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<p><strong>Basic structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/binary+linear+code">binary linear code</a></li> <li><a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></li> <li><a class="existingWikiWord" href="/nlab/show/combinatorial+design">combinatorial design</a></li> <li><a class="existingWikiWord" href="/nlab/show/graph">graph</a></li> <li><a class="existingWikiWord" href="/nlab/show/Latin+square">Latin square</a></li> <li><a class="existingWikiWord" href="/nlab/show/matroid">matroid</a></li> <li><a class="existingWikiWord" href="/nlab/show/partition">partition</a></li> <li><a class="existingWikiWord" href="/nlab/show/permutation">permutation</a></li> <li><a class="existingWikiWord" href="/nlab/show/shuffle">shuffle</a></li> <li><a class="existingWikiWord" href="/nlab/show/tree">tree</a></li> <li><a class="existingWikiWord" href="/nlab/show/Young+diagram">Young diagram</a></li> </ul> <p><strong>Generating functions</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/combinatorial+species">combinatorial species</a></li> <li><a class="existingWikiWord" href="/nlab/show/generating+function">generating function</a></li> <li><a class="existingWikiWord" href="/nlab/show/power+series">power series</a></li> </ul> <p><strong>Proof techniques</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/bijective+proof">bijective proof</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lagrange+inversion">Lagrange inversion</a></li> <li><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+inversion">Möbius inversion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/order+polynomial">order polynomial</a></li> <li><a class="existingWikiWord" href="/nlab/show/zeta+polynomial">zeta polynomial</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/P%C3%B3lya+enumeration+theorem">Pólya enumeration theorem</a></li> </ul> <p><strong>Combinatorial identities</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/binomial+theorem">binomial theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/Catalan+number">Catalan number</a></li> <li><a class="existingWikiWord" href="/nlab/show/Chu%E2%80%93Vandermonde+identity">Chu–Vandermonde identity</a></li> </ul> <p><strong>Polytopes</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/associahedron">associahedron</a></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/combinatorics">combinatorics</a></div></div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%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/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%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/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/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/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/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/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></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></li> <li><a href='#lodays_realization'>Loday’s realization</a></li> <ul> <li><a href='#theorem_loday'>Theorem (Loday)</a></li> </ul> <li><a href='#illustrations'>Illustrations</a></li> <li><a href='#relation_to_other_structures'>Relation to other structures</a></li> <ul> <li><a href='#relation_to_orientals'>Relation to orientals</a></li> <li><a href='#Categorification'>Categorified associahedra</a></li> <li><a href='#tamari_lattice'>Tamari lattice</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>associahedra</em> or <em>Stasheff polytopes</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{K_n\}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a>es that naturally arrange themselves into a topological <a class="existingWikiWord" href="/nlab/show/operad">operad</a> that resolves the standard associative operad: an <a class="existingWikiWord" href="/nlab/show/A-infinity-operad">A-infinity-operad</a>.</p> <p>The vertices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math> correspond to ways in which one can bracket a product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> variables. The edges correspond to rebracketings, the faces relate different sequences of rebracketings that lead to the same result, and so on.</p> <p>The associahedra were introduced by Jim Stasheff in order to describe <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s equipped with a multiplication operation that is associative up to every higher coherent homotopy.</p> <h2 id="definition">Definition</h2> <p>Here is the rough idea, copied, for the moment, verbatim from Markl94 <a href="http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=26">p. 26</a> (for more details see references below):</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> the <strong>associahedron</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-2)</annotation></semantics></math>-dimensional polyhedron whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>-dimensional cells are, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">0 \leq i \leq n-2</annotation></semantics></math>, indexed by all (meaningful) insertions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-i-2)</annotation></semantics></math> pairs of brackets between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> independent indeterminates, with suitably defined incidence maps.