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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Def'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#the_category_of_algebraic_lattices'>The category of algebraic lattices</a></li> <li><a href='#RelationToLocallyFinitelyPresentableCategories'>Relation to locally finitely presentable categories</a></li> <li><a href='#congruence_lattices'>Congruence lattices</a></li> <li><a href='#completely_distributive_lattices'>Completely distributive lattices</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Def">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>An <strong>algebraic lattice</strong> is a <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> which is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>;</p> </li> <li> <p>such that every element is a <a class="existingWikiWord" href="/nlab/show/join">join</a> of <a class="existingWikiWord" href="/nlab/show/compact+elements">compact elements</a>.</p> </li> </ul> </div> <p>An <strong>algebraic lattice</strong> is a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a> (equivalently, a <a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a>, or in different words a <a class="existingWikiWord" href="/nlab/show/poset">poset</a> with the <a class="existingWikiWord" href="/nlab/show/extra+property">property</a> of having arbitrary <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> but with the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of <a class="existingWikiWord" href="/nlab/show/directed+colimits">directed colimits</a>/<a class="existingWikiWord" href="/nlab/show/directed+joins">directed joins</a>) in which every element is the <a class="existingWikiWord" href="/nlab/show/supremum">supremum</a> of the <a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a>s below it (an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is compact if, for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> of the lattice, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is less than or equal to the supremum of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> just in case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is less than or equal to the supremum of some finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>).</p> <p>Here is an alternative formulation:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>An algebraic lattice is a <a class="existingWikiWord" href="/nlab/show/poset">poset</a> which is <a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+category">locally finitely presentable</a> as a category.</p> </div> <p>This formulation suggests a useful way of viewing algebraic lattices in terms of <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a> (but with regard to enrichment in <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>, instead of in <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>).</p> <p>As this last formulation suggests, algebraic lattices typically arise as <a class="existingWikiWord" href="/nlab/show/subobject+lattices">subobject lattices</a> for objects in locally finitely presentable categories. As an example, for any (finitary) <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> <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>, the subobject lattice of an object in <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi></mrow><annotation encoding="application/x-tex">Alg</annotation></semantics></math> is an algebraic lattice (this class of examples explains the origin of the term “algebraic lattice”, which is due to Garrett Birkhoff). In fact, all algebraic lattices arise this way (see Theorem <a class="maruku-ref" href="#GS"></a> below).</p> <p>It is trivial that every finite lattice is algebraic.</p> <h2 id="properties">Properties</h2> <h3 id="the_category_of_algebraic_lattices">The category of algebraic lattices</h3> <p>The <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> most commonly considered between algebraic lattices are the <span class="newWikiWord">finitary functors<a href="/nlab/new/finitary+functors">?</a></span> between them, which is to say, the <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott-continuous</a> functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks <a href="#Def">above</a>).</p> <p>The resulting category <strong>AlgLat</strong> is <a class="existingWikiWord" href="/nlab/show/cartesian+closed">cartesian closed</a> and is dually equivalent to the category whose objects are <a class="existingWikiWord" href="/nlab/show/meet+semilattices">meet semilattices</a> (construed as categories with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>) and whose morphisms are meet-preserving <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> between them (using the convention that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched profunctor from <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> to <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> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">D^{op} \times C \rightarrow V</annotation></semantics></math>; of course, with an opposite convention, one could similarly state a covariant equivalence).</p> <p>There is a <em>full</em> <a class="existingWikiWord" href="/nlab/show/full+embedding">embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>AlgLat</mi><mo>→</mo><msub><mi>Top</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">i \colon AlgLat \to Top_0</annotation></semantics></math></div> <p>to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/separation+axioms">spaces</a>, taking an algebraic lattice <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> to the space whose points are elements of <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>, and whose <a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> are defined by the property that their <a class="existingWikiWord" href="/nlab/show/characteristic+maps">characteristic maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>U</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\chi_U: L \to \mathbf{2}</annotation></semantics></math></div> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\chi_U(a) = 1</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">a \in U</annotation></semantics></math>, else <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\chi_U(a) = 0</annotation></semantics></math>) are poset maps that preserve <a class="existingWikiWord" href="/nlab/show/directed+colimits">directed colimits</a>. The <a class="existingWikiWord" href="/nlab/show/specialization+order">specialization order</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(L)</annotation></semantics></math> is <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> again.