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href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> <h4 id="duality">Duality</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality">duality</a></strong></p> <ul> <li> <p>abstract duality: <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton duality</a></p> </li> <li> <p>concrete duality: <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a>, <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>, <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable object</a>, <a class="existingWikiWord" href="/nlab/show/dualizing+object">dualizing object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a></li> </ul> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p>between <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Langlands+duality">Langlands duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a>, <a class="existingWikiWord" href="/nlab/show/quantum+geometric+Langlands+duality">quantum geometric Langlands duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin+duality">Pontryagin duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartier+duality">Cartier duality</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> for <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+duality">Grothendieck duality</a></p> </li> </ul> <p><strong>In QFT and String theory</strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Seiberg+duality">Seiberg duality</a>, <a class="existingWikiWord" href="/nlab/show/AGT+conjecture">AGT conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electro-magnetic+duality">electro-magnetic duality</a>, <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="stone_duality">Stone duality</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#particular_cases'>Particular cases</a></li> <ul> <li><a href='#locales_and_frames'>Locales and frames</a></li> <li><a href='#topological_spaces'>Topological spaces</a></li> <li><a href='#coherent_spaces_and_distributive_lattices'>Coherent spaces and distributive lattices</a></li> <li><a href='#StoneSpacesAndBooleanAlgebras'>Stone spaces and Boolean algebras</a></li> <ul> <li><a href='#theorem_stone_representation'>Theorem (Stone representation)</a></li> </ul> <li><a href='#StoneSpacesAndProfiniteSets'>Stone spaces and profinite sets</a></li> <li><a href='#stonean_spaces_and_complete_boolean_algebras'>Stonean spaces and complete Boolean algebras</a></li> <li><a href='#profinite_algebras'>Profinite algebras</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>Stone duality is a subject comprising various <a class="existingWikiWord" href="/nlab/show/dualities">dualities</a> between <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a> in the area of <a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a> and topological <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>.</p> <h2 id="particular_cases">Particular cases</h2> <h3 id="locales_and_frames">Locales and frames</h3> <p>Perhaps the most general duality falling under this heading is that between <a class="existingWikiWord" href="/nlab/show/locales">locales</a> (on the <a class="existingWikiWord" href="/nlab/show/space">space</a> side) and <a class="existingWikiWord" href="/nlab/show/frames">frames</a> (on the <a class="existingWikiWord" href="/nlab/show/quantity">quantity</a> side). Of course, this duality is not very deep at all; the category <a class="existingWikiWord" href="/nlab/show/Loc">Loc</a> of locales is simply <em>defined</em> to be the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the category <a class="existingWikiWord" href="/nlab/show/Frm">Frm</a> of frames. But there are several interesting dualities between <a class="existingWikiWord" href="/nlab/show/subcategories">subcategories</a> of these.</p> <h3 id="topological_spaces">Topological spaces</h3> <p>Stone duality is often described for <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> rather than for <a class="existingWikiWord" href="/nlab/show/locales">locales</a>. In this case, the most general duality is that between <a class="existingWikiWord" href="/nlab/show/sober+spaces">sober spaces</a> and frames with enough points (which correspond to <a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a>s). In many cases, one requires the <a class="existingWikiWord" href="/nlab/show/ultrafilter+theorem">ultrafilter theorem</a> (or other forms of the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>) in order for the duality to hold when applied to topological spaces, while the duality holds for locales even in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>.</p> <h3 id="coherent_spaces_and_distributive_lattices">Coherent spaces and distributive lattices</h3> <p>Any <a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a> generates a <a class="existingWikiWord" href="/nlab/show/free+object">free</a> frame. The locales which arise in this way can be characterized as the <a class="existingWikiWord" href="/nlab/show/coherent+locale">coherent locale</a>s, and this gives a duality between distributive lattices and coherent locales. Note that one must additionally restrict to “coherent maps” between coherent locales. Also, at least assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, every coherent locale is <a class="existingWikiWord" href="/nlab/show/topological+locale">topological</a>, so we may say “coherent space” instead.</p> <h3 id="StoneSpacesAndBooleanAlgebras">Stone spaces and Boolean algebras</h3> <p>The duality which is due to <a class="existingWikiWord" href="/nlab/show/Marshall+Stone">Marshall Stone</a>, and which gives its name to the subject, is the duality between <a class="existingWikiWord" href="/nlab/show/Stone+spaces">Stone spaces</a> and <a class="existingWikiWord" href="/nlab/show/Boolean+algebras">Boolean algebras</a>. Specifically, a <a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a> is a Boolean algebra precisely when the free frame it generates is the topology of a Stone space, and any continuous map of Stone spaces is coherent. Therefore, the category of Stone spaces is dual to the category of Boolean algebras. The Boolean algebra corresponding to a Stone space consists of its <a class="existingWikiWord" href="/nlab/show/clopen+sets">clopen sets</a>.