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Consider the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commutative square</a> of <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a>s:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>𝒜</mi></mtd> <mtd><mover><mo>→</mo><mi>Q</mi></mover></mtd> <mtd><mi>ℬ</mi></mtd></mtr> <mtr><mtd><msup><mo /><mi>U</mi></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mi>V</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi></mtd> <mtd><munder><mo>→</mo><mi>R</mi></munder></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{cccc}\mathcal{A} & \overset{Q}{\to} & \mathcal{B} \\ ^{U}\downarrow & & \downarrow^{V} \\ \mathcal{C} & \underset{R}{\to} & \mathcal{D} \end{array} </annotation></semantics></math></div> <p>and suppose that</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic</a>, and</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>s of reflexive pairs.</p> </li> </ul> <p>Then, if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a>, then <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> also has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a>.</p> </div> <p>A detailed proof may be found in Sec. 4.5 of Vol. 2 of <a href='#Borceux'>Borceux</a> (see especially Theorem 4.5.6 on p. 226 and Ex. 4.8.6 on p. 252). Also (<a href='#Johnstone'>Johnstone, prop. 1.1.3</a>) For a sketch of proof, see ahead.</p> <div class='un_cor'> <h6 id='corollary'>Corollary</h6> <p>If the bottom functor <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> of the above square is the identity arrow (so that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>=</mo><mi>V</mi><mo>∘</mo><mi>Q</mi></mrow><annotation encoding='application/x-tex'>U=V\circ Q</annotation></semantics></math>), if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> are monadic, and if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> has coequalizers of reflexive pairs, then <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> is monadic.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The adjoint lifting theorem implies the existence of a left adjoint, and the rest is a straightforward application of the <a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a>.</p> </div> <h2 id='sketch_of_proof'>Sketch of proof</h2> <p>We may assume the situation of the following diagram (with <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>Q</mi><mo>=</mo><mi>R</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>V Q = R U</annotation></semantics></math>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mtd> <mtd><munderover><mo>⇆</mo><mi>Q</mi><mrow><mi>K</mi><mo stretchy='false'>(</mo><mo>?</mo><mo stretchy='false'>)</mo></mrow></munderover></mtd> <mtd><msup><mi>𝒟</mi> <mi>𝕊</mi></msup></mtd></mtr> <mtr><mtd><msup><mo /><mi>U</mi></msup><mo stretchy='false'>↓</mo><msup><mo stretchy='false'>↑</mo> <mi>F</mi></msup></mtd> <mtd /> <mtd><msup><mo /><mi>G</mi></msup><mo stretchy='false'>↑</mo><msup><mo stretchy='false'>↓</mo> <mi>V</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi></mtd> <mtd><munderover><mo>⇆</mo><mi>R</mi><mi>L</mi></munderover></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{cccc}\mathcal{C}^\mathbb{T} & \underoverset{Q}{K(?)}{\leftrightarrows} & \mathcal{D}^\mathbb{S} \\ ^{U}\downarrow \uparrow^{F} & & ^{G}\uparrow\downarrow^{V} \\ \mathcal{C} & \underoverset{R}{L}{\leftrightarrows} & \mathcal{D} \end{array} </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕋</mi><mo>=</mo><mo stretchy='false'>⟨</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo lspace='verythinmathspace'>:</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo>⇒</mo><mi>T</mi><mo>,</mo><mi>μ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\mathbb{T}=\langle T,\varepsilon\colon 1_{\mathcal{C}}\Rightarrow T,\mu \rangle</annotation></semantics></math> is a monad on <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mo>=</mo><mo stretchy='false'>⟨</mo><mi>S</mi><mo>,</mo><mi>ζ</mi><mo lspace='verythinmathspace'>:</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo>⇒</mo><mi>S</mi><mo>,</mo><mi>η</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\mathbb{S}=\langle S,\zeta\colon 1_{\mathcal{D}}\Rightarrow S,\eta\rangle</annotation></semantics></math> is a monad on <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{D}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> are the forgetful functors, and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> are the free algebra functors.