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href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#the_free_suplattice_on_a_poset'>The free suplattice on a poset</a></li> <li><a href='#the_category_of_suplattices'>The category of suplattices</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong>suplattice</strong> is a <a class="existingWikiWord" href="/nlab/show/poset">poset</a> that has <a class="existingWikiWord" href="/nlab/show/joins">joins</a> of arbitrary <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> (and in particular is a join-<a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a>). By the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> for posets, a suplattice necessarily has all <a class="existingWikiWord" href="/nlab/show/meet">meets</a> as well and so is a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>. However, a <strong>suplattice homomorphism</strong> preserves joins, but not necessarily meets. Furthermore, a <em><a class="existingWikiWord" href="/nlab/show/proper+class">large</a></em> semilattice which has all <em>small</em> joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).</p> <p>Dually, an <strong>inflattice</strong> is a poset which has all <a class="existingWikiWord" href="/nlab/show/meets">meets</a>, and an <strong>inflattice homomorphism</strong> is a monotone function that preserves all meets.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/frame">frame</a></strong> (dual to a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>) is a suplattice in which finitary meets distribute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)</p> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/SupLat">SupLat</a> of suplattices and suplattice homomorphisms admits a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> which represents “<a class="existingWikiWord" href="/nlab/show/bilinear+maps">bilinear maps</a>”, i.e. functions which preserve joins separately in each variable. Under this tensor product, the category of suplattices is a <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a> in which the <a class="existingWikiWord" href="/nlab/show/dualizing+object">dualizing object</a> is the suplattice dual to the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/truth-values">truth-values</a>. A <a class="existingWikiWord" href="/nlab/show/semigroup+object">semigroup</a> in this <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is a <strong><a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></strong>, including <a class="existingWikiWord" href="/nlab/show/frames">frames</a> as a special case when the quantale is idempotent and unital. Modules over them are <a class="existingWikiWord" href="/nlab/show/modules+over+quantales">modules over quantales</a> (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck’s descent theory in <a href="#JoyalTierney84">Joyal-Tierney 84</a>).</p> <h2 id="the_free_suplattice_on_a_poset">The free suplattice on a poset</h2> <p>There is a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>SupLat</mi><mo>→</mo><mi>Poset</mi></mrow><annotation encoding="application/x-tex"> U \colon SupLat \to Poset </annotation></semantics></math></div> <p>This has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>Poset</mi><mo>→</mo><mi>SupLat</mi></mrow><annotation encoding="application/x-tex"> F \colon Poset \to SupLat </annotation></semantics></math></div> <p>where 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 suplattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(P)</annotation></semantics></math> is the poset of downsets of <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>, ordered by inclusion. Here a <strong><a class="existingWikiWord" href="/nlab/show/downset">downset</a></strong> of a 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> is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">S \subseteq P</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>s</mi><mo>′</mo><mo>≤</mo><mi>s</mi><mspace width="1em"></mspace><mo>⇒</mo><mspace width="1em"></mspace><mi>s</mi><mo>′</mo><mo>∈</mo><mi>S</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> s \in S, s' \le s \quad \implies \quad s' \in S. </annotation></semantics></math></div> <p>This set of all downsets in <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>, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>P</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{P}</annotation></semantics></math>, is ordered by inclusion, and it’s a suplattice: any union of downsets is a downset. There’s an embedding of <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>P</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{P}</annotation></semantics></math> that sends each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">p \in P</annotation></semantics></math> to its <strong>principal</strong> downset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>s</mi><mo>∈</mo><mi>P</mi><mo>:</mo><mspace width="thickmathspace"></mspace><mi>s</mi><mo>≤</mo><mi>p</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{s \in P : \; s \le p \}</annotation></semantics></math>. (To give a downset is to give an <a class="existingWikiWord" href="/nlab/show/antichain">antichain</a>, and so the free suplattice is sometimes described equivalently in terms of antichains.)</p> <p>To understand this description of the free suplattice on a poset, some <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> is useful. <a class="existingWikiWord" href="/nlab/show/preorder">Preorders</a> are the same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with two objects <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> and one nontrivial morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⇒</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">F \implies T</annotation></semantics></math>, its monoidal structure being “and”. Using this idea, the downsets of a 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> correspond in a one-to-one way with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>P</mi> <mi>op</mi></msup><mo>→</mo><mi>Bool</mi></mrow><annotation encoding="application/x-tex">f \colon P^{op} \to Bool</annotation></semantics></math>, just as <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on a category <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> are functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">f \colon C^{op} \to Set</annotation></semantics></math>. The embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>P</mi><mo>→</mo><mover><mi>P</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">y \colon P \to \hat{P}</annotation></semantics></math> that sends each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">p \in P</annotation></semantics></math> to its principal downset is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-enriched version of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>. So, just as the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> on a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocomplete category</a> on <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>P</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{P}</annotation></semantics></math> is the free cocomplete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-enriched category on <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>. But a cocomplete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-enriched category that happens to be a poset is just the same as a suplattice.