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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> <blockquote> <p>This entry is about the notion of <em>frame</em> in <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>. For other notions, see <a class="existingWikiWord" href="/nlab/show/frame+%28disambiguation%29">frame (disambiguation)</a>.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <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='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#As0Topos'>As a <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>-topos</a></li> <li><a href='#GeneralProperties'>General</a></li> <li><a href='#as_sites'>As sites</a></li> </ul> <li><a href='#formal_duals_locales'>Formal duals: locales</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>frame</em> is a generalization of the notion of <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. A <em>frame</em> is like a category of open subsets in a space possibly more general than a topological space: a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>. This in turn is effectively defined to be anything that has a collection of open subsets that behaves essentially like the open subsets of a topological space do.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>frame</strong> <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">\mathcal{O}</annotation></semantics></math> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/partial+order">poset</a></p> </li> <li> <p>that has</p> <ul> <li> <p>all <a class="existingWikiWord" href="/nlab/show/small+limit">small</a> <a class="existingWikiWord" href="/nlab/show/coproduct"> coproducts</a>, called <a class="existingWikiWord" href="/nlab/show/join">joins</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</annotation></semantics></math></p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, called <a class="existingWikiWord" href="/nlab/show/meet">meets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></p> </li> </ul> </li> <li> <p>and which satisfies the <em>infinite distributive law</em>:</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><msub><mi>y</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>≤</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mi>x</mi><mo>∧</mo><msub><mi>y</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> x \wedge (\bigvee_i y_i) \leq \bigvee_i (x\wedge y_i) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>y</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x, \{y_i\}_i</annotation></semantics></math> in <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></p> <p>(Note that the converse holds in any case, so we have equality.)</p> </li> </ul> <p>A <em>frame homomorphism</em> is a homomorphism of posets that preserves finite meets and arbitrary joins. Frames and frame homomorphisms form the category <a class="existingWikiWord" href="/nlab/show/Frm">Frm</a>.</p> <p>The formal duals of frames, hence the objects in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <a class="existingWikiWord" href="/nlab/show/Locale">Locale</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">:=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Frm">Frm</a><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> are called <a class="existingWikiWord" href="/nlab/show/locale"> locales</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The notation <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">\mathcal{O}</annotation></semantics></math> to supposed to remind us that every frame is like a <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/locale">locale</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>So in particular a frame is a <em><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a></em> which has not only finite <a class="existingWikiWord" href="/nlab/show/joins">joins</a> but all small joins. Being small itself, it is a <a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a>, and hence a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>: it also has all (small) meets.</p> <p>Furthermore, as the distributive law certainly holds when the joins in question are finite, it is a <em><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a></em>.</p> </div> <h2 id="properties">Properties</h2> <p>A useful way to understand frames and <a class="existingWikiWord" href="/nlab/show/locales">locales</a> is as the simplest nontrivial special cases of <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos"> (n,1)-toposes</a>. So we start in</p> <ul> <li><a href="#As0Topos">As (0,1)-toposes</a></li> </ul> <p>with some remarks on this, and only then turn to</p> <ul> <li><a href="#GeneralProperties">General properties</a></li> </ul> <p>of frames, which should make more sense this way.</p> <h3 id="As0Topos">As a <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>-topos</h3> <p>The notion of <em>frame</em> – or rather its formal dual, the notion of <em><a class="existingWikiWord" href="/nlab/show/locale">locale</a></em> – is the special case of the notion of <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos"> (n,1)-toposes</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos"> (0,1)-toposes</a>.</p> <p>The axioms on a frame are nothing but <a class="existingWikiWord" href="/nlab/show/Giraud%27s+axioms">Giraud's axioms</a> on <a class="existingWikiWord" href="/nlab/show/sheaf+topos"> sheaf toposes</a>, restricted to <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-categories">(0,1)-categories</a>:</p> <p>given the existence of finite limits and arbitrary colimits, the <em>infinite distributive law</em> expresses that a frame has <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a>: they are stable under <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>. (For notice that in a poset <a class="existingWikiWord" href="/nlab/show/pullback"> pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/product"> products</a> coincide.)</p> <p>Then a morphism of frames is precisely (the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> of) a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>: a morphism preserving finite limits and arbitrary colimits.</p> <h3 id="GeneralProperties">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A frame is automatically a <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>.</p> </div> <p>In <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> terms this means that it ia a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a>, hence that</p> <p>for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{O}</annotation></semantics></math> the functor</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>𝒪</mi><mo>→</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex"> x \wedge (-) : \mathcal{O} \to \mathcal{O} </annotation></semantics></math></div> <p>that forms the <a class="existingWikiWord" href="/nlab/show/product">product</a> with <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> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>𝒪</mi><mo>→</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex"> x \Rightarrow (-) : \mathcal{O} \to \mathcal{O} </annotation></semantics></math></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This exists by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>, using that there is only a finite number of morphisms between any two objects (one or none) and that finite limits exist in <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">\mathcal{O}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Therefore one may think of a of a frame as a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a> <a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a>, or a cartesian <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a>.