</p> <p>This duality may be realized via a <a class="existingWikiWord" href="/nlab/show/dualizing+object">dualizing object</a> as follows. The two-element <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a> may be regarded as a Boolean algebra object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CH</mi></mrow><annotation encoding="application/x-tex">CH</annotation></semantics></math>. Thus, for each finitary Boolean algebra operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo lspace="verythinmathspace">:</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>n</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\theta\colon \mathbf{2}^n \to \mathbf{2}</annotation></semantics></math>, there is a corresponding operation on the <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>:</mo><msup><mi>CH</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">CH(-, \mathbf{2}): CH^{op} \to Set</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>≅</mo><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mn>2</mn></mstyle> <mi>n</mi></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mover><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CH(-, \mathbf{2})^n \cong CH(-, \mathbf{2}^n) \stackrel{CH(-, \theta)}{\to} CH(-, \mathbf{2})</annotation></semantics></math></div> <p>and therefore we obtain a lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CH</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>:</mo><msup><mi>CH</mi> <mi>op</mi></msup><mo>→</mo><mi>Bool</mi></mrow><annotation encoding="application/x-tex">CH(-, \mathbf{2}): CH^{op} \to Bool</annotation></semantics></math></div> <p>A <em><a class="existingWikiWord" href="/nlab/show/Stone+space">Stone space</a></em> is by definition a <a class="existingWikiWord" href="/nlab/show/totally+disconnected+topological+space">totally disconnected</a> <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stone</mi><mo>↪</mo><mi>CH</mi></mrow><annotation encoding="application/x-tex">Stone \hookrightarrow CH</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of Stone spaces.</p> <div class="num_theorem"> <h6 id="theorem_stone_representation">Theorem (Stone representation)</h6> <p>The representable functor restricts to an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Stone</mi> <mi>op</mi></msup><mo>→</mo><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Stone^{op} \to Bool</annotation></semantics></math>.</p> </div> <p>This important theorem can be exploited to give a third description of the free Boolean algebra on a set <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>CH</mi><mo stretchy="false">(</mo><msup><mn>2</mn> <mi>X</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bool(X) \cong CH(2^X, \mathbf{2})</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> denotes the 2-element compact Hausdorff space, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">2^X</annotation></semantics></math> the product space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>X</mi></msub><mn>2</mn></mrow><annotation encoding="application/x-tex">\prod_X 2</annotation></semantics></math>. Indeed, the inverse equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Bool</mi> <mi>op</mi></msup><mo>→</mo><mi>Stone</mi></mrow><annotation encoding="application/x-tex">Bool^{op} \to Stone</annotation></semantics></math></div> <p>takes a Boolean algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>, i.e., the space of Boolean algebra maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bool(B, 2)</annotation></semantics></math> (<em>this</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> is the two-element Boolean algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>!) equipped with the <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a>. Applied to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mi>Bool</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B = Bool(X)</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≅</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mn>2</mn> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Bool(B, 2) \cong Set(X, 2) = 2^X</annotation></semantics></math></div> <p>where the Zariski topology coincides with the <a class="existingWikiWord" href="/nlab/show/product+topology">product topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">2^X</annotation></semantics></math>. By the equivalence, we therefore retrieve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bool(X)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CH</mi><mo stretchy="false">(</mo><msup><mn>2</mn> <mi>X</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CH(2^X, \mathbf{2})</annotation></semantics></math>. This in turn is identified with the Boolean algebra of <a class="existingWikiWord" href="/nlab/show/clopen+subset">clopen subset</a>s of the generalised <a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">2^X</annotation></semantics></math>.