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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="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> </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> <ul> <li><a href='#sets_groupoids_or_categories'>Sets, groupoids, or categories?</a></li> </ul> <li><a href='#morphisms_of_ionads'>Morphisms of ionads</a></li> <li><a href='#the_topos_of_opens'>The topos of opens</a></li> <li><a href='#bases_of_ionad_structures'>Bases of ionad structures</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>As a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> is a <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a> <a class="existingWikiWord" href="/nlab/show/locale">locale</a>, so an ionad is a categorified <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. While the opens are primary in toposes and locales, the points are primary in ionads and topological spaces.</p> <p><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a> developed the theory of ionads, in which the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math> plays an role analogous to that of the <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{2}^X</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/power+set">power set</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>) in the theory of topological spaces. Intuitively, we are <a class="existingWikiWord" href="/nlab/show/categorification">categorifying</a> the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟚</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathbb{2}\in Set</annotation></semantics></math> to the “categorified subobject classifier” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Set\in Cat</annotation></semantics></math> (which classifies <a class="existingWikiWord" href="/nlab/show/discrete+opfibrations">discrete opfibrations</a>.</p> <p>The word ‘ionad’ is Irish for a location, place, or site; ‘Ionad’ often translates ‘Centre’ in titles of institutions. It is pronounced /ˈɪnəd/ (roughly ‘INN-ad’ or ‘UNN-ad’, not ‘i-NAD’ or ‘yonad’; ‘ЫН-ад’ in a North Slavic language), or more precisely ˈɨ̞n̪ˠəd̪ˠeast in Munster), following <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Irish_phonology">Wikipedia</a>. The plural (which you can use if you like to use ‘topoi’ too) is ‘ionaid’ (/ˈɪnəɟ/, ‘INN-adge’ or ‘UNN-adge’, ‘ЫН-адь’, [nd]). We could go on to decline it out of the nominative case, but now it's getting silly.</p> <h2 id="definition">Definition</h2> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> 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> can be defined by giving its interior operator, an operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo lspace="verythinmathspace">:</mo><msup><mi>𝟚</mi> <mi>X</mi></msup><mo>→</mo><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Int\colon \mathbb{2}^X \to \mathbb{2}^X</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟚</mi></mrow><annotation encoding="application/x-tex">\mathbb{2}</annotation></semantics></math> is the poset of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>) such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊇</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \supseteq Int(A)</annotation></semantics></math>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo stretchy="false">(</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊇</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Int(Int(A)) \supseteq Int(A)</annotation></semantics></math>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Int(X) = X</annotation></semantics></math>, and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Int(A \cap B) = Int(A) \cap Int(B)</annotation></semantics></math></li> </ul> <p>for all <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A, B</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>.</p> <p>In more sophisticated language, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi></mrow><annotation encoding="application/x-tex">Int</annotation></semantics></math> is a finite <a class="existingWikiWord" href="/nlab/show/meet">meet</a>-preserving <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> on the <a class="existingWikiWord" href="/nlab/show/power+set">power set</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>.</p> <p><a class="existingWikiWord" href="/nlab/show/categorification">Categorifying</a> (mostly), we have this:</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>An <strong>ionad</strong> is 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> together with a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">finite limit-preserving</a> <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Int</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">Int_X</annotation></semantics></math> on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math>. An ionad is <strong>bounded</strong> if the comonad is <a class="existingWikiWord" href="/nlab/show/accessible+monad">accessible</a>.</p> </div> <p>Although Garner does not require an ionad to be bounded, the nicest results hold for them, and all of his applications involve only bounded ionads. In fact, Garner writes, ‘Indeed, the existence of unbounded ionads is a problem that seems to be independent of the axioms of <a class="existingWikiWord" href="/nlab/show/Zermelo-Fraenkel+set+theory">Zermelo-Fraenkel set theory</a>.’ (Section 3.8).</p> <div class="un_defn"> <h6 id="remark_garner_remark_39">Remark (Garner Remark 3.9)</h6> <p>Bounded ionads give rise to exactly the Grothendieck toposes with <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a> as their topos of coalgebras. (See below)</p> </div> <h3 id="sets_groupoids_or_categories">Sets, groupoids, or categories?</h3> <p>It may seem odd 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 be a <em>set</em> rather than something else such as a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> or a <a class="existingWikiWord" href="/nlab/show/category">category</a>. An analogous definition can be given where <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 groupoid or a category, of course, but the reason for taking it to be a set is that it makes the analogy to classical topological spaces closer. Consider the following three notions:</p> <ol> <li>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> together with a finite-limit-preserving comonad on its powerset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{2}^X</annotation></semantics></math>.