</p> <p><a class="existingWikiWord" href="/nlab/show/simplicial+set">Simplicially</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow></mrow></mrow><annotation encoding="application/x-tex">{ }</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/subdivision">subdivided</a> associahedra (complete with simplicial <a class="existingWikiWord" href="/nlab/show/operad">operadic</a> structure) can be presented efficiently in terms of an abstract <a class="existingWikiWord" href="/nlab/show/bar+construction">bar construction</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>:</mo><mi>Set</mi><mo stretchy="false">/</mo><mi>ℕ</mi><mo>→</mo><mi>Set</mi><mo stretchy="false">/</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}: Set/\mathbb{N} \to Set/\mathbb{N}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/monad">monad</a> which takes a <a class="existingWikiWord" href="/nlab/show/graded+set">graded set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the non-permutative <a class="existingWikiWord" href="/nlab/show/non-unital+operad">non-unital operad</a> freely generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, with monad multiplication denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>𝒪</mi><mi>𝒪</mi><mo>→</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">m: \mathcal{O}\mathcal{O} \to \mathcal{O}</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">t_+</annotation></semantics></math> be the graded set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>n</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_n\}_{n \geq 0}</annotation></semantics></math> that is <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 0, 1</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>; this carries a unique non-unital non-permutative operad structure, via a structure map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>t</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\alpha: \mathcal{O}t_+ \to t_+</annotation></semantics></math>. The bar construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo>,</mo><mi>𝒪</mi><mo>,</mo><msub><mi>t</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(\mathcal{O}, \mathcal{O}, t_+)</annotation></semantics></math> is an (augmented) simplicial graded set (an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo>×</mo><mi>ℕ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Delta^{op} \times \mathbb{N}}</annotation></semantics></math>) whose face maps take the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>…</mi><mi>𝒪</mi><mi>𝒪</mi><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub><mover><munder><mo>→</mo><mrow><mi>𝒪</mi><mi>𝒪</mi><mi>α</mi></mrow></munder><mover><munder><mo>→</mo><mrow><mi>𝒪</mi><mi>m</mi><msub><mi>t</mi> <mo>+</mo></msub></mrow></munder><mover><mo>→</mo><mrow><mi>m</mi><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub></mrow></mover></mover></mover><mi>𝒪</mi><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub><mover><munder><mo>→</mo><mrow><mi>𝒪</mi><mi>α</mi></mrow></munder><mover><mo>→</mo><mrow><mi>m</mi><msub><mi>t</mi> <mo>+</mo></msub></mrow></mover></mover><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub><mover><mo>→</mo><mi>α</mi></mover><msub><mi>t</mi> <mo>+</mo></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">\ldots \mathcal{O}\mathcal{O}\mathcal{O}t_+ \stackrel{\stackrel{\overset{m\mathcal{O} t_+}{\to}}{\underset{\mathcal{O}m t_+}{\to}}}{\underset{\mathcal{O}\mathcal{O}\alpha}{\to}} \mathcal{O}\mathcal{O}t_+ \stackrel{\overset{m t_+}{\to}}{\underset{\mathcal{O}\alpha}{\to}} \mathcal{O}t_+ \stackrel{\alpha}{\to} t_+. </annotation></semantics></math></div> <p>Intuitively, the (graded set of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><msub><mi>t</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}t_+</annotation></semantics></math> consists of planar trees where each inner node has two or more incoming edges, with trees graded by number of leaves; the extreme points are binary trees [corresponding to complete binary bracketings of words], whereas other trees are barycenters of higher-dimensional faces of Stasheff polytopes. The construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo>,</mo><mi>𝒪</mi><mo>,</mo><msub><mi>t</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(\mathcal{O}, \mathcal{O}, t_+)</annotation></semantics></math> carries a simplicial (non-permutative non-unital) operad structure, where the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the simplicial set at grade (or <a class="existingWikiWord" href="/nlab/show/arity">arity</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> defines the barycentric subdivision of the Stasheff polytope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math>. As the operad structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo>,</mo><mi>𝒪</mi><mo>,</mo><msub><mi>t</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(\mathcal{O}, \mathcal{O}, t_+)</annotation></semantics></math> is expressed in <a class="existingWikiWord" href="/nlab/show/doctrine">finite product logic</a> and geometric realization preserves finite products, the (simplicially subdivided) associahedra form in this way the components of a topological operad.</p> <h2 id="lodays_realization">Loday’s realization</h2> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a> gave a simple formula for realizing the Stasheff polytopes as a convex hull of integer coordinates in Euclidean space <a href="#Loday04">(Loday 2004)</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Y_n</annotation></semantics></math> denote the set of (rooted planar) binary trees with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> leaves (and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> internal vertices). For any binary tree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t \in Y_n</annotation></semantics></math>, enumerate the leaves by left-to-right order, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>ℓ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\ell_1, \ldots, \ell_{n+1}</annotation></semantics></math>, and enumerate the internal vertices as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">v_1, \ldots, v_n</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math> is the closest common ancestor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\ell_i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\ell_{i+1}</annotation></semantics></math>. Define a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">M(t) \in \mathbb{R}^n</annotation></semantics></math> whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th coordinate is the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><msub><mi>b</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i b_i</annotation></semantics></math> of the number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math> of leaves that are left descendants of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math> and the number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math> of leaves that are right descendants of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math>.