text:
\section{Introduction}

It is a classic result in modal logic, known as \emph{J\'onsson-Tarski duality}, that the category ${\sf{MA}}$ of modal algebras is dually equivalent to the category ${\sf{DFr}}$ of descriptive frames. This result can be traced back to the work of J\'onsson-Tarski~\cite{JT51}, Halmos~\cite{Hal56}, and Kripke~\cite{Kri63}. In the modern form it was proved by Esakia~\cite{Esa74} and Goldblatt~\cite{Gol76}.\footnote{We point out that Esakia phrased it for the subcategory of descriptive frames where the relation $R$ is reflexive and transitive. Consequently, he worked with the subcategory of modal algebras consisting of closure algebras of McKinsey and Tarski~\cite{MT44}.}

J\'onsson-Tarski duality is a generalization of the celebrated Stone duality between the category ${\sf{BA}}$ of boolean algebras and the category ${\sf{Stone}}$ of Stone spaces. It was observed by Abramsky~\cite{Abr88} and Kupke, Kurz, and Venema~\cite{KKV04} that J\'onsson-Tarski duality can be proved by lifting Stone duality using algebra/coalgebra methods. This can be done by utilizing the classic Vietoris construction (see, e.g.,~\cite[Ch.~III.4]{Joh82}). Indeed, associating with each Stone space $X$ its Vietoris space $\mathcal{V}(X)$ gives rise to an endofunctor $\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ such that ${\sf{DFr}}$ is isomorphic to the category ${\sf{Coalg}}(\mathcal{V})$ of coalgebras for $\mathcal{V}$. Let ${\sf{SL}}$ be the category of meet-semilattices with top. Then the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$ has a left adjoint $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$. Letting $\mathcal{K}=\mathcal{L}\mathcal{U}$ gives an endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ such that ${\sf{MA}}$ is isomorphic to the category ${\sf{Alg}}(\mathcal{K})$ of algebras for $\mathcal{K}$. Moreover, the following diagram commutes up to natural isomorphism, yielding that Stone duality lifts to a dual equivalence between ${\sf{Alg}}(\mathcal{K})$ and ${\sf{Coalg}}(\mathcal{V})$. This provides an alternate proof of J\'onsson-Tarski duality.

\begin{center}
\begin{tikzcd}[column sep = 8pc]
{\sf{BA}} \arrow[r, <->, "\textrm{Stone duality}"] \arrow[d, "\mathcal{K}"'] &  {\sf{Stone}} \arrow[d, "\mathcal{V}"] \\
{\sf{BA}}  \arrow[r, <->, "\textrm{Stone duality}"'] & {\sf{Stone}}
\end{tikzcd}
\end{center}

Descriptive frames can be thought of as Kripke frames $(X,R)$ equipped with a Stone topology compatible with the relation $R$.
In~\cite{Tho75} Thomason proved a ``discrete version" of J\'onsson-Tarski duality, establishing that the category ${\sf{KFr}}$ of Kripke frames is dually equivalent to the category ${\sf{CAMA}}$ of complete atomic modal algebras whose modal operator is completely multiplicative (see Sections~\ref{sec:free} and~\ref{sec:coalgebra} for all the undefined notions).

The same way J\'onsson-Tarski duality generalizes Stone duality, Thomason duality generalizes Tarski duality between the category $\caba$ of complete atomic boolean algebras and the category ${\sf{Set}}$ of sets. It is natural to try to obtain Thomason duality from Tarski duality using algebra/coalgebra methods in the same vein J\'onsson-Tarski duality was obtained from Stone duality in~\cite{Abr88,KKV04}. Surprisingly, such an approach has not yet been undertaken. Our aim is to fill in this gap.

For this purpose, it is natural to replace the Vietoris endofunctor $\mathcal{V}$ on ${\sf{Stone}}$ with the powerset endofunctor $\mathcal{P}$ on ${\sf{Set}}$. It is known (see, e.g.,~\cite[Sec.~9]{Ven07}) that ${\sf{KFr}}$ is isomorphic to ${\sf{Coalg}}(\mathcal{P})$. Thus, the key is to construct an endofunctor $\mathcal{H}$ on $\caba$ that is an analogue of the endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$. We recall that $\mathcal{K}=\mathcal{L}\mathcal{U}$ where $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ is left adjoint to the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$. A natural analogue of ${\sf{SL}}$ in the complete case is the category ${\sf{CSL}}$ of complete meet-semilattices.
Our main contribution is to show that the forgetful functor $\mathcal{U}:\caba\to{\sf{CSL}}$ has a left adjoint $\mathcal{L}:{\sf{CSL}}\to\caba$. We then define $\mathcal{H}:\caba\to\caba$ as the composition $\mathcal{H}=\mathcal{L}\mathcal{U}$, and prove that ${\sf{Alg}}(\mathcal{H})$ is dually equivalent to ${\sf{Coalg}}(\mathcal P)$. Since ${\sf{CAMA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal P)$ is isomorphic to ${\sf{KFr}}$, Thomason duality follows.