</p> <p>Let us write <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi><mo lspace='verythinmathspace'>:</mo><mi>F</mi><mi>U</mi><mo>⇒</mo><msub><mn>1</mn> <mrow><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mrow></msub></mrow><annotation encoding='application/x-tex'>\tau\colon F U\Rightarrow 1_{\mathcal{C}^{\mathbb{T}}}</annotation></semantics></math> for the counit of the adjunction <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>F\dashv U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo lspace='verythinmathspace'>:</mo><mi>G</mi><mi>V</mi><mo>⇒</mo><msub><mn>1</mn> <mrow><msup><mi>𝒟</mi> <mi>𝕊</mi></msup></mrow></msub></mrow><annotation encoding='application/x-tex'>\sigma\colon G V\Rightarrow 1_{\mathcal{D}^{\mathbb{S}}}</annotation></semantics></math> for the counit of the adjunction <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>⊣</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>G\dashv V</annotation></semantics></math>. As usual, we have <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo>=</mo><mi>U</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>T = U F</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>=</mo><mi>V</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>S = V G</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi><mo>=</mo><mi>U</mi><mi>τ</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>\mu=U\tau F</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>=</mo><mi>V</mi><mi>σ</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>\eta=V\sigma G</annotation></semantics></math>.</p> <p>Finally, let <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be a left adjoint to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> (which exists by assumption), and let <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo lspace='verythinmathspace'>:</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo>⇒</mo><mi>R</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>\alpha\colon 1_{\mathcal{D}}\Rightarrow R L</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mo lspace='verythinmathspace'>:</mo><mi>LR</mi><mo>⇒</mo><msub><mn>1</mn> <mi>𝒞</mi></msub></mrow><annotation encoding='application/x-tex'>\beta\colon LR\Rightarrow 1_{\mathcal{C}}</annotation></semantics></math> be the unit and counit (respectively) of the adjunction <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>⊣</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>L\dashv R</annotation></semantics></math>.</p> <p>We would like to construct a functor <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo lspace='verythinmathspace'>:</mo><msup><mi>𝒟</mi> <mi>𝕊</mi></msup><mo>→</mo><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mrow><annotation encoding='application/x-tex'>K\colon \mathcal{D}^{\mathbb{S}}\to \mathcal{C}^{\mathbb{T}}</annotation></semantics></math>. To get a hint of what <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> should look like, let us assume for a moment that such a <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> already exists. In this case, we have <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>G</mi><mo>⊣</mo><mi>V</mi><mi>Q</mi><mo stretchy='false'>(</mo><mo>=</mo><mi>R</mi><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K G\dashv V Q(=R U)</annotation></semantics></math>. But <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>F L</annotation></semantics></math> is also left adjoint to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>R U</annotation></semantics></math>, and by the uniqueness of a left adjoint we must have <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>G</mi><mo>=</mo><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>K G = F L</annotation></semantics></math> (at least up to a natural isomorphism).</p> <p>From this, we already know how to define <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> on free algebras. Also, being a left adjoint, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> preserves in particular all coequalizers. But every <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'>\mathbb{S}</annotation></semantics></math>-algebra <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo lspace='verythinmathspace'>:</mo><mi>SD</mi><mo>→</mo><mi>D</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle D,\xi\colon SD\to D\rangle</annotation></semantics></math> is the (object part) of a reflexive coequalizer, namely, the <a class='existingWikiWord' href='/nlab/show/diff/canonical+presentation'>canonical