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\begin{document}

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\section*{SupLat}

\hypertarget{}{}\section*{{$Sup Lat$}}\label{}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_weak_foundations}{In weak foundations}\dotfill \pageref*{in_weak_foundations} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\textbf{$Sup Lat$} is the [[category]] whose [[objects]] are [[suplattices]] and whose [[morphisms]] are suplattice [[homomorphisms]], that is [[functions]] which preserve all [[joins]] (including the [[bottom element]]). Analogously, \textbf{$Inf Lat$} is the category whose objects are [[inflattices]] and whose morphisms are inflattice homomorphisms, which preserve all [[meets]].

Actually, $Sup Lat$ and $Inf Lat$ are [[equivalence of categories|equivalent]]; the difference between the two is merely the notational choice between $\leq$ and $\geq$. However, this choice corresponds to using either of two [[inclusion functor]]s representing $Sup Lat$ and $Inf Lat$ as [[replete subcategories]] of [[Pos]]; similarly, [[CompLat]] can be viewed as a replete [[wide subcategory]] of $Sup Lat$ and $Inf Lat$ in two different ways.

One can write \textbf{$Comp Semi Lat$} (meaning the category of [[complete semilattices]]) if one wishes to remain ambiguous about the notation.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

$Sup Lat$ is given by a [[variety of algebras]], or equivalently by an [[algebraic theory]], so it is an [[equationally presented category]]; however, it requires operations of arbitrarily large arity. Nevertheless, it is a [[monadic category]] (over [[Set]]), because it has [[free objects]]. Specifically, the \textbf{free suplattice} on a [[set]] $X$ is the [[power set]] $\mathcal{P}X$ of $X$ with the operation of [[union]]; an element $a$ of $X$ appears as the [[singleton subset]] $\{a\}$ in $\mathcal{P}X$.

The \textbf{free inflattice} on $X$ is slightly less natural; of course, we can take it to be $\mathcal{P}X$ with the operation of union again, but then the order on the elements is the opposite of the usual order. However, we can also take it to be $\mathcal{P}X$ with the operation of [[intersection]]; this uses the fact that [[complementation]] is an [[automorphism]] of $\mathcal{P}X$. Then the generator $a$ appears as $X \setminus \{a\}$ in the lattice.

$Sup Lat$ is a [[monoidal category]]; it admits a [[tensor product]] which represents [[binary morphisms]]: functions which preserve joins separately in each variable. A [[monoid object|monoid]] in $Sup Lat$ is a \textbf{[[quantale]]}, including [[frames]] as a special case.

In fact, $Sup Lat$ is even a [[star-autonomous category]], and \emph{a fortiori} a [[linearly distributive category]]. The dual of a suplattice is its opposite poset, which is also a suplattice since every suplattice is also an inflattice.

\hypertarget{in_weak_foundations}{}\subsection*{{In weak foundations}}\label{in_weak_foundations}

For all practical purposes, $Sup Lat$ is not available in [[predicative mathematics]]. The definition goes through, but we cannot prove that $Sup Lat$ has any [[infinite set|infinite]] objects. (More precisely, the [[power set]] of any nontrivial small suplattice must be small.) Generally speaking, predicative mathematics treats infinite suplattices only as [[proper class|large]] objects. Although they are of little interest, we can ask which of the facts above hold predicatively; the answer is that $Comp Lat$ is not wide as a subcategory of $Sup Lat$, and $Sup Lat$ is not monadic (since $\mathcal{P}X$ is generally large).

In impredicative [[constructive mathematics]], we cannot intepret $\mathcal{P}X$ with intersection as the free inflattice on $X$, since [[complementation]] is not an automorphism. Everything else goes through, however, including the interpretation of $\mathcal{P}X$ with reverse inclusion as the free inflattice. In particular, $Sup Lat$ (and hence $Sup Inf$) is still a monadic category.

category: category

[[!redirects Sup]] [[!redirects SupLat]] [[!redirects Sup Lat]] [[!redirects Inf]] [[!redirects InfLat]] [[!redirects Inf Lat]] [[!redirects CompSemiLat]] [[!redirects Comp Semi Lat]]

[[!redirects free suplattice]] [[!redirects free sup-lattice]] [[!redirects free sup lattice]] [[!redirects free inflattice]] [[!redirects free inf-lattice]] [[!redirects free inf lattice]] [[!redirects free complete semilattice]]



\end{document}