</p> <p>A second description of the inverse equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Bool</mi> <mi>op</mi></msup><mo>→</mo><mi>Stone</mi></mrow><annotation encoding="application/x-tex">Bool^{op} \to Stone</annotation></semantics></math> comes about through the yoga of <a class="existingWikiWord" href="/nlab/show/ambimorphic+object">ambimorphic objects</a>. Namely, the Boolean compact Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math> can equally well be seen as a <a href="/nlab/show/compact+Hausdorff+object#comphaus">compact Hausdorff object</a> in the category of Boolean algebras. Thus, the representable functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>:</mo><msup><mi>Bool</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Bool(-, \mathbf{2}): Bool^{op} \to Set</annotation></semantics></math> lifts canonically to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Bool</mi> <mi>op</mi></msup><mo>→</mo><mi>CH</mi></mrow><annotation encoding="application/x-tex">Bool^{op} \to CH</annotation></semantics></math></div> <p>and in fact part of the Stone representation theorem is that this factors through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stone</mi><mo>↪</mo><mi>CH</mi></mrow><annotation encoding="application/x-tex">Stone \hookrightarrow CH</annotation></semantics></math> as the inverse equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Bool</mi> <mi>op</mi></msup><mo>→</mo><mi>Stone</mi></mrow><annotation encoding="application/x-tex">Bool^{op} \to Stone</annotation></semantics></math>. In particular this lift determines the topology, providing an description alternative to the description in terms of the Zariski topology (although they are of course the same).</p> <p>An extension of the classical Stone duality to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor) (see the references below).</p> <h3 id="StoneSpacesAndProfiniteSets">Stone spaces and profinite sets</h3> <p>Note that a finite <a class="existingWikiWord" href="/nlab/show/Stone+space">Stone space</a> is necessarily <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete</a>, and these correspond to the finite Boolean algebras, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinSet</mi><mo>≃</mo><mi>FinStoneTop</mi><mo>≃</mo><msup><mi>FinBool</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">FinSet \simeq FinStoneTop \simeq FinBool^{op}</annotation></semantics></math>. However, since Boolean algebras form a <a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+category">locally finitely presentable category</a>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>FinBool</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>FinSet</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Bool \simeq Ind(FinBool) \simeq Pro(FinSet)^{op}</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a> and <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a>). In consequence, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StoneTop</mi><mo>≃</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>FinSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">StoneTop \simeq Pro(FinSet)</annotation></semantics></math>: i.e. <a class="existingWikiWord" href="/nlab/show/Stone+spaces">Stone spaces</a> are equivalent to <em><a class="existingWikiWord" href="/nlab/show/profinite+sets">profinite sets</a></em>, in this context then often called <em><a class="existingWikiWord" href="/nlab/show/profinite+spaces">profinite spaces</a></em>.</p> <p>One way of explaining this classical Stone duality is hence via the following sequence of <a class="existingWikiWord" href="/nlab/show/equivalences+of+categories">equivalences of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Bool</mi><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>FinBool</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><msup><mi>FinSet</mi> <mi>op</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>FinSet</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Bool \simeq Ind(FinBool) \simeq Ind(FinSet^{op}) \simeq Pro(FinSet)^{op} \,, </annotation></semantics></math></div> <p>where “<a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a>” is the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a>, “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi></mrow><annotation encoding="application/x-tex">Ind</annotation></semantics></math>” stands for <a class="existingWikiWord" href="/nlab/show/ind-objects">ind-objects</a>, “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi></mrow><annotation encoding="application/x-tex">Pro</annotation></semantics></math>” for <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">{}^{op}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> and the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>FinSet</mi> <mi>op</mi></msup><mo>≃</mo><mi>FinBool</mi></mrow><annotation encoding="application/x-tex">FinSet^{op} \simeq FinBool</annotation></semantics></math> is that discussed at <em><a href="FinSet#OppositeCategory">FinSet – Opposite category</a></em>.</p> <h3 id="stonean_spaces_and_complete_boolean_algebras">Stonean spaces and complete Boolean algebras</h3> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Stonean+spaces">Stonean spaces</a>, i.e., compact <a class="existingWikiWord" href="/nlab/show/extremally+disconnected">extremally disconnected</a> Hausdorff topological spaces equipped with open continuous maps as morphisms, is contravariantly equivalent to the category of <a class="existingWikiWord" href="/nlab/show/complete+Boolean+algebras">complete Boolean algebras</a> and continuous Boolean homomorphisms as morphisms.</p> <p>See <a class="existingWikiWord" href="/nlab/show/complete+Boolean+algebra">complete Boolean algebra</a> for more information.