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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> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#a_criterion_for_grothendieck_toposes'>A criterion for Grothendieck toposes</a></li> <li><a href='#RelationToSiteMorphisms'>Relation to morphisms of (co)sites</a></li> <li><a href='#some_morphism_calculus'>Some morphism calculus</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#logical_functors_and_etale_geometric_morphisms'>Logical functors and etale geometric morphisms</a></li> <li><a href='#locally_connected_toposes'>Locally connected toposes</a></li> <li><a href='#tiny_objects'>Tiny objects</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">f : E \to F</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of the underlying categories that is consistent with the interpretation of <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> and <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> as generalized <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>Set</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = Set = Sh(*)</annotation></semantics></math> is the terminal <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">E \to Set</annotation></semantics></math> is essential if <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 a <strong><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a></strong>. In general, <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> being essential is a necessary (but not sufficient) condition to ensure that <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> behaves like a map of topological spaces whose <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>s are locally connected: that it is a <a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">(f^* \dashv f_*) : E \to F</annotation></semantics></math>, it is an <strong>essential geometric morphism</strong> if the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> functor <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> has not only the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math>, but also a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>E</mi><mover><mover><munder><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></mover></mover><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}} F \,. </annotation></semantics></math></div></div> <p>A <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">point of a topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">x : Set \to E</annotation></semantics></math> which is given by an essential geometric morphism is called an <strong>essential point</strong> of <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>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>There are various further conditions that can be imposed on a geometric morphism:</p> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> can be made into an <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>-<a class="existingWikiWord" href="/nlab/show/indexed+functor">indexed functor</a> and <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> satisfies some extra conditions, the geometric morphism <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 a <a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a> (see there for details).</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> preserves finite <a class="existingWikiWord" href="/nlab/show/products">products</a> 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 called <strong>connected surjective</strong>.</p> </li> <li> <p>If in addition to the above <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 a <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a> in that there is a further functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>!</mo></msup><mo>:</mo><mi>F</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f^! : F \to E</annotation></semantics></math> which is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>!</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_* \dashv f^!)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful</a> then the geometric morphism <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 called <strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive</a></strong>.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Since for a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <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> the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math> of an essential point is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_E(P,-)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>≇</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">P\ncong\emptyset</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a> and <a class="existingWikiWord" href="/nlab/show/connected">connected</a>, objects satisfying these three conditions are sometimes called <em>essential objects</em> (cf. <a href="#JTT77">Johnstone 1977, p.255</a>).</p> </div> <h2 id="Properties">Properties</h2> <h3 id="a_criterion_for_grothendieck_toposes">A criterion for Grothendieck toposes</h3> <p>The inverse image <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^\ast</annotation></semantics></math> of an essential geometric morphism preserves small limits since it is a right adjoint. Hence, this provides a minimal requirement to satisfy for a general geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f^\ast\dashv f_\ast</annotation></semantics></math> in order to qualify for being essential. In case, the toposes involved are <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a> this condition is not only necessary but sufficient.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f^\ast\dashv f_\ast:\mathcal{E}\to\mathcal{F}</annotation></semantics></math> a geometric morphism between Grothendieck toposes. 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 essential iff <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^\ast</annotation></semantics></math> preserves small limits iff <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^\ast</annotation></semantics></math> preserves small products.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Grothendieck toposes are <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a> and <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^\ast</annotation></semantics></math> has rank i.e. it preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-filtered colimits for some <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> since it is a left adjoint. But by (<a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Borceux vol. 2</a> Thm.5.5.7, p.275) a functor between two locally presentable categories has a left adjoint precisely if it has rank and preserves small limits.