</p> <p>But notice that the frame <a class="existingWikiWord" href="/nlab/show/homomorphism"> homomorphisms</a> are not required to preserve the Heyting implication.</p> </div> <h3 id="as_sites">As sites</h3> <p>A frame is naturally equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/site">site</a>:</p> <p>a family of morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/covering">covering</a> precisely if <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> is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>U</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \bigvee_i U_i = U \,. </annotation></semantics></math></div> <p>For more on this see <a class="existingWikiWord" href="/nlab/show/locale">locale</a>.</p> <h2 id="formal_duals_locales">Formal duals: locales</h2> <p>The <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> to the category <a class="existingWikiWord" href="/nlab/show/Frm">Frm</a> is the category <a class="existingWikiWord" href="/nlab/show/Loc">Loc</a> of <a class="existingWikiWord" href="/nlab/show/locale"> locales</a> (possibly slightly generalized <a class="existingWikiWord" href="/nlab/show/topological+space"> topological spaces</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Loc</mi><mo>:</mo><mo>=</mo><msup><mi>Frm</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> Loc := Frm^{op} </annotation></semantics></math></div> <p>Conversely, any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> has a <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of open subsets</a>. (In fact, one definition of a topological space is a set equipped with a subframe of its <a class="existingWikiWord" href="/nlab/show/power+set">power set</a>.)</p> <h2 id="examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> 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> is a frame: the <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/set">set</a>, the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(X)</annotation></semantics></math> 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> is the <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a> corresponding to the <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a> 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> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/poset+of+truth+values">poset of truth values</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/frame">frame</a> equivalent to the <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a> corresponding to the <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a> on a <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>For <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">\mathcal{E}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/geometric+category">geometric 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> an <a class="existingWikiWord" href="/nlab/show/object">object</a> in <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">\mathcal{E}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/subobject+poset">subobject poset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Sub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Sub}(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/frame">frame</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/sup-lattice">complete</a> <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable</a> <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a> is a frame.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>For any <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> and collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i</annotation></semantics></math>, the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><mi>x</mi><mo>∧</mo><msub><mi>y</mi> <mi>i</mi></msub><mo>≤</mo><mi>x</mi><mo>∧</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\bigvee_i x \wedge y_i \leq x \wedge \bigvee_i y_i</annotation></semantics></math> holds automatically. For the reverse inequality, we note this follows trivially in case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\bigvee_i y_i \leq x</annotation></semantics></math>, since in that case we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>i</mi></msub><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y_i \leq x</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, whence</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><msub><mi>y</mi> <mi>i</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><msub><mi>y</mi> <mi>i</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><mi>x</mi><mo>∧</mo><msub><mi>y</mi> <mi>i</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">x \wedge \bigvee_i y_i = \bigvee_i y_i = \bigvee_i x \wedge y_i.</annotation></semantics></math></div> <p>Otherwise we are in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo><</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x \lt \bigvee_i y_i</annotation></semantics></math>, where we must show the inequality</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><msub><mi>y</mi> <mi>i</mi></msub><mo>=</mo><mi>x</mi><mo>≤</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></munder><mi>x</mi><mo>∧</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x \wedge \bigvee_i y_i = x \leq \bigvee_i x \wedge y_i</annotation></semantics></math></div> <p>But this inequality must hold, else <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><mi>x</mi><mo>∧</mo><msub><mi>y</mi> <mi>i</mi></msub><mo><</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\bigvee_i x \wedge y_i \lt x</annotation></semantics></math> which would imply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>i</mi></msub><mo><</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y_i \lt x</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, whence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\bigvee_i y_i \leq x</annotation></semantics></math>, contradiction.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></li> <li><a class="existingWikiWord" href="/nlab/show/coframe">coframe</a></li> </ul> <h2 id="references">References</h2> <p>Frames in <a class="existingWikiWord" href="/nlab/show/univalent+foundations">univalent foundations</a>:</p> <ul> <li>Ayberk Tosun, <em>Formal Topology in Univalent Foundations</em>, (<a href="https://odr.chalmers.se/handle/20.500.12380/301098">pdf</a>, <a href="https://www.cs.bham.ac.uk/~axt978/talks/lab-lunch-formal-topology.pdf">slides</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 13, 2024 at 22:32:33. 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