presentation</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mi>S</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><munderover><mo>⇉</mo><mrow><mi>G</mi><mi>ξ</mi></mrow><mrow><msub><mi>σ</mi> <mrow><mi>G</mi><mi>D</mi></mrow></msub></mrow></munderover><mi>G</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>σ</mi> <mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow></msub><mo>=</mo><mi>ξ</mi></mrow></mover><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'> G S(D)\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows}G(D)\overset{\sigma_{\langle D,\xi\rangle}=\xi}{\rightarrow}\langle D,\xi \rangle </annotation></semantics></math></div> <p>Applying <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> and using <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>G</mi><mo>=</mo><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>K G=F L</annotation></semantics></math>, we see that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>K\langle D,\xi\rangle</annotation></semantics></math> should be the object part of a reflexive coequalizer in <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{\mathbb{T}}</annotation></semantics></math> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>S</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><munderover><mo>⇉</mo><mrow><mi>F</mi><mi>L</mi><mi>ξ</mi></mrow><mo>?</mo></munderover><mi>F</mi><mi>L</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>x</mi></mover><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'> F L S(D)\underoverset{F L\xi}{?}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle </annotation></semantics></math></div> <p>(recall that we assume that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{\mathbb{T}}</annotation></semantics></math> has coequalizers of reflexive pairs).</p> <p>To eventually define <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>K\langle D,\xi \rangle</annotation></semantics></math> as a coequalizer (as above), we first need some reasonable guess for the ∞-arrow. For this, we will need a lemma.</p> <div class='un_lemma'> <h6 id='lemma'>Lemma</h6> <p>There exists a natural transformation <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>λ</mi><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mi>R</mi><mo>⇒</mo><mi>R</mi><mi>T</mi></mrow><annotation encoding='application/x-tex'>\lambda\colon S R \Rightarrow R T</annotation></semantics></math> for which the following diagram of functors and natural transformations is commutative:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center center center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>R</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>ζ</mi><mi>R</mi></mrow></mover></mtd> <mtd><mi>S</mi><mi>R</mi></mtd> <mtd /> <mtd><mover><mo>←</mo><mrow><mi>η</mi><mi>R</mi></mrow></mover></mtd> <mtd /> <mtd><mi>S</mi><mi>S</mi><mi>R</mi></mtd></mtr> <mtr><mtd /> <mtd><msup><mo /><mrow><mi>R</mi><mi>ε</mi></mrow></msup><mo>↘</mo></mtd> <mtd><msup><mo /><mi>λ</mi></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><msup><mo /><mrow><mi>S</mi><mi>λ</mi></mrow></msup><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>R</mi><mi>T</mi></mtd> <mtd><mover><mo>←</mo><mrow><mi>R</mi><mi>μ</mi></mrow></mover></mtd> <mtd><mi>R</mi><mi>T</mi><mi>T</mi></mtd> <mtd><mover><mo>←</mo><mrow><mi>λ</mi><mi>T</mi></mrow></mover></mtd> <mtd><mi>S</mi><mi>R</mi><mi>T</mi></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{ccccccc} R & \overset{\zeta R}{\to} & S R & & \overset{\eta R}{\leftarrow}& & S S R \\ & ^{R\varepsilon}\searrow & ^{\lambda}\downarrow & & & & ^{S\lambda}\downarrow\\ & & R T & \overset{R\mu}{\leftarrow} & R T T & \overset{\lambda T}{\leftarrow} & S R T \end{array} </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Define <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>λ</mi><mo>:</mo><mo>=</mo><mi>V</mi><mi>σ</mi><mi>Q</mi><mi>F</mi><mo>∘</mo><mi>V</mi><mi>G</mi><mi>R</mi><mi>ε</mi></mrow><annotation encoding='application/x-tex'>\lambda := V\sigma Q F \circ V G R\varepsilon</annotation></semantics></math>, so that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>λ</mi><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mi>R</mi><mo>=</mo><mi>V</mi><mi>G</mi><mi>R</mi><mover><mo>→</mo><mrow><mi>V</mi><mi>G</mi><mi>R</mi><mi>ε</mi></mrow></mover><mi>V</mi><mi>G</mi><mi>R</mi><mi>T</mi><mo>=</mo><mi>V</mi><mi>G</mi><mi>R</mi><mi>U</mi><mi>F</mi><mo>=</mo><mi>V</mi><mi>G</mi><mi>V</mi><mi>Q</mi><mi>F</mi><mover><mo>→</mo><mrow><mi>V</mi><mi>σ</mi><mi>Q</mi><mi>F</mi></mrow></mover><mi>V</mi><mi>Q</mi><mi>F</mi><mo>=</mo><mi>R</mi><mi>U</mi><mi>F</mi><mo>=</mo><mi>R</mi><mi>T</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \lambda \colon S R = V G R\overset{V G R\varepsilon}{\to}V G R T = V G R U F = V G V Q F\overset{V\sigma Q F}{\to} V Q F = R U F = R T. </annotation></semantics></math></div> <p>The required commutativity may be verified by using the commutative diagrams in the definitions of a monad and an EM-algebra, naturality, and the triangular identities. For details, see the proof of Lemma 4.5.1 of <a href='#Borceux'>Borceux</a>, pp. 222-223. (Note that this lemma does not depend on the existence of a left adjoint for the bottom horizontal arrow, nor on the existence of coequalizers. Only the commutativity is required.)