider the following characterization</p> <div class='num_lemma'> <h6 id='lemma_3'>Lemma</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a> whose <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-objects</a> are all <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complexes</a>, regard the <a class='existingWikiWord' href='/nlab/show/diff/interval+category'>interval category</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>:</mo><mo>=</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\Delta[1] := \{0 \to 1\}</annotation></semantics></math> as an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-category in the obvious way using the embedding <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>const</mi><mo>:</mo><mi>Set</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>const : Set \hookrightarrow sSet</annotation></semantics></math> and consider an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-enriched functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>K \to \Delta[1]</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><msub><mi>K</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>C := K_0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>:</mo><mo>=</mo><msub><mi>K</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>D := K_1</annotation></semantics></math> be the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-enriched categories that are the fibers of this. Then under the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a> <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo>:</mo><mi>sSet</mi><mi>Cat</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>N : sSet Cat \to sSet</annotation></semantics></math> the morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>N</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> N(p) : N(K) \to \Delta[1] </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a> precisely if for all objects <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math> there exists a morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>f : c \to d</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> such that postcomposition with this morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C(c',f ) : C(c',c) = K(c',c) \to K(c',d) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a> of <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es for all objects <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>c' \in C'</annotation></semantics></math>.</p> </div> <p>This appears as <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 5.2.2.4</a>.</p> <div class='proof'> <h6 id='proof_7'>Proof</h6> <p>The statement follows from the characterization of <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a>s under homotopy coherent nerves (<a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 2.4.1.10</a>), which says that for an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-enriched functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>p : C \to D</annotation></semantics></math> between Kan-complex enriched categories that is <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a>-wise a <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a>, a morphim <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>c</mi><mo>′</mo><mo>→</mo><mi>c</mi><mo>″</mo></mrow><annotation encoding='application/x-tex'>f : c' \to c''</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(p)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a> if for all objects <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>″</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>p</mi> <mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo></mrow></msub></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>p</mi> <mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>″</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>D</mi><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>D</mi><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>″</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ C(c,c') &\stackrel{C(c,f)}{\to}& C(c,c'') \\ \downarrow^{\mathrlap{p_{c,c'}}} && \downarrow^{\mathrlap{p_{c,c''}}} \\ D(p(c),p(c')) &\stackrel{D(p(c),p(f))}{\to}& D(p(c), p(c'')) } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a> in the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-categories'>model structure on sSet-categories</a>.</p> <p>For the case under consideration the functor in question is <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>p : K \to \Delta[1]</annotation></semantics></math> and the above diagram becomes</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>″</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ K(c,c') &\stackrel{K(c,f)}{\to}& K(c,c'') \\ \downarrow && \downarrow \\ * &\to& * } \,. </annotation></semantics></math></div> <p>This is clearly a homotopy pullback precisely if the top morphism is an equivalence.</p> </div> <p>Using this, we get the following.</p> <div class='num_prop'> <h6 id='proposition_7'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a> whose hom-objects are all <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complexes</a>, the image</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow></munder><mover><mo>⟶</mo><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow></mover></munderover><mi>N</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> N(C) \underoverset {\underset{N(R)}{\longleftarrow}} {\overset{N(L)}{\longrightarrow}} {\bot} N(D) </annotation></semantics></math></div> <p>under the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a> of an <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-enriched adjunction between <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><munder><mo>←</mo><mi>R</mi></munder><mover><mo>→</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding='application/x-tex'> C \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} D </annotation></semantics></math></div> <p>is an adjunction of <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a>.</p> <p>Moreover, if <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> are equipped with the structure of a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a> then the quasi-categorically <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><msup><mi>C</mi> <mo>∘</mo></msup><mo stretchy='false'>)</mo><mover><munder><mo>←</mo><mi>R</mi></munder><mover><mo>→</mo><mi>L</mi></mover></mover><mi>N</mi><mo stretchy='false'>(</mo><msup><mi>D</mi> <mo>∘</mo></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> N(C^\circ) \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} N(D^\circ) </annotation></semantics></math></div> <p>form an adjunction of quasi-categories.