</p> <h2 id="the_category_of_suplattices">The category of suplattices</h2> <p>The category of suplattices is <a class="existingWikiWord" href="/nlab/show/monadic+adjunction">monadic</a> over the category of posets, and each algebra structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>:</mo><mover><mi>P</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\xi: \hat{P} \to P</annotation></semantics></math> is left adjoint to the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mi>P</mi><mo>→</mo><mover><mi>P</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">y: P \to \hat{P}</annotation></semantics></math>. This makes suplattices the same thing (up to equivalence) as <a class="existingWikiWord" href="/nlab/show/total+categories">total categories</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bool</mi></mrow><annotation encoding="application/x-tex">Bool</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched</a> sense. Notice that algebra structure maps, being left adjoints, are cocontinuous and therefore suplattice morphisms. This makes the monad a <a class="existingWikiWord" href="/nlab/show/commutative+monad">commutative monad</a>, and therefore according to general theory, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SupLat</mi></mrow><annotation encoding="application/x-tex">SupLat</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+closed+category">symmetric monoidal closed category</a> where the internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(P, Q)</annotation></semantics></math> between two suplattices is the suplattice of cocontinuous maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \to Q</annotation></semantics></math>, which are the same as left adjoints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \to Q</annotation></semantics></math> according to the poset version of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SupLat</mi></mrow><annotation encoding="application/x-tex">SupLat</annotation></semantics></math> is also monadic over <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>, where the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">P: Set \to Set</annotation></semantics></math> is the covariant <a class="existingWikiWord" href="/nlab/show/power+set">power set</a> functor. It therefore is a complete and cocomplete <a class="existingWikiWord" href="/nlab/show/Barr-exact+category">Barr-exact category</a>. The <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a> of this monad is equivalent to <a class="existingWikiWord" href="/nlab/show/Rel">Rel</a>, the category of sets and relations. Thus the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rel</mi></mrow><annotation encoding="application/x-tex">Rel</annotation></semantics></math> may be thought of as the free suplattices.</p> <p>As stated above, the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+closed+category">symmetric monoidal closed category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SupLat</mi></mrow><annotation encoding="application/x-tex">SupLat</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a> where the star-<a class="existingWikiWord" href="/nlab/show/involution">involution</a> takes a suplattice <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> to the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">P^{op}</annotation></semantics></math>. In part this says that a suplattice is also an <a class="existingWikiWord" href="/nlab/show/inflattice">inflattice</a>, a fact which holds internally in any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> (where we use an internal covariant power-object functor to form an appropriate monad). Thus the tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊗</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \otimes Q</annotation></semantics></math> may be formed as the suplattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msup><mi>Q</mi> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Hom(P, Q^{op})^{op}</annotation></semantics></math>. The presence of the equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>:</mo><msup><mi>SupLat</mi> <mi>op</mi></msup><mo>→</mo><mi>SupLat</mi></mrow><annotation encoding="application/x-tex">\ast = (-)^{op}: SupLat^{op} \to SupLat</annotation></semantics></math></div> <p>(which takes a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f: P \to Q</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mi>op</mi></msup><mo>:</mo><msup><mi>Q</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>P</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">g^{op}: Q^{op} \to P^{op}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is right adjoint to <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>) also means that colimits may be formed as appropriate limits, which are in turn formed pointwise by monadicity over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> <h2 id="references">References</h2> <ul> <li id="JoyalTierney84"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Miles+Tierney">Miles Tierney</a>, <em>An extension of the Galois theory of Grothendieck</em>, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp. (<a href="https://bookstore.ams.org/memo-51-309/">ISBN: 978-1-4704-0722-3</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 6, 2024 at 12:12:25. See the <a href="/nlab/history/suplattice" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/suplattice" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9664/#Item_13">Discuss</a><span class="backintime"><a href="/nlab/revision/suplattice/23" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/suplattice" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/suplattice" accesskey="S" class="navlink" id="history" rel="nofollow">History (23 revisions)</a> <a href="/nlab/show/suplattice/cite" style="color: black">Cite</a> <a href="/nlab/print/suplattice" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/suplattice" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>