</p> <h3 id="profinite_algebras">Profinite algebras</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> on <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>, we can talk about <em>Stone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras</em>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras with a compatible Stone topology, and compare the resulting category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Alg</mi><mo stretchy="false">(</mo><mi>Stone</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T Alg(Stone)</annotation></semantics></math> with the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>Fin</mi><mi>T</mi><mi>Alg</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(Fin T Alg)</annotation></semantics></math> of pro-(finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras). The previous duality says that these categories are equivalent when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the identity theory. It is also true in many other cases, such as:</p> <ul> <li>the theory of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>, resulting in the rich theory of <a class="existingWikiWord" href="/nlab/show/profinite+groups">profinite groups</a></li> <li>the theories of <a class="existingWikiWord" href="/nlab/show/semigroups">semigroups</a> and <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a></li> <li>the theory of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> (with or without 1)</li> <li>the theories of <a class="existingWikiWord" href="/nlab/show/distributive+lattices">distributive lattices</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebras">Heyting algebras</a>, and <a class="existingWikiWord" href="/nlab/show/Boolean+algebras">Boolean algebras</a></li> <li>the theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-sets, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a finite monoid</li> <li>the theories of <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>-modules and <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>-algebras, where <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 finite ring</li> </ul> <p>However it is false for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, such as:</p> <ul> <li>the theory 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">\mathbb{N}</annotation></semantics></math>-sets, i.e. sets equipped with an endomorphism</li> <li>the theory of <a class="existingWikiWord" href="/nlab/show/J%C3%B3nsson-Tarski+algebras">Jónsson-Tarski algebras</a></li> <li>the theory of <a class="existingWikiWord" href="/nlab/show/lattices">lattices</a></li> </ul> <p>All of these can be found in chapter VI of Johnstone’s book cited below.</p> <p>The corresponding fact is also notably false for <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gpd</mi><mo stretchy="false">(</mo><mi>Stone</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gpd(Stone)</annotation></semantics></math> is not equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>FinGpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(FinGpd)</annotation></semantics></math>, in contrast to the case for groups. (Of course, groupoids are not described by a Lawvere theory.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntax+-+semantics+duality">syntax - semantics duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abstract+Stone+duality">abstract Stone duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone+gamut">Stone gamut</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Stone+Spaces">Stone Spaces</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olivia+Caramello">Olivia Caramello</a>, <em>A topos-theoretic approach to Stone-type dualities</em>, <a href="http://arxiv.org/abs/1103.3493">arxiv/1103.3493</a> 158 pp.</p> </li> <li> <p>G. D. Dimov, Some generalizations of the Stone Duality Theorem, Publ. Math. Debrecen 80/3-4 (2012), 255–293.</p> </li> <li> <p>G. Dimov, E. Ivanova-Dimova, <a class="existingWikiWord" href="/nlab/show/W.+Tholen">W. Tholen</a>, <em>Categorical extension of dualities: From Stone to de Vries and beyond, I.</em> Appl. Categ. Struct. 30, 287–329 (2022) <a href="https://doi.org/10.1007/s10485-021-09658-6">doi</a></p> </li> <li> <p>H. P. Doctor, <em>The categories of Boolean lattices, Boolean rings and Boolean spaces</em>, Canad. Math. Bulletin 7 (1964), 245–252 <a href="https://doi.org/10.4153/CMB-1964-022-6">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section A.1.1 of <em><a class="existingWikiWord" href="/nlab/show/Spectral+Algebraic+Geometry">Spectral Algebraic Geometry</a></em>.</p> </li> <li id="CCGM24"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Cherubini">Felix Cherubini</a>, <a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>, <a class="existingWikiWord" href="/nlab/show/Freek+Geerligs">Freek Geerligs</a>, <a class="existingWikiWord" href="/nlab/show/Hugo+Moeneclaey">Hugo Moeneclaey</a>, <em>A Foundation for Synthetic Stone Duality</em> (<a href="https://arxiv.org/abs/2412.03203">arXiv:2412.03203</a>)</p> </li> </ul> <p>There is a version in model theory, <a class="existingWikiWord" href="/nlab/show/Makkai+duality">Makkai duality</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/M.+Makkai">M. Makkai</a>, <em>Stone duality for first-order logic</em>, Adv. Math. <strong>65</strong> (1987) no. 2, 97–170, <a href="http://dx.doi.org/10.1016/0001-8708(87)90020-X">doi</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=900266">MR89h:03067</a>; <em>Duality and definability in first order logic</em>, Mem. Amer. Math. Soc. <strong>105</strong> (1993), no. 503</li> </ul> <p>Other variants are in</p> <ul> <li> <p>Henrik Forssell, <em>First-order logical duality</em>, Ph.D. thesis, Carnegie Mellon U. 2008, <a href="http://www.andrew.cmu.edu/user/awodey/students/forssell.pdf">pdf</a></p> </li> <li> <p>Spencer Breiner, <em>Scheme representation for first-order logic</em>, Ph.D. thesis, Carnegie Mellon U. 2014, <a href="https://www.andrew.cmu.edu/user/awodey/students/breiner.pdf">pdf</a></p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry">E-∞ geometry</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section A of <em><a class="existingWikiWord" href="/nlab/show/Proper+Morphisms%2C+Completions%2C+and+the+Grothendieck+Existence+Theorem">Proper Morphisms, Completions, and the Grothendieck Existence Theorem</a></em></li> </ul> <p>Discussion of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>-version of Stone duality is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 3.5.3 of <a class="existingWikiWord" href="/nlab/show/Spectral+Algebraic+Geometry">Spectral Algebraic Geometry</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 6, 2024 at 14:13:42. 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