</p> <div class="un_thm"> <h6 id="theorem_loday">Theorem (Loday)</h6> <p>The convex hull of the points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>M</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>t</mi><mo>∈</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ M(t) \in \mathbb{R}^n \mid t \in Y_n \}</annotation></semantics></math> is a realization of the Stasheff polytope of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math>, and is included in the affine hyperplane <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>+</mo><mi>…</mi><mo>+</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>=</mo><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(x_1, \ldots, x_n): x_1 + \ldots + x_n = \binom{n}{2}\}</annotation></semantics></math>.</p> </div> <h2 id="illustrations">Illustrations</h2> <ul> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">K_1</annotation></semantics></math></strong> is the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a>, a degenerate case not usually considered.</p> </li> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">K_2</annotation></semantics></math></strong> is simply the shape of a binary operation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> x \otimes y ,</annotation></semantics></math></div> <p>which we interpret here as a single <a class="existingWikiWord" href="/nlab/show/point">point</a>.</p> </li> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">K_3</annotation></semantics></math></strong> is the shape of the usual <a class="existingWikiWord" href="/nlab/show/associator">associator</a> or associative law</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi><mo>→</mo><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> (x \otimes y) \otimes z \to x \otimes (y \otimes z) ,</annotation></semantics></math></div> <p>consisting of a single <a class="existingWikiWord" href="/nlab/show/interval">interval</a>.</p> </li> <li id="K4"> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math></strong> The fourth associahedron <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon</a> which expresses the different ways a product of four elements may be bracketed</p> </li> </ul> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="541.497" height="317.98" viewBox="0 0 541.497 317.98"> <defs> <g> <g id="a4FjenFyZwM9R98g1SfXtszSbZA=-glyph-0-0"> </g> <g id="a4FjenFyZwM9R98g1SfXtszSbZA=-glyph-0-1"> <path d="M 5.265625 4.203125 C 5.265625 4.1875 5.265625 4.15625 5.21875 4.09375 C 4.421875 3.28125 2.3125 1.078125 2.3125 -4.28125 C 2.3125 -9.65625 4.390625 -11.84375 5.234375 -12.703125 C 5.234375 -12.71875 5.265625 -12.765625 5.265625 -12.8125 C 5.265625 -12.859375 5.21875 -12.890625 5.140625 -12.890625 C 4.953125 -12.890625 3.5 -11.625 2.65625 -9.734375 C 1.796875 -7.8125 1.546875 -5.953125 1.546875 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xlink:href="#a4FjenFyZwM9R98g1SfXtszSbZA=-glyph-3-2" x="494.261487" y="244.812"></use> <use xlink:href="#a4FjenFyZwM9R98g1SfXtszSbZA=-glyph-3-5" x="497.513302" y="244.812"></use> </g> </svg> <p>One can also think of this as the top-level structure of the 4th <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a>. This controls in particular the <em>pentagon identity</em> in the definition of <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, as discussed there.</p> <ul> <li><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">K_5</annotation></semantics></math></strong> is the <a href="http://en.wikipedia.org/wiki/Dual_polyhedron">dual polyhedron</a> to the <a href="http://en.wikipedia.org/wiki/Triaugmented_triangular_prism">triaugmented triangular prism</a></li> </ul> <div style="text-align:center"> <p><img src="https://ncatlab.org/nlab/files/K5associahedron.png" alt="" /></p> <p><br /></p> <p>(image from the <a href="http://commons.wikimedia.org/wiki/File:Polytope_K3.svg">Wikimedia Commons</a>)</p> </div> <ul> <li> <p>One can rotate and explore Stasheff polyhedra in <a href="https://ltrujello.github.io/associahedron/">this interactive associahedron app</a>.</p> </li> <li> <p>Illustrations of some polytopes, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">K_5</annotation></semantics></math>, can also be found <a href="http://irma.math.unistra.fr/~chapoton/galerie.html">here</a>.</p> </li> <li> <p>A template which can be cut out and assembled into a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">K_5</annotation></semantics></math> can be found in Appendix B of <a href="#ChengLauda2004">ChengLauda2004</a>.</p> </li> </ul> <h2 id="relation_to_other_structures">Relation to other structures</h2> <h3 id="relation_to_orientals">Relation to orientals</h3> <p>The above list shows that the first few Stasheff polytopes are nothing but the first few <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a>s. This doesn’t remain true as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> increases. The orientals are free <strong>strict</strong> <a class="existingWikiWord" href="/nlab/show/omega-category">omega-categories</a> on <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>es as parity complexes. This means that certain interchange cells (e.g., Gray tensorators) show up as thin in the oriental description.</p> <p>The first place this happens is the sixth oriental: where there are three tensorator squares and six pentagons in Stasheff’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">K_5</annotation></semantics></math>, the corresponding tensorator squares coming from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(5)</annotation></semantics></math> are collapsed.</p> <p>It was when <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a> made this point to <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a> that Street began to think about using associahedra to define weak <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a>.