</p> </div> <p>We may now return to our task of defining the ∞-arrow in the diagram preceding the lemma. We would like to get from <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>S</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F L S(D)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F L(D)</annotation></semantics></math>, and for this, we will construct a natural transformation <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>S</mi><mo>⇒</mo><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>F L S\Rightarrow F L</annotation></semantics></math> in the following way. First, we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mover><mo>→</mo><mrow><mi>S</mi><mi>α</mi></mrow></mover><mi>S</mi><mi>R</mi><mi>L</mi><mover><mo>→</mo><mrow><mi>λ</mi><mi>L</mi></mrow></mover><mi>R</mi><mi>T</mi><mi>L</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> S\overset{S\alpha}{\to} S R L \overset{\lambda L}{\to} R T L. </annotation></semantics></math></div> <p>Applying <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> and composing with <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mi>T</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>\beta T L</annotation></semantics></math>, we get</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LS</mi><mover><mo>⟶</mo><mrow><mi>L</mi><mi>λ</mi><mi>L</mi><mo>∘</mo><mi>LS</mi><mi>α</mi></mrow></mover><mi>L</mi><mi>R</mi><mi>T</mi><mi>L</mi><mover><mo>→</mo><mrow><mi>β</mi><mi>TL</mi></mrow></mover><mi>TL</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> LS\overset{L\lambda L\circ LS\alpha}{\longrightarrow}L R T L \overset{\beta TL}{\to} TL. </annotation></semantics></math></div> <p>Applying <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and composing with <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>\tau F L</annotation></semantics></math>, we finally get</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>S</mi><mover><mo>⟶</mo><mrow><mi>F</mi><mi>β</mi><mi>T</mi><mi>L</mi><mo>∘</mo><mi>F</mi><mi>L</mi><mi>λ</mi><mi>L</mi><mo>∘</mo><mi>F</mi><mi>L</mi><mi>S</mi><mi>α</mi></mrow></mover><mi>F</mi><mi>T</mi><mi>L</mi><mo>=</mo><mi>F</mi><mi>U</mi><mi>F</mi><mi>L</mi><mover><mo>→</mo><mrow><mi>τ</mi><mi>F</mi><mi>L</mi></mrow></mover><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'> F L S\overset{F\beta T L\circ F L\lambda L\circ F L S\alpha}{\longrightarrow}F T L = F U F L\overset{\tau F L}{\to} F L </annotation></semantics></math></div> <p>Let us call the resulting natural transformation <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>, that is,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>:</mo><mo>=</mo><mi>τ</mi><mi>F</mi><mi>L</mi><mo>∘</mo><mi>F</mi><mi>β</mi><mi>T</mi><mi>L</mi><mo>∘</mo><mi>F</mi><mi>L</mi><mi>λ</mi><mi>L</mi><mo>∘</mo><mi>F</mi><mi>L</mi><mi>S</mi><mi>α</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \omega:=\tau F L \circ F\beta T L\circ F L\lambda L\circ F L S\alpha. </annotation></semantics></math></div> <p>Now we take the sought for ∞-arrow to be <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>\omega_D</annotation></semantics></math>, and <em>define</em> <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>K\langle D,\xi\rangle</annotation></semantics></math> as the object of some fixed coequalizer of <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>\omega_D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>ξ</mi></mrow><annotation encoding='application/x-tex'>F L\xi</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><mi>S</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><munderover><mo>⇉</mo><mrow><mi>F</mi><mi>L</mi><mi>ξ</mi></mrow><mrow><msub><mi>ω</mi> <mi>D</mi></msub></mrow></munderover><mi>F</mi><mi>L</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>x</mi></mover><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> F L S(D)\underoverset{F