</p> </div> <div class='proof'> <h6 id='proof_8'>Proof</h6> <p>The first part is <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, cor. 5.2.4.5</a>, the second <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 5.2.4.6</a>.</p> <p>To get the first part, let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> be the <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-category which is the join of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>: its set of objects is the disjoint union of the sets of objects of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, and the <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a>s are</p> <ul> <li> <p>for <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c,c' \in C</annotation></semantics></math>: <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(c,c') := C(c,c')</annotation></semantics></math>;</p> </li> <li> <p>for <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d,d' \in D</annotation></semantics></math>: <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(d,d') := D(d,d')</annotation></semantics></math>;</p> </li> <li> <p>for <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math>: <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(c,d) := C(L(c),d) = D(c,R(d))</annotation></semantics></math>;</p> <p>and</p> <p><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>K(d,c) = \emptyset</annotation></semantics></math></p> </li> </ul> <p>and equipped with the evident composition operation.</p> <p>Then for every <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math> there is the morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow></msub><mo>∈</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Id_{R(d)} \in K(R(d),d)</annotation></semantics></math>, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>N(K) \to \Delta[1]</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a>.</p> <p>By the analogous dual argument, we find that it is also a coCartesian fibration and hence an adjunction.</p> <p>For the second statement, we need to refine the above argument just slightly to pass to the full <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-subcategories on fibrant cofibrant objects:</p> <p>let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> be as before and let <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>K</mi> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>K^\circ</annotation></semantics></math> be the full <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-subcategory on objects that are fibrant-cofibrant (in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_313' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> or in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_314' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, respectively). Then for any fibrant cofibrant <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math>, we cannot just use the identity morphism <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow></msub><mo>∈</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Id_{R(d)} \in K(R(d),d)</annotation></semantics></math> since the right Quillen functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_317' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is only guaranteed to respect fibrations, not cofibrations, and so <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>R(d)</annotation></semantics></math> might not be in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_319' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>K</mi> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>K^\circ</annotation></semantics></math>. But we can use the <a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a> to obtain a functorial cofibrant replacement functor <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>Q : C \to C</annotation></semantics></math>, such that <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Q(R(d))</annotation></semantics></math> is cofibrant and there is an acyclic fibration <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Q(R(d)) \to R(d)</annotation></semantics></math>. Take this to be the morphism in <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_323' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>Q</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(Q(R(d)), d)</annotation></semantics></math> that we pick for a given <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_324' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>. Then this does induce a homotopy equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_325' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>Q</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C(c', Q(R(d))) \to C(c',R(d)) = K(c',d) </annotation></semantics></math></div> <p>because in an <a class='existingWikiWord' href='/nlab/show/diff/enriched+model+category'>enriched model category</a> the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.</p> </div> <h3 id='localizations'>Localizations</h3> <p>A pair of adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_326' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-functors <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_327' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mover><mo>↪</mo><mo>←</mo></mover><mi>D</mi></mrow><annotation encoding='application/x-tex'>(L \dashv R) : C \stackrel{\leftarrow}{\hookrightarrow} D</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_328' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/fully+faithful+%28infinity%2C1%29-functor'>full and faithful (∞,1)-functor</a> exhibits <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_329' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective (∞,1)-subcategory</a> of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_330' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>. This subcategory and the composite <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_331' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>∘</mo><mi>L</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>R \circ L : D \to D</annotation></semantics></math> are a <a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>localization</a> of <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_332' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/adjoint+triple'>adjoint triple</a>, <a class='existingWikiWord' href='/nlab/show/diff/adjoint+quadruple'>adjoint quadruple</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proadjoint'>proadjoint</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hopf+adjunction'>Hopf adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-adjunction'>2-adjunction</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/biadjunction'>biadjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/lax+2-adjunction'>lax 2-adjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/pseudoadjunction'>pseudoadjunction</a></p> </li> <li> <p><strong>adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_333' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-functor</strong></p> </li> </ul> <h2 id='references'>References</h2> <p>The suggestion that a pair of adjoint <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_334' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-functors should just be an <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-category+of+%28%E2%88%9E%2C1%29-categories'>homotopy 2-category of $\infty$-categories</a> was originally stated, briefly, in:</p> <ul> <li id='Joyal08'><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, p. 159 (11 of 348) in: <em>The theory of quasicategories and its applications</em>, lectures at: <em><a href='https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html'>Advanced Course on Simplicial Methods in Higher Categories</a></em>, Quadern <strong>45</strong> 2, Centre de Recerca Matemàtica, Barcelona 2008 (<a class='existingWikiWord' href='/nlab/files/JoyalTheoryOfQuasiCategories.pdf' title='pdf'>pdf</a>)</li> </ul> <p>The definition as an isofibration of <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasicategories</a> over <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_335' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta[1]</annotation></semantics></math> is due to:</p> <ul> <li id='Lurie'><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, Section 5.2 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em>, Annals of Mathematics Studies 170, Princeton University Press 2009 (<a href='https://press.princeton.edu/titles/8957.html'>pup:8957</a>, <a href='https://www.math.ias.edu/~lurie/papers/HTT.pdf'>pdf</a>)</li> </ul> <p>The original suggestion of <a href='#Joyal08'>Joyal 2008</a> was then much expanded on (and generalized to <a class='existingWikiWord' href='/nlab/show/diff/infinity-cosmos'>∞-cosmoi</a>), in the spirit of <a class='existingWikiWord' href='/nlab/show/diff/formal+%28infinity%2C1%29-category+theory'>formal $\infty$-category theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, §18.6 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Categorical+Homotopy+Theory'>Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), <a href='http://www.math.jhu.edu/~eriehl/cathtpy.pdf'>pdf</a>]</p> </li> <li id='RiehlVerity15'> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a>, <em>The 2-category theory of quasi-categories</em>, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 (<a href='http://arxiv.org/abs/1306.5144'>arXiv:1306.5144</a>, <a href='https://doi.org/10.1016/j.aim.2015.04.021'>doi:10.1016/j.aim.2015.04.021</a>),</p> </li> <li id='RiehlVerity16'> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a>, <em>Homotopy coherent adjunctions and the formal theory of monads</em>, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 (<a href='http://arxiv.org/abs/1310.8279'>arXiv:1310.8279</a>, <a href='https://doi.org/10.1016/j.aim.2015.09.011'>doi:10.1016/j.aim.2015.09.011</a>)</p> </li> <li id='RiehlVerity20'> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a>, Def. 1.1.2 in: <em>Infinity category theory from scratch</em>, Higher Structures Vol 4, No 1 (2020) (<a href='https://arxiv.org/abs/1608.05314'>arXiv:1608.05314</a>, <a href='http://www.math.jhu.edu/~eriehl/scratch.pdf'>pdf</a>)</p> </li> </ul> <p>That the two definitions (of <a href='#Joyal08'>Joyal 2008</a> and <a href='#Lurie'>Lurie 2009</a>) are in fact equivalent is first indicated in <a href='#RiehlVerity15'>Riehl-Verity 15, Rem. 4.4.5</a> and then made fully explicit in:</p> <ul> <li id='RVElements'><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Elements+of+%E2%88%9E-Category+Theory'>Elements of ∞-Category Theory</a></em>, Cambridge studies in advanced mathematics <strong>194</strong>, Cambridge University Press (2022) <math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_336' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.1017/9781108936880'>doi:10.1017/9781108936880</a>, ISBN:978-1-108-83798-9, <a href='https://emilyriehl.github.io/files/elements.pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_1b909e28597e49464b01329cc4906bd8b4c15f69_337' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></li> </ul> <p>A proof that a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> induces an adjunction between <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> (in the sense of <a href='#Lurie'>Lurie 2009</a>) is recorded in:</p> <ul> <li id='Hinich14'><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Hinich'>Vladimir Hinich</a>, <em>Dwyer-Kan Localization Revisited</em>, Homology, Homotopy and Applications Volume 18 (2016) Number 1 (<a href='https://arxiv.org/abs/1311.4128'>arXiv:1311.4128</a>, <a href='https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3'>doi:10.4310/HHA.2016.v18.n1.a3</a>)</li> </ul> <p>and also in</p> <ul> <li id='MazelGee15'><a class='existingWikiWord' href='/nlab/show/diff/Aaron+Mazel-Gee'>Aaron Mazel-Gee</a>, <em>Quillen adjunctions induce adjunctions of quasicategories</em>, New York Journal of Mathematics Volume 22 (2016) 57-93 (<a href='https://arxiv.org/abs/1501.03146'>arXiv:1501.03146</a>, <a href='http://nyjm.albany.edu/j/2016/22-4.html'>publisher</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p><span /></p><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 21, 2023 at 13:44:34. See the <a href="/nlab/history/adjoint+%28infinity%2C1%29-functor" 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/adjoint+%28infinity%2C1%29-functor" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1041/#Item_37">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/adjoint+%28infinity%2C1%29-functor/62" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/adjoint+%28infinity%2C1%29-functor" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/adjoint+%28infinity%2C1%29-functor" accesskey="S" class="navlink" id="history" rel="nofollow">History (62 revisions)</a> <a href="/nlab/show/adjoint+%28infinity%2C1%29-functor/cite" style="color: black">Cite</a> <a href="/nlab/print/adjoint+%28infinity%2C1%29-functor" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/adjoint+%28infinity%2C1%29-functor" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>