</li> <li>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> equipped with an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> (which we can regard as a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> that happens to be symmetric) together with a finite-limit-preserving comonad on the hom-preorder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{2}^X</annotation></semantics></math>.</li> <li>A <a class="existingWikiWord" href="/nlab/show/preorder">preorder</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> together with a finite-limit-preserving comonad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{2}^X</annotation></semantics></math>.</li> </ol> <p>All three of these <em>induce</em> a topology on the underlying set 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>. But it is (1) that is <em>exactly</em> a topological space: (2) and (3) include the extra data of a (perhaps symmetric) preorder 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> that maps bijectively-on-objects into the <a class="existingWikiWord" href="/nlab/show/specialization+preorder">specialization preorder</a> of that topology.</p> <p>However, as in other cases such as <a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a>/<a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a> and <a class="existingWikiWord" href="/nlab/show/generalized+multicategories">generalized multicategories</a>, another way to “get rid of extra data” is to force it to duplicate data that’s already present (a “completeness” condition). Thus we could consider instead (still in the uncategorified case):</p> <ul> <li>A structure as in (2) above, but such that the given equivalence relation coincides with the relation “<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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are in all the same open sets” (which it automatically <em>implies</em>).</li> <li>A structure as in (3) above, but such that the given preorder coincides with the specialization preorder.</li> </ul> <p>These would give equivalent definitions to (1), but may be better-behaved in some ways. In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> without <a class="existingWikiWord" href="/nlab/show/sets+cover">sets cover</a>, they would no longer be equivalent, but the groupoidal/preorder versions might be better. For ionads the corresponding definitions would be</p> <ul> <li>A groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a finite-limit-preserving comonad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math> such that the induced functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the category of points of the resulting topos is <a class="existingWikiWord" href="/nlab/show/pseudomonic+functor">pseudomonic</a>.</li> <li>A category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a finite-limit-preserving comonad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math> such that the induced functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the category of points of the resulting topos is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a>.</li> </ul> <p>(Asking that these functors also be surjective on objects would be a <a class="existingWikiWord" href="/nlab/show/sober+space">sobriety</a> condition on an ionad.)</p> <p>When categorifying further to “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ionads” and “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-ionads”, the possible options multiply further; but that is probably a topic for another page. For discussion of these questions, see the <a href="https://nforum.ncatlab.org/discussion/5857">nForum thread</a>.</p> <h2 id="morphisms_of_ionads">Morphisms of ionads</h2> <p>Recall that, given two topological spaces <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> from <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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> (on the <a class="existingWikiWord" href="/nlab/show/underlying+set">underlying set</a>s) such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊆</mo><mi>Int</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^*(Int(A)) \subseteq Int(f^*(A))</annotation></semantics></math></li> </ul> <p>for every subset <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> takes the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a>.</p> <p>Categorifying (and adding a coherence law), we have this:</p> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>Given ionads <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, a <strong>continuous map</strong> from <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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> consists of a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> (on the underlying sets of points) together with a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Int</mi> <mi>f</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>∘</mo><msub><mi>Int</mi> <mi>Y</mi></msub><mo>→</mo><msub><mi>Int</mi> <mi>X</mi></msub><mo>∘</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>,</mo></mrow><annotation encoding="application/x-tex"> Int_f\colon f^* \circ Int_Y \to Int_X \circ f^* ,</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msup><mi>Set</mi> <mi>Y</mi></msup><mo>→</mo><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">f^*\colon Set^Y \to Set^X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>f</mi></msup></mrow><annotation encoding="application/x-tex">Set^f</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Int</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Int_f</annotation></semantics></math> ‘respects the comonad structures’.</p> </div> <div class="query"> <p>I need to figure out exactly what this last clause means.</p> </div> <p>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Int</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Int_f</annotation></semantics></math> is part of the <a class="existingWikiWord" href="/nlab/show/extra+structure">structure</a> here; it is not merely a property. (In other words, the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> from ionads to topological spaces is not <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a>.)