</p> <h3 id="Categorification">Categorified associahedra</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">categorification</a> of associahedra discussed in</p> <ul> <li>S. Saneblidze, R. Umble, Diagonals on the permutahedra, multiplihedra and associahedra, Homology Homotopy Appl. 6(1) (2004) 363–411.</li> <li id="Forcey12"><a class="existingWikiWord" href="/nlab/show/Stefan+Forcey">Stefan Forcey</a>, <em>Quotients of the multiplihedron as categorified associahedra</em>, Homology Homotopy Appl. <strong>10</strong>:2 (2008) 227–256 (<a href="http://projecteuclid.org/euclid.hha/1251811075">Euclid</a>)</li> </ul> <h3 id="tamari_lattice">Tamari lattice</h3> <p>The associahedron is closely related to a structure known as the <em>Tamari lattice</em>, which is especially well-studied in <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a>. The Tamari lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">T_n</annotation></semantics></math> can be defined as the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of all parenthesizations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> variables with the order generated by rightwards reassociation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>c</mi><mo>≤</mo><mi>a</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a b)c \le a(b c)</annotation></semantics></math>, or equivalently as the poset of all binary trees with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> internal nodes (and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> leaves), with the order generated by rightwards tree rotation. (Note the off-by-one offset from the convention for associahedra: the Tamari lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">T_n</annotation></semantics></math> corresponds to the associahedron <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">K_{n+1}</annotation></semantics></math>.) It was originally introduced by Dov Tamari in his thesis “Monoïdes préordonnés et chaînes de Malcev” (Université de Paris, 1951), around a decade before Jim Stasheff’s work.<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup></p> <p>As the name suggests, the Tamari lattice also carries the structure of a <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>. This property was originally established by Haya Friedman and Tamari (1967), and later simplified by Samuel Huang and Tamari (1972).</p> <h2 id="references">References</h2> <p>The original articles that define associahedra and in which the operad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> that gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(\infty)</annotation></semantics></math>-topological spaces is implicit are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Homotopy associativity of H-spaces I</em>, Trans. Amer. Math. Soc. 108 (1963), 275–312. (<a href="http://www.jstor.org/stable/1993608">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Homotopy associativity of H-spaces II</em>, Trans. Amer. Math. Soc. 108 (1963), 293–312. (<a href="http://www.jstor.org/stable/1993609">web</a>)</p> </li> </ul> <p>A textbook discussion (slightly modified) is in section 1.6 of the book</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, <a class="existingWikiWord" href="/nlab/show/Steven+Shnider">Steven Shnider</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Operads in Algebra, Topology and Physics</em> (<a href="http://books.google.de/books?id=fMhZjT9lQo0C&pg=PA56&lpg=PA56&dq=Stasheff+associahedra&source=bl&ots=ZuGXjT4zbp&sig=V-taGG2LHS0msHK-PTxmUXXCvEY&hl=de#PPP1,M1">web</a>)</li> </ul> <p>Loday’s original article on the Stasheff polytope is</p> <ul> <li id="Loday04"><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, Realization of the Stasheff polytope, <em>Archiv der Mathematik</em> 83 (2004), 267-278. (<a href="https://dx.doi.org/10.1007%2Fs00013-004-1026-y">doi</a>)</li> </ul> <p>Further explanations and references are collected at</p> <ul> <li> <p><a href="http://www.ams.org/featurecolumn/archive/associahedra.html">AMS entry on associahedra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Postnikov">Alexander Postnikov</a>, <em>Permutohedra, associahedra and beyond</em>, <a href="http://arxiv.org/abs/math/0507163">math.CO/0507163</a> <a href="http://math.mit.edu/~apost/papers/permutohedron.pdf">pdf</a></p> </li> </ul> <p>The connection to Tamari lattices as well as other developments are in</p> <ul> <li id="TomariFestschrift">Folkert Müller-Hoissen, Jean Marcel Pallo, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a> (editors), <em>Associahedra, Tamari Lattices, and Related Structures: Tamari Memorial Festschrift</em>, Birkhäuser, 2012. (<a href="https://books.google.fr/books?id=Y01d6g5UemQC&lpg=PP1&pg=PP1#v=onepage&q&f=false">google books</a>)</li> </ul> <p>For a template of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">K_5</annotation></semantics></math>, see Appendix B of the following.</p> <ul> <li id="ChengLauda2004"><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, <a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, <em>Higher-dimensional categories: an illustrated guidebook</em>, 2004, <a href="http://eugeniacheng.com/guidebook/">available here</a>.</li> </ul> <p>For the combinatorics of associahedra, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Viviane+Pons">Viviane Pons</a>, <em>Combinatorics of the Permutahedra, Associahedra, and Friends</em> (<a href="https://arxiv.org/abs/2310.12687">arXiv:2310.12687</a>)</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/combinatorics">combinatorics</a></div><div class="footnotes"><hr /><ol><li id="fn:1"> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a> comments on this in an essay titled “How I ‘met’ Dov Tamari” <a href="#TomariFestschrift">(Tamari Memorial Festschrift 2012)</a>, writing that the “so-called Stasheff polytope … more accurately should be called the Tamari or Tamari-Stasheff polytope”. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 19:02:14. 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