L\xi}{\omega_D}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle. </annotation></semantics></math></div> <p>In order to do this, we must first verify that the parallel arrows above have a common section (since we only assume that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>𝕋</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{\mathbb{T}}</annotation></semantics></math> has coequalizers of reflexive pairs). To find a guess for a common section, note that the common section for the parallel pair in the above canonical presentation in <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒟</mi> <mi>𝕊</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{D}^{\mathbb{S}}</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><msub><mi>ζ</mi> <mrow><mi>V</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow></msub><mo>=</mo><mi>G</mi><msub><mi>ζ</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>G\zeta_{V\langle D,\xi\rangle}=G\zeta_D</annotation></semantics></math>, and if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> exists, then applying <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> gives <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>G</mi><msub><mi>ζ</mi> <mi>D</mi></msub><mo>=</mo><mi>F</mi><mi>L</mi><msub><mi>ζ</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>K G\zeta_D=F L\zeta_D</annotation></semantics></math>. Having this guess, it is now straightforward to verify that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>L</mi><msub><mi>ζ</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>F L\zeta_D</annotation></semantics></math> is indeed a common section, as required.</p> <p>So, we have defined an object function of a would be left adjoint <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>. To make it into a functor left adjoint to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math>, we will build a universal arrow from <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle D,\xi\rangle</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math>, whose object part is <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>K\langle D,\xi\rangle</annotation></semantics></math> (Theorem IV.1.2(ii) of <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories Work</a>).</p> <p>To get an arrow <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo><mo>→</mo><mi>Q</mi><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle D,\xi\rangle\to Q K\langle D,\xi\rangle</annotation></semantics></math> , suppose for a moment that we have a natural transformation <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi><mo lspace='verythinmathspace'>:</mo><mi>G</mi><mo>⇒</mo><mi>Q</mi><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>\varphi\colon G\Rightarrow Q F L</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi><mo>∘</mo><mi>σ</mi><mi>G</mi><mo>=</mo><mi>Q</mi><mi>ω</mi><mo>∘</mo><mi>φ</mi><mi>S</mi></mrow><annotation encoding='application/x-tex'>\varphi\circ \sigma G = Q\omega\circ \varphi S</annotation></semantics></math>. Then the left square in the following diagram commutes:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>G</mi><mi>S</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mtd> <mtd><munderover><mo>⇉</mo><mrow><mi>G</mi><mi>ξ</mi></mrow><mrow><msub><mi>σ</mi> <mrow><mi>G</mi><mi>D</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>G</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>σ</mi> <mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow></msub><mo>=</mo><mi>ξ</mi></mrow></mover></mtd> <mtd><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mtd></mtr> <mtr><mtd><msup><mo /><mrow><msub><mi>φ</mi> <mrow><mi>S</mi><mi>D</mi></mrow></msub></mrow></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo /><mrow><msub><mi>φ</mi> <mi>D</mi></msub></mrow></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo /><mi>χ</mi></msup><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>Q</mi><mi>F</mi><mi>L</mi><mi>S</mi><mi>D</mi></mtd> <mtd><munderover><mo>⇉</mo><mrow><mi>Q</mi><mi>F</mi><mi>L</mi><mi>ξ</mi></mrow><mrow><mi>Q</mi><msub><mi>ω</mi> <mi>D</mi></msub></mrow></munderover></mtd> <mtd><mi>Q</mi><mi>F</mi><mi>L</mi><mi>D</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>Q</mi><mi>x</mi></mrow></mover></mtd> <mtd><mi>Q</mi><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{ccccc}G S(D)& \underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows} & G(D) &\overset{\sigma_{\langle D,\xi\rangle}=\xi}{\rightarrow}&\langle D,\xi \rangle \\ ^{\varphi_{S D}}\downarrow && ^{\varphi_D}\downarrow && ^{\chi}\downarrow\\ Q F L S D &\underoverset{Q F L\xi}{Q\omega_D}{\rightrightarrows} & Q F L D &\overset{Q x}{\rightarrow}&Q K\langle D,\xi\rangle \end{array} </annotation></semantics></math></div> <p>Since both rows are forks, it follows that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>x</mi><mo>∘</mo><msub><mi>φ</mi> <mi>D</mi></msub></mrow><annotation encoding='application/x-tex'>Q x\circ \varphi_D</annotation></semantics></math> has the same composition with the arrows of the upper<br />parallel pair, and hence there exists a unique arrow <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo><mo>→</mo><mi>Q</mi><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\chi\colon \langle D,\xi\rangle\to Q K\langle D,\xi\rangle</annotation></semantics></math> making the right square commutative (recall that the upper row is a coequalizer).</p> <p>It is now possible to prove that the pair <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>K</mi><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo><mo>,</mo><mi>χ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle K\langle D,\xi\rangle,\chi \rangle</annotation></semantics></math> is a universal arrow from <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>D</mi><mo>,</mo><mi>ξ</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle D,\xi\rangle</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math>, showing that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> is indeed (the object function) of a left adjoint (for details, see the proof of Theorem 4.5.6, pp. 226-227 in <a href='#Borceux'>Borceux</a>).</p> <p>But we still have to prove the existence of a natural transformation <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi><mo lspace='verythinmathspace'>:</mo><mi>G</mi><mo>⇒</mo><mi>Q</mi><mi>F</mi><mi>L</mi></mrow><annotation encoding='application/x-tex'>\varphi\colon G\Rightarrow Q F L</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi><mo>∘</mo><mi>σ</mi><mi>G</mi><mo>=</mo><mi>Q</mi><mi>ω</mi><mo>∘</mo><mi>φ</mi><mi>S</mi></mrow><annotation encoding='application/x-tex'>\varphi\circ \sigma G = Q\omega\circ \varphi S</annotation></semantics></math>. For this, we define <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi><mo>:</mo><mo>=</mo><mi>σ</mi><mi>Q</mi><mi>F</mi><mi>L</mi><mo>∘</mo><mi>G</mi><mi>R</mi><mi>ε</mi><mi>L</mi><mo>∘</mo><mi>G</mi><mi>α</mi></mrow><annotation encoding='application/x-tex'>\varphi:=\sigma Q F L \circ G R\varepsilon L \circ G\alpha</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is faithful, to prove the required property of <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>φ</mi></mrow><annotation encoding='application/x-tex'>\varphi</annotation></semantics></math>, it is enough to prove that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>φ</mi><mo>∘</mo><mi>V</mi><mi>σ</mi><mi>G</mi><mo>=</mo><mi>V</mi><mi>Q</mi><mi>ω</mi><mo>∘</mo><mi>V</mi><mi>φ</mi><mi>S</mi></mrow><annotation encoding='application/x-tex'>V\varphi \circ V\sigma G = V Q\omega\circ V\varphi S</annotation></semantics></math>, and this is a long, yet straightforward, computation (noting that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>φ</mi><mo>=</mo><mi>λ</mi><mi>L</mi><mo>∘</mo><mi>VG</mi><mi>α</mi></mrow><annotation encoding='application/x-tex'>V\varphi = \lambda L \circ VG\alpha</annotation></semantics></math> and using the commutative diagram from Lemma 1; see Lemma 4.5.3, p. 224 of <a href='#Borceux'>Borceux</a>).</p> <h2 id='examples'>Examples</h2> <h3 id='forgetful_functors_between_varieties_of_algebras'>Forgetful functors between varieties of algebras</h3> <p>Since varieties of algebras are <a class='existingWikiWord' href='/nlab/show/diff/cocompleteness+of+varieties+of+algebras'>cocomplete</a> and monadic over <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Set</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{Set}</annotation></semantics></math>, the corollary implies that forgetful functors between varieties of algebras (e.g., the forgetful functor <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Rng</mi></mstyle><mo>→</mo><mstyle mathvariant='bold'><mi>Ab</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{Rng}\to\mathbf{Ab}</annotation></semantics></math>) are monadic.