</p> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math>-space <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> occurs as a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of some space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(L)</annotation></semantics></math> associated with an algebraic lattice. Explicitly, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(X)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a> of the underlying set of the <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">P{|\mathcal{O}(X)|}</annotation></semantics></math>, and define</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><mo stretchy="false">(</mo><mi>i</mi><mo>∘</mo><mi>L</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to (i\circ L)(X)</annotation></semantics></math></div> <p>to take <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">{</mo><mi>U</mi><mo>∈</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>x</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}</annotation></semantics></math>. This gives a topological embedding of <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(L(X))</annotation></semantics></math>.</p> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>On similar grounds, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>AlgLat</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U \colon AlgLat \to Set</annotation></semantics></math> is the forgetful functor, then the <a href="http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property#a_factorisation_system_14">2-image</a> of the projection functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi \colon Set\downarrow U \to Set</annotation></semantics></math> is the category of topological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>. In more nuts-and-bolts terms, an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>L</mi><mo>,</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, L, f \colon S \to U(L))</annotation></semantics></math> gives a space with underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and open sets those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(O)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> ranges over the Scott topology on <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>. Notice that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>S</mi><mo>′</mo><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>L</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f \colon S \to S', g \colon L \to L')</annotation></semantics></math> is a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">Set \downarrow U</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous with respect to these topologies. Therefore the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi \colon Set \downarrow U \to Set</annotation></semantics></math> factors through the faithful forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Top \to Set</annotation></semantics></math>. Thus, working in the factorization system (eso+full, faithful) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, we have a faithful functor <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">im(\pi) \to Top</annotation></semantics></math> filling in as the diagonal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Top</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>2</mn><mtext>-</mtext><mi>im</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Set</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ Set \downarrow U & \to & Top \\ \downarrow & \nearrow & \downarrow \\ 2\text{-}im(\pi) & \to & Set. } </annotation></semantics></math></div> <p>But notice also that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Set \downarrow U \to Top</annotation></semantics></math> is <a href="http://ncatlab.org/nlab/show/ternary+factorization+system#examples_9">eso and full</a>. It is eso because any topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(S)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> can be reconstituted from the triple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>,</mo><mi>x</mi><mo>↦</mo><mi>N</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|})</annotation></semantics></math>. We claim it is full as well. For, every continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X \to X'</annotation></semantics></math> between topological spaces induces a continuous map between their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> reflections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>′</mo></mrow><annotation encoding="application/x-tex">X_0 \to X_{0}'</annotation></semantics></math>, and since algebraic lattices like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">P{|\mathcal{O}(X)|}</annotation></semantics></math> (being continuous lattices) are <a class="existingWikiWord" href="/nlab/show/injective+objects">injective objects</a> in the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> spaces, we are able to complete to a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi><mrow><mo stretchy="false">|</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\ \downarrow & & \downarrow & & \downarrow \\ X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|} } </annotation></semantics></math></div> <p>where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>N</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \mapsto N(x)</annotation></semantics></math>). Finally, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo stretchy="false">↓</mo><mi>U</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Set \downarrow U \to Top</annotation></semantics></math> is eso and full, it follows that <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">im(\pi) \to Top</annotation></semantics></math> is eso, full, and faithful, and therefore an equivalence of categories.</p> <p>This connection is explored in more depth with the category of <a class="existingWikiWord" href="/nlab/show/equilogical+spaces">equilogical spaces</a>, which can be seen either as a category of (set-theoretic) <a class="existingWikiWord" href="/nlab/show/equivalence+relation">partial equivalence relations</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AlgLat</mi></mrow><annotation encoding="application/x-tex">AlgLat</annotation></semantics></math>, or equivalently of (set-theoretic) total <a class="existingWikiWord" href="/nlab/show/equivalence+relations">equivalence relations</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> topological spaces.</p> </div> <h3 id="RelationToLocallyFinitelyPresentableCategories">Relation to locally finitely presentable categories</h3> <p>One of our definitions of algebraic lattice is: a poset <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> which is locally finitely presentable when viewed as a category. The completeness of <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> means that right adjoints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">L \to Set</annotation></semantics></math> are representable, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">L(p, -) \colon L \to Set</annotation></semantics></math>, and we are particularly interested in those representable functors that preserve <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>. These correspond precisely to finitely presentable objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.