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\begin{document}

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\section*{suplattice}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{suplattice} is a [[partial order|poset]] which has all [[joins]] (and in particular is a join-[[semilattice]]). By the [[adjoint functor theorem]] for posets, a suplattice necessarily has all [[meet|meets]] as well and so is a [[complete lattice]]. However, a \textbf{suplattice homomorphism} preserves joins, but not necessarily meets. Furthermore, a \emph{[[proper class|large]]} semilattice which has all \emph{small} joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).

Dually, an \textbf{inflattice} is a poset which has all [[meets]], and an \textbf{inflattice homomorphism} in a monotone function that preserves all meets.

A \textbf{[[frame]]} (dual to a [[locale]]) is a suplattice in which finitary meets distribute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)

The [[category]] [[SupLat]] of suplattices and suplattice homomorphisms admits a [[tensor product]] which represents ``bilinear maps,'' i.e. functions which preserve joins separately in each variable. Under this tensor product, the category of suplattices is a [[star-autonomous category]] in which the dualizing object is the suplattice dual to the object $TV$ of [[truth-values]]. A [[semigroup object|semigroup]] in this [[monoidal category]] is a \textbf{[[quantale]]}, including [[frames]] as a special case when the quantale is idempotent and unital. Modules over them are [[modules over quantales]] (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck's descent theory in Joyal and Tierney).

\begin{itemize}%
\item [[André Joyal]], M. Tierney, \emph{An extension of the Galois theory of Grothendieck}, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

\end{itemize}
\hypertarget{the_free_suplattice_on_a_poset}{}\subsection*{{The free suplattice on a poset}}\label{the_free_suplattice_on_a_poset}

There is a forgetful functor

\begin{displaymath}
U \colon SupLat \to Poset
\end{displaymath}
This has a left adjoint

\begin{displaymath}
F \colon Poset \to SupLat
\end{displaymath}
where for any poset $P$, the suplattice $F(P)$ is the poset of downsets of $P$, ordered by inclusion. Here a \textbf{downset} of a poset $P$ is a subset $S \subseteq P$ such that

\begin{displaymath}
s \in S, s' \le s \quad  \implies \quad s' \in S.
\end{displaymath}
This set of all downsets in $P$, say $\hat{P}$, is ordered by inclusion, and it's a suplattice: any union of downsets is a downset. There's an embedding of $P$ in $\hat{P}$ that sends each $p \in P$ to its \textbf{principal} downset $\{s \in P : \; s \le p \}$. (To give a downset is to give an [[antichain]], and so the free suplattice is sometimes described equivalently in terms of antichains.)

To understand this description of the free suplattice on a poset, some [[enriched category theory]] is useful. [[preorder|Preorders]] are the same as $Bool$-enriched categories, where $Bool$ is the monoidal category with two objects $F$, $T$ and one nontrivial morphism $F \implies T$, its monoidal structure being ``and''. Using this idea, the downsets of a poset $P$ correspond in a one-to-one way with $Bool$-enriched functors $f \colon P^{op} \to Bool$, just as presheaves on a category $C$ are functors $f \colon C^{op} \to Set$. The embedding $y \colon P \to \hat{P}$ that sends each $p \in P$ to its principal downset is the $Bool$-enriched version of the Yoneda embedding. So, just as the category of presheaves on a category $C$ is the [[free cocompletion|free cocomplete category]] on $C$, $\hat{P}$ is the free cocomplete $Bool$-enriched category on $P$. But a cocomplete $Bool$-enriched category that happens to be a poset is just the same as a suplattice.

\hypertarget{the_category_of_suplattices}{}\subsection*{{The category of suplattices}}\label{the_category_of_suplattices}

The category of suplattices is monadic over the category of posets, and each algebra structure $\xi: \hat{P} \to P$ is left adjoint to the Yoneda embedding $y: P \to \hat{P}$. This makes suplattices the same thing (up to equivalence) as [[total categories]] in the $Bool$-[[enriched category theory|enriched]] sense. Notice that algebra structure maps, being left adjoints, are cocontinuous and therefore suplattice morphisms. This makes the monad a [[commutative monad]], and therefore according to general theory, $SupLat$ is a [[symmetric monoidal closed category]] where the internal hom $Hom(P, Q)$ between two suplattices is the suplattice of cocontinuous maps $P \to Q$, which are the same as left adjoints $P \to Q$ according to the poset version of the [[adjoint functor theorem]].

$SupLat$ is also monadic over $Set$, where the monad $P: Set \to Set$ is the covariant [[power set]] functor. It therefore is a complete and cocomplete [[Barr-exact category]].

As stated above, the symmetric monoidal closed category $SupLat$ is a [[star-autonomous category]] where the star-[[involution]] takes a suplattice $P$ to the [[opposite category|opposite]] poset $P^{op}$. In part this says that a suplattice is also an [[inflattice]], a fact which holds internally in any [[topos]] (where we use an internal covariant power-object functor to form an appropriate monad). Thus the tensor product $P \otimes Q$ may be formed as the suplattice $Hom(P, Q^{op})^{op}$. The presence of the equivalence

\begin{displaymath}
\ast = (-)^{op}: SupLat^{op} \to SupLat
\end{displaymath}
(which takes a morphism $f: P \to Q$ to $g^{op}: Q^{op} \to P^{op}$, where $g$ is right adjoint to $f$) also means that colimits may be formed as appropriate limits, which are in turn formed pointwise by monadicity over $Set$.

[[!redirects suplattice]] [[!redirects sup lattice]] [[!redirects sup-lattice]] [[!redirects suplattices]] [[!redirects sup lattices]] [[!redirects sup-lattices]]

[[!redirects inflattice]] [[!redirects inf lattice]] [[!redirects inf-lattice]] [[!redirects inflattices]] [[!redirects inf lattices]] [[!redirects inf-lattices]]

[[!redirects complete semilattice]] [[!redirects complete semilattices]]



\end{document}