</p> <p>Since limits can be constructed from products and equalizers and <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^\ast</annotation></semantics></math> preserves the latter, it preserves small limits precisely when it preserves small products.</p> </div> <h3 id="RelationToSiteMorphisms">Relation to morphisms of (co)sites</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <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> <a class="existingWikiWord" href="/nlab/show/small+categories">small categories</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,Set]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Set]</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a> <a class="existingWikiWord" href="/nlab/show/presheaf+topos">toposes</a>. (If we think of the <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">D^{op}</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/sites">sites</a> equipped with the trivial <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, then these are the corresponding <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a>.)</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>This construction extends to a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><msubsup><mi>Cat</mi> <mi>small</mi> <mi>co</mi></msubsup><mo>→</mo><msub><mi>Topos</mi> <mi>ess</mi></msub></mrow><annotation encoding="application/x-tex"> [-,Set] : Cat_{small}^{co} \to Topos_{ess} </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>small</mi></msub></mrow><annotation encoding="application/x-tex">{}_{small}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a>s reversed) to the sub-2-category of <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a> on essential geometric morphisms, where a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f : C \to D</annotation></semantics></math> is sent to the essential geometric morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mover><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mo>=</mo><msub><mi>Ran</mi> <mi>f</mi></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>:</mo><mo>=</mo><msub><mi>Lan</mi> <mi>f</mi></msub></mrow></mover></mover><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_*) : [C,Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f}{\to}}} [D,Set] \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Lan_f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">Ran_f</annotation></semantics></math> denote the left and right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> along <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>, respectively.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>This 2-functor is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a> when restricted to <a class="existingWikiWord" href="/nlab/show/Cauchy+complete+categories">Cauchy complete categories</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><msubsup><mi>Cat</mi> <mi>CauchyComp</mi> <mi>co</mi></msubsup><mo>↪</mo><msub><mi>Topos</mi> <mi>ess</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess} \,. </annotation></semantics></math></div> <p>For all small categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>,</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C,D</annotation></semantics></math> we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mover><mi>C</mi><mo>¯</mo></mover><mo>,</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>≃</mo><msub><mi>Topos</mi> <mi>ess</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Func(\overline{C},\overline{D})^{op} \simeq Topos_{ess}([C,Set], [D,Set]) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> between the <a class="existingWikiWord" href="/nlab/show/Cauchy+completion">Cauchy completion</a>s 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> and <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> and the the category of essential geometric morphisms between the copresheaf toposes and <a class="existingWikiWord" href="/nlab/show/geometric+transformation">geometric transformation</a>s between them.</p> </div> <p>In particular, since every <a class="existingWikiWord" href="/nlab/show/poset">poset</a> – when regarded as a <a class="existingWikiWord" href="/nlab/show/category">category</a> – is Cauchy complete, we have</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>The <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mi>Poset</mi><mo>→</mo><msub><mi>Topos</mi> <mi>ess</mi></msub></mrow><annotation encoding="application/x-tex"> [-,Set] : Poset \to Topos_{ess} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Sometimes it is useful to decompose this statement as follows.</p> <p>There is a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mo>:</mo><mi>Poset</mi><mo>→</mo><mi>Locale</mi></mrow><annotation encoding="application/x-tex"> Alex : Poset \to Locale </annotation></semantics></math></div> <p>which assigns to each <a class="existingWikiWord" href="/nlab/show/poset">poset</a> a <a class="existingWikiWord" href="/nlab/show/locale">locale</a> called its <a class="existingWikiWord" href="/nlab/show/Alexandroff+locale">Alexandroff locale</a>. By a theorem discussed there, a morphisms of locales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> is in the image of this functor precisely if its inverse image morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>Op</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^* Op(Y) \to Op(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/frame">frame</a>s has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> in the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Locale">Locale</a>.