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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#of_groups_and_monoids'>Of groups and monoids</a></li> <li><a href='#of_rings'>Of rings</a></li> <li><a href='#of_lie_algebras'>Of Lie algebras</a></li> <li><a href='#OfCategories'>Of (higher) categories</a></li> <ul> <li><a href='#of_general_categories'>Of general categories</a></li> <li><a href='#of_abelian_categories'>Of abelian categories</a></li> <li><a href='#of_higher_categories_2'>Of higher categories</a></li> </ul> <li><a href='#of_groups'>Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In general, the <em>center</em> (or <em>centre</em>) of an <a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the collection of <a class="existingWikiWord" href="/nlab/show/elements">elements</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which “commute with all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.” This has a number of specific incarnations.</p> <h2 id="definitions">Definitions</h2> <h3 id="of_groups_and_monoids">Of groups and monoids</h3> <p>The original example is the <strong>center</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(G)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, which is defined to be the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> consisting of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g\in G</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">h\in G</annotation></semantics></math> the equality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>h</mi><mo>=</mo><mi>h</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">g h=h g</annotation></semantics></math> holds. The center is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> subgroup, but not every abelian subgroup is in the center. See also <a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>.</p> <p>This notion of center of a group can be generalized to the center of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in an obvious way.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an object in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>. The center of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(C)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> of <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> of the identity morphism, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>C</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">id_C : C \rightarrow C</annotation></semantics></math>.</p> </div> <p>One can invoke the <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a> to prove that vertical and horizontal composition agree on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(C)</annotation></semantics></math> and are commutative.</p> <h3 id="of_rings">Of rings</h3> <p>The <strong>center</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(R)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is defined to be the <a class="existingWikiWord" href="/nlab/show/multiplicative+subset">multiplicative subset</a> consisting of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">s \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>⋅</mo><mi>s</mi><mo>=</mo><mi>s</mi><mo>⋅</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">r \cdot s = s \cdot r</annotation></semantics></math> is true. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(R)</annotation></semantics></math>.</p> <h3 id="of_lie_algebras">Of Lie algebras</h3> <p>The <strong>center of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is an abelian Lie subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(L)</annotation></semantics></math>, consisting of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex"> z\in L</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[l,z]=0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">l\in L</annotation></semantics></math>. There are generalizations for some other kinds of algebras.</p> <h3 id="OfCategories">Of (higher) categories</h3> <h4 id="of_general_categories">Of general categories</h4> <p>The notion of <em><a class="existingWikiWord" href="/nlab/show/center">center</a> of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></em> has a <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> to a notion of <em><a class="existingWikiWord" href="/nlab/show/center+of+a+category">center of a category</a></em>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a category, its center is defined to be the <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>C</mi></msub><mo>,</mo><msub><mi>Id</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Z(C) \;\coloneqq\; [C,C](Id_C,Id_C) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/endomorphism">endo</a>-<a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Id_C \,\colon\, C \to C</annotation></semantics></math>, i.e. the <a class="existingWikiWord" href="/nlab/show/endomorphism+monoid">endomorphism monoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">Id_C</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,C]</annotation></semantics></math>.</p> <p>It is straightforward to check that this reduces to the usual definition of the center of monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = \mathbf{B}(A,\cdot)</annotation></semantics></math> is the corresponding <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>.</p> <ul> <li> <p>For a <a class="existingWikiWord" href="/nlab/show/generator">generator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> there is an embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(\mathcal{C})</annotation></semantics></math> into the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(G,G)</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>↦</mo><msub><mi>η</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">\eta\mapsto\eta _G</annotation></semantics></math>. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(G,G)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(Hom(G,G))</annotation></semantics></math> is trivial, as happens e.g. for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">G=\ast</annotation></semantics></math>, then so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(\mathcal{C})</annotation></semantics></math> [<a href="#Hoffmann75">Hoffmann (1975)</a>]</p> </li> <li> <p>For <a class="existingWikiWord" href="/nlab/show/Cauchy+completion">Cauchy complete</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> the idempotent elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(\mathcal{C})</annotation></semantics></math> correspond precisely to the <em>quintessential localizations</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> [<a href="#Johnstone96">Johnstone (1996)</a>]</p> </li> </ul> <h4 id="of_abelian_categories">Of abelian categories</h4> <p>If a category carries further <a class="existingWikiWord" href="/nlab/show/structure">structure</a> this may be inherited by its center. Notably the center of an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> is not just a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> but a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (the <a class="existingWikiWord" href="/nlab/show/endomorphism+ring">endomorphism ring</a> of its <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a>).