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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #15 to #16: <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='higher_algebra'>Higher algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal algebra</a></p> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-operad'>(∞,1)-operad</a></p> </li> </ul> <h2 id='algebras_and_modules'>Algebras and modules</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category'>commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/deformation+theory'>deformation theory</a></li> </ul> </li> </ul> <h2 id='model_category_presentations'>Model category presentations</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>model structure on algebras over an operad</a></p> </li> </ul> <h2 id='geometry_on_formal_duals_of_algebras'>Geometry on formal duals of algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/higher+algebra+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#MonadicityTheorem'>Barr-Beck monadicity theorem</a></li><li><a href='#HomotopyCoherence'>Homotopy coherence</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 <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_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>-monad is the <a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>vertical categorification</a> of that of <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> from the context of <a class='existingWikiWord' href='/nlab/show/diff/category'>categories</a> to that of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a>.</p> <p>They relate to <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunctions</a> as <a class='existingWikiWord' href='/nlab/show/diff/monad'>monads</a> relate to <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunctions</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='MonadicityTheorem'>Barr-Beck monadicity theorem</h3> <div class='num_prop' id='CanonicalMonadicAdjunction'> <h6 id='proposition'>Proposition</h6> <p>Given an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a> <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> on an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, there is an <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_269923edb99792022684838a527da1a1f8fe8162_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>Alg</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>T</mi><mo stretchy='false'>)</mo><mover><munder><mo>⟶</mo><mi>U</mi></munder><mover><mo>↔</mo><mi>F</mi></mover></mover><mi>𝒞</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> (F \dashv U) \;\colon\; Alg_{\mathcal{C}}(T) \stackrel{\overset{F}{\leftrightarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Alg</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>T</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Alg_{\mathcal{C}}(T)</annotation></semantics></math> is the (Eilenberg-Moore) <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>(∞,1)-category of algebras over the (∞,1)-monad</a> and where <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> that remembers the underlying <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> of <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>.</p> </div> <p>This appears in (<a href='#RiehlVerity13'><span> Riehl-Verity 13, def.<del class='diffmod'> 6.1.15</del><ins class='diffmod'> 6.1.14</ins></span></a>).</p> <p>The following is the refinement to <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> of the classical <a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>Barr-Beck monadicity theorem</a> which states sufficient conditions for recognizing an <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a> as being canonically <a class='existingWikiWord' href='/nlab/show/diff/equivalence'>equivalent</a> to the one in prop. <a class='maruku-ref' href='#CanonicalMonadicAdjunction'>1</a>, hence to be a <em><a class='existingWikiWord' href='/nlab/show/diff/monadic+adjunction'>monadic adjunction</a></em>.</p> <div class='num_theorem' id='InfinityBarrBeckTheorem'> <h6 id='theorem'>Theorem</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_8' 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 pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint (∞,1)-functors</a> such that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_9' 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/conservative+%28%E2%88%9E%2C1%29-functor'>conservative (∞,1)-functor</a>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> of <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> admits <a class='existingWikiWord' href='/nlab/show/diff/geometric+realization'>geometric realization</a> (<a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-colimit</a>) of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object+in+an+%28infinity%2C1%29-category'>simplicial objects</a>;</p> </li> <li> <p>and <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> preserves these</p> </li> </ol> <p>then for <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo>≔</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>T \coloneqq R \circ L</annotation></semantics></math> the essentially unique <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_13' 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>-monad structure on the composite endofunctor, there is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of (∞,1)-categories</a> identifying the <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a> of <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>(∞,1)-category of algebras over an (∞,1)-monad</a> <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Alg</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>T</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Alg_{\mathcal{C}}(T)</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> itself as the canonical <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> from prop. <a class='maruku-ref' href='#CanonicalMonadicAdjunction'>1</a>.</p> </div> <p>This appears as (<a class='existingWikiWord' href='/nlab/show/diff/Higher+Algebra'>Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5</a>, <a href='#RiehlVerity13'>Riehl-Verity 13, section 7</a>)</p> <h3 id='HomotopyCoherence'>Homotopy coherence</h3> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>An <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a> <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_19' 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 lspace='verythinmathspace'>:</mo><mi>𝒞</mi><mo>↔</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D}</annotation></semantics></math> is uniquely determined already by its image in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-category'>homotopy 2-category</a> (<a href='#RiehlVerity13'>Riehl-Verity 13, theorem 5.4.14</a>). This is not in general true for <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_20' 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>-monads <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo lspace='verythinmathspace'>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>T \colon \mathcal{C} \to \mathcal{C}</annotation></semantics></math>. As these are <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoids in an (∞,1)-category</a> of <a class='existingWikiWord' href='/nlab/show/diff/endomorphism'>endomorphisms</a>, they in general have relevant <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>coherence</a> data all the way up in degree. However, by the previous statement and the monadicity theorem <a class='maruku-ref' href='#InfinityBarrBeckTheorem'>1</a>, for <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_22' 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>-monads given via specified <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunctions</a> as <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo>≃</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>T \simeq R \circ L</annotation></semantics></math> are determined by less (further) coherence data (<a class='existingWikiWord' href='/nlab/show/diff/Higher+Algebra'>Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3</a>, <a href='#RiehlVerity13'>Riehl-Verity 13, page 6</a>). (Of course there is, instead, extra data/information carried by the choice of <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{D}</annotation></semantics></math>.) This should justify the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a>-theoretic discussion in (<a href='#Hess10'>Hess 10</a>) in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+monadic+descent'>higher monadic descent</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/Lawvere+theory'>Lawvere theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-monad'>2-monad</a>/ <a class='existingWikiWord' href='/nlab/show/diff/doctrine'>doctrine</a> / <strong><math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_25' 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>-monad</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+%28infinity%2C1%29-monad'>idempotent (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modal+type+theory'>modal type theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-operad'>(∞,1)-operad</a></p> </li> </ul> <h2 id='references'>References</h2> <p>A general treatment of <math class='maruku-mathml' display='inline' id='mathml_269923edb99792022684838a527da1a1f8fe8162_26' 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>-monads in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, section 3 of <em>Noncommutative algebra</em> (<a href='http://arxiv.org/abs/math/0702299'>math.CT/0702299</a>)</li> </ul> <p>later absorbed as</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, section 6.2 of <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Algebra'>Higher Algebra</a></em></li> </ul> <p>More explict discussion in terms of <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a>:</p> <ul> <li id='RiehlVerity13'> <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>)</li> </ul> <p>Some homotopy theory of (<a class='existingWikiWord' href='/nlab/show/diff/enriched+functor'>enriched</a>) monads on (<a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial</a>) <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> is discussed (with an eye towards <a class='existingWikiWord' href='/nlab/show/diff/higher+monadic+descent'>higher monadic descent</a>) in</p> <ul> <li id='Hess10'><a class='existingWikiWord' href='/nlab/show/diff/Kathryn+Hess'>Kathryn Hess</a>, <em>A general framework for homotopic descent and codescent</em> (<a href='http://arxiv.org/abs/1001.1556'>arXiv/1001.1556</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 22, 2020 at 13:58:49. 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