</p> <p>Moreover, for any poset <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> the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alex</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">Alex P</annotation></semantics></math> is naturally equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>P</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[P,Set]</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>Sh</mi><mo>∘</mo><mi>Alex</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [-,Set] \simeq Sh \circ Alex \,. </annotation></semantics></math></div> <p>With this, the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mi>Poset</mi><mo>→</mo><mi>Topos</mi></mrow><annotation encoding="application/x-tex">[-,Set] : Poset \to Topos</annotation></semantics></math> hits precisely the essential geometric morphisms follows with the basic fact about <a class="existingWikiWord" href="/nlab/show/localic+reflection">localic reflection</a>, which says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo>:</mo><mi>Locale</mi><mo>→</mo><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Sh : Locale \to Topos</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a>.</p> </div> <h3 id="some_morphism_calculus">Some morphism calculus</h3> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">f : E \to F</annotation></semantics></math> be an essential geometric morphism.</p> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi : X \to f^* f_* A</annotation></semantics></math> 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">E</annotation></semantics></math> the 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><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{\phi}{\to}& f^* f_* A \\ \downarrow && \downarrow \\ f^* f_! X &\stackrel{}{\to}& A } </annotation></semantics></math></div> <p>commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> under the composite adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^* f_! \dashv f^* f_*)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi : X \to f^* f_* A</annotation></semantics></math> is the component of 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><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>X</mi></mover></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>A</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>ϕ</mi></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ *&&\overset{X}{\to}&& E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow \\ E &\underset{f_*}{\to}&F } \,. </annotation></semantics></math></div> <p>The composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>ϕ</mi></mover><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\phi}{\to} f^* f_* A \to A</annotation></semantics></math> is the component of this composed with the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mo>⇒</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">f^* f_* \Rightarrow Id</annotation></semantics></math>.</p> <p>We may insert the 2-identity given by the zig-zag law</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>X</mi></mover></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>A</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>ϕ</mi></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></msup></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder></mtd> <mtd><mi>F</mi></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd> <mtd></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \;\;\; = \;\;\; \array{ *&&\overset{X}{\to}&& E && = && E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!} &\Downarrow& \nearrow_{\mathrlap{f^*}} \\ E &\underset{f_*}{\to}&F &&=&& F } \,. </annotation></semantics></math></div> <p>Composing this with the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>*</mo></msub><mo>⇒</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">f^* f_* \Rightarrow Id</annotation></semantics></math> produces the transformation whose component is manifestly the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to f^* f_! X \to A</annotation></semantics></math>.</p> </div> <h2 id="examples">Examples</h2> <h3 id="logical_functors_and_etale_geometric_morphisms">Logical functors and etale geometric morphisms</h3> <p>A <a class="existingWikiWord" href="/nlab/show/logical+functor">logical functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">E\to F</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> has automatically also a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> whence is the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> part of an essential geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">F\to E</annotation></semantics></math>.</p> <p>A particularly important instance of this situation is the following:</p> <p>For any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A\to B</annotation></semantics></math> in a topos <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>, the induced geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi><mo>→</mo><mi>E</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon E/A \to E/B</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es is essential. Here, the logical functor is given by the pullback functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>E</mi><mo stretchy="false">/</mo><mi>B</mi><mo>→</mo><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f^*:E/B\to E/A</annotation></semantics></math> of course.</p> <p>In the special case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">B = *</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, the essential geometric morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \pi : E/A \to E </annotation></semantics></math></div> <p>is also called an <a class="existingWikiWord" href="/nlab/show/etale+geometric+morphism">etale geometric morphism</a>.</p> <p>Conversely, (almost) any essential geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\pi^*</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/logical+functor">logical functor</a> has the form of an <a class="existingWikiWord" href="/nlab/show/etale+geometric+morphism">etale geometric morphism</a>:</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>If the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">\pi_!</annotation></semantics></math> of a logical functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mo>*</mo></msup><mo>:</mo><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\pi^*:E\to F</annotation></semantics></math> furthermore preserves equalizers, then the corresponding essential geometric morphism is up to equivalence an <a class="existingWikiWord" href="/nlab/show/etale+geometric+morphism">etale geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>F</mi><mo>≃</mo><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\pi : F\simeq E/A \to E</annotation></semantics></math> for an object <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> 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">E</annotation></semantics></math> that is determined up to isomorphism.</p> </div> <p>For a proof see e.g. Johnstone <a href="#JTT77">(1977, p.37)</a>.