</p> <p>For more on this see at <em><a class="existingWikiWord" href="/nlab/show/center+of+an+abelian+category">center of an abelian category</a></em>.</p> <h4 id="of_higher_categories_2">Of higher categories</h4> <p>The notion of center also has a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a>: It is easy to categorify the notion of center of a category as defined above: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a>, then its <em>center</em> is the monoidal <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>C</mi></msub><mo>,</mo><msub><mi>Id</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C,C](Id_C,Id_C)</annotation></semantics></math> of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of <em>braided</em> monoidal <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-category, just as the center of a category is actually a commutative monoid, not merely a monoid.</p> <p>For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">C = \mathbf{B}_\otimes \mathcal{C}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, then this center is called the <em><a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, \otimes)</annotation></semantics></math>.</p> <p>Generally, we can now obtain a notion of the center of a monoidal <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>-category by regarding it as a one-object <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category, according to the <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>. It follows that the center of a monoidal <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>-category should naturally be a braided monoidal <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>-category. This is known to be true when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math> (the center of a monoid is a commutative monoid) and also 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=1</annotation></semantics></math> and <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=2</annotation></semantics></math>.</p> <p>Note that a monoidal <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>-category has two different centers: if we regard it as a one-object <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category, then its center is a braided monoidal <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>-category, but if we regard it merely as an <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>-category, then its center is a braided monoidal <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal <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>-category has three different centers, depending on whether we regard it as an <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>-category, a connected <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category, or a 2-connected <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>-category, and so on (a <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>-tuply monoidal <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>-category has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math> different centers).</p> <p>It seems that in applications, however, one is usually most interested in the sort of center of a monoidal <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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> obtained by regarding it as a one-object <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>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category, thereby obtaining a braided monoidal <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>-category. It is in this case, and seemingly this case only, that the center comes with a natural forgetful functor to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, corresponding to the classical inclusion of the center of a monoid. (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></mrow><annotation encoding="application/x-tex">n\gt 0</annotation></semantics></math>, however, this functor will not be an inclusion; the objects of the center of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> equipped with additional <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a>.)</p> <p>Moreover, one expects that if we perform this “canonical” operation on a <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k\ge 1</annotation></semantics></math>), the resulting braided monoidal <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>-category will actually be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-tuply monoidal. This is known to be true in the cases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n\le 4</annotation></semantics></math>: the center of a braided monoidal category is symmetric monoidal, the center of a braided monoidal 2-category is sylleptic, and the center of a sylleptic monoidal 2-category is symmetric.</p> <p>Finally, if we decategorify further, we find that the center of a <a class="existingWikiWord" href="/nlab/show/set">set</a> (i.e. a <a class="existingWikiWord" href="/nlab/show/0-category">0-category</a>) is a monoidal <a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a>, i.e. the <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a> “true.” This is what we ought to expect, since when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a set, there is precisely one endo-transformation of its identity endofunction (namely, the identity).</p> <h3 id="of_groups">Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/center+of+an+%E2%88%9E-group">center of an ∞-group</a>.</em></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/center+of+a+category">center of a category</a></p> <p><a class="existingWikiWord" href="/nlab/show/center+of+an+additive+category">center of an additive category</a></p> <p><a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+product+of+groups">central product of groups</a></p> </li> </ul> <h2 id="references">References</h2> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Center_(group_theory)">Center (group theory)</a></em></li> </ul> <p>On the notion of <a class="existingWikiWord" href="/nlab/show/center+of+a+category">center of a category</a>:</p> <ul> <li id="Hoffmann75"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf-E.+Hoffmann">Rudolf-E. Hoffmann</a>, <em>Über das Zentrum einer Kategorie</em>, Math. Nachr. <strong>68</strong> (1975) 299-306 [<a href="https://doi.org/10.1002/mana.19750680122">doi:10.1002/mana.19750680122</a>]</p> </li> <li id="Johnstone96"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Remarks on Quintessential and Persistent Localizations</em>, TAC <strong>2</strong> 8 (1996) 90-99 [<a href="http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html">tac:2-08</a>, <a href="http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf">pdf</a>]</p> </li> </ul> <p>and of an <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Harald+Lindner">Harald Lindner</a>. <em>Center and trace</em>, Archiv der Mathematik 35 (1980): 476-496. (<a href="https://doi.org/10.1007/BF01235372">doi:10.1007/BF01235372</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 20, 2023 at 10:53:29. 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