</p> <h3 id='sufficient_conditions_for_cocompleteness_of_monadic_categories'>Sufficient conditions for cocompleteness of monadic categories</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒥</mi></mrow><annotation encoding='application/x-tex'>\mathcal{J}</annotation></semantics></math> be an arbitrary category, and consider the commutative diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>𝒜</mi></mtd> <mtd><mover><mo>→</mo><mi>Δ</mi></mover></mtd> <mtd><msup><mi>𝒜</mi> <mi>𝒥</mi></msup></mtd></mtr> <mtr><mtd><msup><mo /><mi>U</mi></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mrow><msup><mi>U</mi> <mi>𝒥</mi></msup></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi></mtd> <mtd><munder><mo>→</mo><mi>Δ</mi></munder></mtd> <mtd><msup><mi>𝒞</mi> <mi>𝒥</mi></msup></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{ccccc}\mathcal{A} & \overset{\Delta}{\to} & \mathcal{A}^\mathcal{J} \\ ^{U}\downarrow & & \downarrow^{U^{\mathcal{J}}} \\ \mathcal{C} & \underset{\Delta}{\to} & \mathcal{C}^\mathcal{J} \end{array} </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> is monadic, <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi></mrow><annotation encoding='application/x-tex'>\Delta</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+functor'>diagonal functor</a> and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>U</mi> <mi>𝒥</mi></msup><mo>=</mo><mi>U</mi><mo>∘</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo></mrow><annotation encoding='application/x-tex'>U^{\mathcal{J}}=U\circ -</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is left adjoint <br />to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mi>𝒥</mi></msup></mrow><annotation encoding='application/x-tex'>F^\mathcal{J}</annotation></semantics></math> is left adjoint to <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>U</mi> <mi>𝒥</mi></msup></mrow><annotation encoding='application/x-tex'>U^{\mathcal{J}}</annotation></semantics></math> (using the unit and counit of the original adjunction, one can construct appropriate natural transformations that satisfy the triangular identities, see, e.g., p. 119 of <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories Work</a>). Also, the conditions of the monadicity theorem for <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> imply those for <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>U</mi> <mi>𝒥</mi></msup></mrow><annotation encoding='application/x-tex'>U^\mathcal{J}</annotation></semantics></math> (basically because the definition of a split fork involves only compositions and identities, and because natural transformations are composed componentwise).</p> <p>Now, if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒥</mi></mrow><annotation encoding='application/x-tex'>\mathcal{J}</annotation></semantics></math>-cocomplete (so that the bottom horizontal functor has a left adjoint) and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> has coequalizers of reflexive pairs, then the adjoint lifting theorem implies that <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒥</mi></mrow><annotation encoding='application/x-tex'>\mathcal{J}</annotation></semantics></math>-cocomplete. In particular, if <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> has coequalizers of reflexive pairs and <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is small-cocomplete, then <math class='maruku-mathml' display='inline' id='mathml_b10bf5f044896dfc1793ee61eda411816a9827e6_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is small cocomplete.</p> <h2 id='related_pages'>Related pages</h2> <ul> <li><span><del class='diffdel'> The</del><del class='diffdel'> adjoint</del><del class='diffdel'> lifting</del><del class='diffdel'> theorem</del><del class='diffdel'> is</del><del class='diffdel'> a</del><del class='diffdel'> corollary</del><del class='diffdel'> of</del><del class='diffdel'> the</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/adjoint+triangle+theorem'>adjoint triangle theorem</a></del><ins class='diffmod'><p>The adjoint lifting theorem is a corollary of the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+triangle+theorem'>adjoint triangle theorem</a>.</p></ins><span><del class='diffdel'> .</del></span></li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li></ins> </ul> <h2 id='references'>References</h2> <ul id='Borceux'> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Barr'>Michael Barr</a>, <a class='existingWikiWord' href='/nlab/show/diff/Charles+Wells'>Charles Wells</a><span> ,<ins class='diffins'> section</ins><ins class='diffins'> 3.7,</ins><ins class='diffins'> pp.131</ins><ins class='diffins'> in:</ins></span><em><del class='diffmod'>Toposes, Triples and Theories</del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Toposes%2C+Triples%2C+and+Theories'>Toposes, Triples, and Theories</a></ins></em><span><del class='diffdel'> </del> , Springer<del class='diffmod'> Heidelberg</del><ins class='diffmod'> (1985),</ins><del class='diffmod'> 1985.