</p> <h3 id="locally_connected_toposes">Locally connected toposes</h3> <p>A <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> <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 one where the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma : E \to Set</annotation></semantics></math> is essential.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>E</mi><mover><mover><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{LConst}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set \,. </annotation></semantics></math></div> <p>In this case, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mo>!</mo></msub><mo>=</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>:</mo><mi>E</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma_! = \Pi_0 : E \to Set</annotation></semantics></math> sends each object to its set of connected components. More on this situation is at <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy groups in an (∞,1)-topos</a>.</p> <p>Note, though that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">p\colon E\to S</annotation></semantics></math> is an arbitrary geometric morphism through which we regard <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> as an <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>-topos, i.e. a topos “in the world 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>,” the condition 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">E</annotation></semantics></math> to be locally connected as an <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>-topos is not just that <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> is essential, but that the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">p_!</annotation></semantics></math> can be made into an <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>-<a class="existingWikiWord" href="/nlab/show/indexed+functor">indexed functor</a> (which is automatically true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p_*</annotation></semantics></math>). This is automatically the case for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>-toposes (at least, when our <a class="existingWikiWord" href="/nlab/show/foundation">foundation</a> is <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a>—and if our foundation is <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a>, then our large categories and functors all need to be assumed to be <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>-indexed anyway). For more see <a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a>.</p> <h3 id="tiny_objects">Tiny objects</h3> <p>The <a class="existingWikiWord" href="/nlab/show/tiny+object">tiny object</a>s of a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,Set]</annotation></semantics></math> are precisely the essential points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Set \to [C,Set]</annotation></semantics></math>. See <a class="existingWikiWord" href="/nlab/show/tiny+object">tiny object</a> for details.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+subtopos">essential subtopos</a> / <a class="existingWikiWord" href="/nlab/show/level+of+a+topos">level of a 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/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+connected+%28%E2%88%9E%2C1%29-topos">totally connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="references">References</h2> <p>Like many other things, it all started as an exercise in</p> <ul> <li id="SGA4"><a class="existingWikiWord" href="/nlab/show/M.+Artin">M. Artin</a>, <a class="existingWikiWord" href="/nlab/show/A.+Grothendieck">A. Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/J.+L.+Verdier">J. L. Verdier</a>, <em>Théorie des Topos et Cohomologie Etale des Schémas (<a class="existingWikiWord" href="/nlab/show/SGA4">SGA4</a>)</em>, LNM <strong>269</strong> Springer Heidelberg 1972. (exposé IV, ex.7.6, pp.414-417)</li> </ul> <p>Speaking of exercises, consider the results of <a class="existingWikiWord" href="/nlab/show/Jan+Erik+Roos">Roos</a> on essential points reported in exercise 7.3 of</p> <ul> <li id="JTT77"><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos Theory</em> , Academic Press New York 1977. (Dover reprint 2014, pp.254f)</li> </ul> <p>The case of sheaves valued in <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> is considered in</p> <ul> <li>J. Haigh, <em>Essential geometric morphisms between toposes of finite sets</em> , Math. Proc. Phil. Soc. <strong>87</strong> (1980) pp.21-24.</li> </ul> <p>The standard reference for <em>essential localizations</em> <sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup>, aka <a class="existingWikiWord" href="/nlab/show/level+of+a+topos">levels</a>, is</p> <ul> <li id="KL89"><a class="existingWikiWord" href="/nlab/show/G.+M.+Kelly">G. M. Kelly</a>, <a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <em>On the Complete Lattice of Essential Localizations</em> , Bull. Soc. Math. de Belgique <strong>XLI</strong> (1989) 289-319 [<a class="existingWikiWord" href="/nlab/files/Kelly-Lawvere_EssentialLocalizations.pdf" title="pdf">pdf</a>]</li> </ul> <p>For more on this see also</p> <ul> <li id="Lima15"><a class="existingWikiWord" href="/nlab/show/Guilherme+Frederico+Lima">Guilherme Frederico Lima</a>, <em>From Essential Inclusions to Local Geometric Morphisms</em>, talk at <a href="https://indico.math.cnrs.fr/event/747/">Topos à l’IHES</a>, November 2015, Paris (<a href="https://www.youtube.com/watch?v=YsoGN91Rh_s">video</a>)</li> </ul> <p>The definition of essential geometric morphisms appears before Lemma A.4.1.5 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a> vol. I</em> , Oxford UP 2002.</li> </ul> <p>Connected surjective and local geometric morphisms are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Local maps of toposes</em> , Proceedings London Mathematical Society <strong>58</strong> (1989), pp.281-305.</li> </ul> <p>Further refinements are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em>Axiomatic cohesion</em> Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (<a href="http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf">pdf</a>)</li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>See at <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a> for further references on essential localizations. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on March 27, 2023 at 07:20:19. 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