</p> <p>There is an obvious notion of <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a>, which turns out to be trivial (but probably would not be if <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> were allowed to be a groupoid). However, the category of ionads is presumably (like <a class="existingWikiWord" href="/nlab/show/Top">Top</a>) still a <a class="existingWikiWord" href="/nlab/show/locally+prosetal+2-category">locally prosetal 2-category</a> under the <a class="existingWikiWord" href="/nlab/show/specialisation+order">specialisation order</a>.</p> <h2 id="the_topos_of_opens">The topos of opens</h2> <p>As every topological space has a <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a> (in fact a <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, or dually a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>) of open subsets, so every ionad has a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> (in fact a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>, if it is bounded) of opens.</p> <div class="un_defn"> <h6 id="definition_4">Definition</h6> <p>Given an ionad <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 <strong>open</strong> in <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 simply a <span class="newWikiWord">coalgebra<a href="/nlab/new/algebra+of+a+monad">?</a></span> of the comonad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Int</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">Int_X</annotation></semantics></math>.</p> </div> <p>The opens 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> form a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math>, and we have a <a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective</a> <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math>. In fact, an ionad may be defined as 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> together with a surjective geometric morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Set^X</annotation></semantics></math> to some topos, much as a topological space may be defined as 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> together with a surjective locale morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝟚</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{2}^X</annotation></semantics></math> to some <a class="existingWikiWord" href="/nlab/show/locale">locale</a>. The topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math> is essentially unique by surjectivity, and this also shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/enough+points">enough points</a>.</p> <p>Just as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/frame">frame</a> whenever <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 topological space, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math> should be a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> when <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 ionad. In fact, this holds only for <em>bounded</em> ionads; an ionad may be defined to be bounded if and only if its topos of opens is <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a> (or equivalently a Grothendieck topos).</p> <p>A continous map between topological spaces may be given by a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> and a commuting square</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>Ω</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>𝟚</mi> <mi>Y</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msup><mi>𝟚</mi> <mi>X</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { \Omega(Y) & \to & \Omega(X) \\ \downarrow & & \downarrow \\ \mathbb{2}^Y & \overset{f^*}\to & \mathbb{2}^X } </annotation></semantics></math></div> <p>Similarly, a continuous map between ionads may be given by a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> and a commuting square</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>Ω</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>Set</mi> <mi>Y</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msup><mi>Set</mi> <mi>X</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { \Omega(Y) & \to & \Omega(X) \\ \downarrow & & \downarrow \\ Set^Y & \overset{f^*}\to & Set^X } </annotation></semantics></math></div> <p>Without loss of generality, we may require this square to commute on the nose; this is related to the triviality of <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>-morphisms in the category of ionads.</p> <p>Note that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(Y) \to \Omega(X)</annotation></semantics></math> must be the preimage half of a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(Y)</annotation></semantics></math>; we may also define a continous map <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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to be such a geometric morphism together with a compatible map between the generating points of the toposes.</p> <h2 id="bases_of_ionad_structures">Bases of ionad structures</h2> <p>…</p> <h2 id="references">References</h2> <p>Ionads have been introduced and studied in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <em>Ionads</em>, J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747. (<a href="http://arxiv.org/abs/0912.1415">arXiv</a>) (<a href="http://dx.doi.org/10.1016/j.jpaa.2012.02.013">doi</a>) (<a href="http://comp.mq.edu.au/~rgarner/Papers/Ionads.pdf">author-archived version of published copy</a>)</li> </ul> <p>Some basic aspects of the theory are developed there, and applications to <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, <a class="existingWikiWord" href="/nlab/show/logic">logic</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> are discussed.</p> <p>Further aspects of the theory of ionads, with a focus on their logical importance were studied in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Di+Liberti">Ivan Di Liberti</a>, <em>Towards Higher Topology</em>, J. Pure Appl. Algebra 226 (2022)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Di+Liberti">Ivan Di Liberti</a>, <em>Formal Model Theory & Higher Topology</em>, ArXiv 2020.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 24, 2023 at 21:23:35. 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