</del><ins class='diffmod'> Reprints</ins><del class='diffmod'> (Reprinted</del><ins class='diffmod'> in</ins><del class='diffmod'> as</del><ins class='diffmod'> Theories</ins><ins class='diffins'> and</ins><ins class='diffins'> Applications</ins><ins class='diffins'> of</ins><ins class='diffins'> Categories</ins></span><del class='diffmod'><a href='http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html'>TAC reprint no.12</a></del><ins class='diffmod'><strong>12</strong></ins><span> <del class='diffmod'> (2005);</del><ins class='diffmod'> (2005)</ins><del class='diffmod'> section</del><ins class='diffmod'> 1-287</ins><del class='diffmod'> 3.7,</del><ins class='diffmod'> [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]</ins><del class='diffdel'> pp.131ff)</del></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a> II</em> , Cambridge UP 1994. (section 4.5, pp.221ff)</p> </li> </ul> <ul> <li id='Johnstone'> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+Johnstone'>Peter Johnstone</a>, <em>Adjoint lifting theorems for categories of algebras</em><span><del class='diffdel'> </del> , Bull. London Math. Soc.</span><strong>7</strong><span> (1975)<del class='diffmod'> pp.294-297.</del><ins class='diffmod'> 294-297</ins><ins class='diffins'> [[doi:10.1112/blms/7.3.294](https://doi.org/10.1112/blms/7.3.294)]</ins></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+Johnstone'>Peter Johnstone</a><span> ,<ins class='diffins'> section</ins><ins class='diffins'> A1.1,</ins><ins class='diffins'> p.5</ins><ins class='diffins'> in:</ins></span><em><a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Sketches of an Elephant</a> I</em><span><del class='diffdel'> </del> , Oxford UP<del class='diffmod'> 2002.</del><ins class='diffmod'> 2002</ins><del class='diffdel'> (section</del><del class='diffdel'> A1.1,</del><del class='diffdel'> p.5)</del></span></p> </li> </ul> <p><span><del class='diffmod'> The</del><ins class='diffmod'> On</ins><ins class='diffins'> the</ins> dual theorem for<del class='diffdel'> comonads</del><del class='diffdel'> is</del><del class='diffdel'> also</del><del class='diffdel'> in</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/comonad'>comonads</a></ins><ins class='diffins'>:</ins></p> <ul> <li> <p>William F. Keigher, <em>Adjunctions and comonads in differential algebra</em><span> , Pacific J. Math.<del class='diffdel'> 59,</del><del class='diffdel'> n.</del><del class='diffdel'> 1</del><del class='diffdel'> (1975)</del><del class='diffdel'> 99-112</del></span><del class='diffmod'><a href='http://projecteuclid.org/euclid.pjm/1102905501'>euclid</a></del><ins class='diffmod'><strong>59,</strong></ins><ins class='diffins'> 1 (1975) 99-112 [[euclid:pjm/1102905501](http://projecteuclid.org/euclid.pjm/1102905501)]</ins></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Power'>John Power</a>, <em>A unified approach to the lifting of adjoints</em><span><del class='diffdel'> </del> ,<del class='diffdel'> Cah.</del><del class='diffdel'> Top.</del><del class='diffdel'> Géom.</del><del class='diffdel'> Diff.</del><del class='diffdel'> Cat.</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/Cahiers'>Cahiers</a></ins><ins class='diffins'> </ins><strong>XXIX</strong><span> <del class='diffmod'> no.1</del><ins class='diffmod'> 1</ins> (1988)<del class='diffmod'> pp.67-77.</del><ins class='diffmod'> 67-77</ins><del class='diffmod'> (</del><ins class='diffmod'> [[numdam](http://www.numdam.org/item/CTGDC_1988__29_1_67_0)]</ins></span><del class='diffdel'><a href='http://www.numdam.org/item/CTGDC_1988__29_1_67_0'>numdam</a></del><del class='diffdel'>)</del></p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> </div> <div class="revisedby"> <p> Last revised on November 16, 2023 at 08:50:14. See the <a href="/nlab/history/adjoint+lifting+theorem" 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+lifting+theorem" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2286/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/adjoint+lifting+theorem/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/adjoint+lifting+theorem" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/adjoint+lifting+theorem" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/adjoint+lifting+theorem/cite" style="color: black">Cite</a> <a href="/nlab/print/adjoint+lifting+theorem" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/adjoint+lifting+theorem" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>