</p> <p>By <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a>, <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 determined from the join-semilattice of compact elements <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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≅</mo><mi>Lex</mi><mo stretchy="false">(</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \cong Lex(K^{op}, Set)</annotation></semantics></math>. Since the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math> are subterminal, we can also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≅</mo><mi>Lex</mi><mo stretchy="false">(</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \cong Lex(K^{op}, 2)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>=</mo><mi>Sub</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 = Sub(1)</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Porst)</strong></p> <p>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 a <a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+category">locally finitely presentable category</a> and <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> is an object 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>, then</p> <ul> <li> <p>The lattice of subobjects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sub(X)</annotation></semantics></math>,</p> </li> <li> <p>the lattice of quotient objects (equivalence classes of epis sourced at <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>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Quot</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Quot(X)</annotation></semantics></math>,</p> </li> <li> <p>the lattice of congruences (internal equivalence relations) on <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> </li> </ul> <p>are all algebraic lattices.</p> </div> <p>This is due to <a href="#Porst">Porst</a>. Of course 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 the category of algebras of an Lawvere theory, then the lattice of quotient objects of an algebra is isomorphic to its congruence lattice, as such <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/exact+category">exact category</a>.</p> <h3 id="congruence_lattices">Congruence lattices</h3> <p>The following result is due to Grätzer and Schmidt:</p> <div class="num_theorem" id="GS"> <h6 id="theorem_2">Theorem</h6> <p>Every algebraic lattice is isomorphic to the congruence lattice of some <a class="existingWikiWord" href="/nlab/show/model">model</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> of some finitary algebraic theory.</p> </div> <p>In particular, since every finite lattice is algebraic, every finite lattice arises this way. Remarkably, it is not known at this time whether every finite lattice arises as the congruence lattice of a <em>finite</em> algebra <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>. It has been conjectured that this is in fact <strong>false</strong>: see this <a href="http://mathoverflow.net/a/196074/2926">MO discussion</a>.</p> <p>Another problem which had long remained open is the congruence lattice problem: is every <em>distributive</em> algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice <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>? The answer is negative, as shown by Wehrung in 2007: see this <a href="http://en.m.wikipedia.org/wiki/Congruence_lattice_problem">Wikipedia article</a>.</p> <h3 id="completely_distributive_lattices">Completely distributive lattices</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The category of <a class="existingWikiWord" href="/nlab/show/Alexandroff+locales">Alexandroff locales</a> is equivalent to that of <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive</a> algebraic lattices.</p> </div> <p>This appears as (<a href="#Caramello">Caramello, remark 4.3</a>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive</a> algebraic lattices form a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> of that of all distributive lattices. The reflector is called <em><a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <p>See also <a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a>, <span class="newWikiWord">compact element in a locale<a href="/nlab/new/compact+element+in+a+locale">?</a></span>.</p> <div> <p><strong>Locally presentable categories:</strong> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">Cocomplete</a> possibly-<a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> generated under <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> by <a class="existingWikiWord" href="/nlab/show/small+object">small</a> <a class="existingWikiWord" href="/nlab/show/generators">generators</a> under <a class="existingWikiWord" href="/nlab/show/small+colimit">small</a> <a class="existingWikiWord" href="/nlab/show/relations">relations</a>. Equivalently, <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/reflective+localizations">reflective localizations</a> of <a class="existingWikiWord" href="/nlab/show/free+cocompletions">free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> localization.</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/toposes">toposes</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locales">locales</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+lattices">algebraic lattices</a></td><td style="text-align: left;"><a href="algebraic+lattice#RelationToLocallyFinitelyPresentableCategories">Porst’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/powerset">powerset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/poset">poset</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a></td><td style="text-align: left;"><a href="locally+presentable+category#AsLocalizationsOfPresheafCategories">Gabriel–Ulmer’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a></td><td style="text-align: left;">global <a class="existingWikiWord" href="/nlab/show/model+structures+on+simplicial+presheaves">model structures on simplicial presheaves</a></td><td style="text-align: left;">n/a</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="locally+presentable+infinity-category#Definition">Simpson’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-categories">(∞,1)-presheaf (∞,1)-categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <ul id="Caramello"> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <a class="existingWikiWord" href="/nlab/show/Lars+Birkedal">Lars Birkedal</a>, <a class="existingWikiWord" href="/nlab/show/Dana+Scott">Dana Scott</a>, <em>Equilogical Spaces</em>, Theoretical Computer Science, 315(1):35-59, 2004. (<a href="http://math.andrej.com/2002/07/05/equilogical-spaces/">web</a>)</p> </li> <li> <p>Olivia Caramello, <em>A topos-theoretic approach to Stone-type dualities</em> (<a href="http://arxiv.org/abs/1103.3493">arXiv:1103.3493</a>)</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a> is discussed in</p> <ul id="Porst"> <li><a class="existingWikiWord" href="/nlab/show/Hans+Porst">Hans Porst</a>, <em>Algebraic lattices and locally finitely presentable categories</em>, Algebra Univers. <strong>65</strong>, 285–298 (2011). <a href="https://doi.org/10.1007/s00012-011-0129-0">https://doi.org/10.1007/s00012-011-0129-0</a> (<a href="http://www.math.uni-bremen.de/~porst/dvis/PORST_AlgebraicLattices_revfinAU.pdf">pdf</a>)</li> </ul> <p>That every algebraic lattice is a congruence lattice is proved in</p> <ul> <li>G. Grätzer and E. T. Schmidt, <em>Characterizations of congruence lattices of abstract algebras</em>, Acta Sci. Math. (Szeged) 24 (1963), 34–59.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 16, 2020 at 06:50:54. 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