The paper is organized as follows. In Section~\ref{sec:free} we recall J\'onsson-Tarski duality and how it can be obtained by lifting Stone duality. For this we need to work with the left adjoint $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ of the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$.
The standard approach to constructing $\mathcal{L}$ is to take the free boolean algebra over the underlying set of a meet-semilattice with top and quotient it by the relations defining a modal operator $\Box$ (see~\cite[Prop.~3.12]{KKV04}).
As we will show in Theorem~\ref{prop: H is a reflector}, $\mathcal{L}$ can alternatively be constructed by utilizing Pontryagin duality for semilattices~\cite{HMS74}. Let $\sf StoneSL$ be the category of topological meet-semilattices, where the topology is a Stone topology. Then ${\sf{SL}}$ is dually equivalent to $\sf StoneSL$ (see~\cite[Thm.~3.9]{HMS74} or~\cite[p.~251]{Joh82}).
We will show that $\mathcal{L}$ can be constructed by taking the boolean algebra of clopen subsets of the Pontryagin dual $M^*:= \hom_{\sf{SL}}(M,2)$ of $M \in {\sf{SL}}$. This shows that $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ can be constructed either purely algebraically, by utilizing the existence of free algebras in ${\sf{BA}}$, or using duality, as the boolean algebra of clopens of the Pontryagin dual of a meet-semilattice with top.

In Section~\ref{sec:coalgebra} we show that the same approach applies to the forgetful functor $\mathcal{U}:\caba\to{\sf{CSL}}$. Its left adjoint $\mathcal{L}:{\sf{CSL}}\to\caba$ can be constructed by taking the free object in $\caba$ over the underlying set of a complete meet-semilattice and then taking the quotient of it by the relations defining a completely multiplicative modal operator $\Box$. We point out that care is needed in constructing the free object in $\caba$ since it is well known that free objects do not exist in the category of complete boolean algebras (see~\cite{Gai64,Hal64}). Nevertheless, free objects in $\caba$ do exist. This can be seen by observing that the Eilenberg-Moore algebras of the double contravariant powerset monad are exactly the objects of $\caba$~\cite{Tay02}, and that categories of algebras for monads have free objects~\cite[Prop.~20.7(2)]{AHS06}. A more concrete construction of free objects in $\caba$ can be given by utilizing the theory of canonical extensions of J\'onsson and Tarski~\cite{JT51}. Indeed, in Theorem~\ref{prop: free caba} we will prove that the free object in $\caba$ over a set $X$ is the canonical extension $F^\sigma$ of the free boolean algebra $F$ over $X$. We then quotient $F^\sigma$ by the complete congruence generated by the relations defining a completely multiplicative modal operator, yielding the desired $\mathcal{L}:{\sf{CSL}}\to\caba$ (see Theorem~\ref{prop: H is a reflector Venema def}).

An alternate construction of $\mathcal{L}:{\sf{CSL}}\to\caba$ that parallels the alternate construction of $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ can be given by taking the powerset of $\hom_{\sf{CSL}}(M,2)$ for each $M\in{\sf{CSL}}$. Since $\hom_{\sf{CSL}}(M,2)$ is isomorphic to the order-dual of $M$, this amounts to taking the powerset of $M$.\footnote{We thank one of the referees for suggesting this approach.} Therefore, we again obtain that the left adjoint can be constructed either purely algebraically, utilizing that free objects exist in $\caba$, or else using the powerset construction. Thus, we arrive at the following diagram, which parallels the constructions of the left adjoints $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ and $\mathcal{L}:{\sf{CSL}}\to\caba$:

\medskip

\begin{center}
\begin{tabular}{|l|l|l|} \hline
\begin{tabular}{l}Forgetful functor\end{tabular} & \begin{tabular}{l}Left adjoint\end{tabular} & \begin{tabular}{l}Location\end{tabular} \\ \hline
\begin{tabular}{l}$\mathcal{U} : {\sf{BA}} \to {\sf{SL}}$ \\  \phantom{algebraic} \\  \phantom{dual} \end{tabular}
&
\begin{tabular}{l}$\mathcal{L} : {\sf{SL}} \to {\sf{BA}}$ \\ algebraic construction:  ${\sf Free}_{{\sf{BA}}}(M)/{\equiv}$ \\ dual construction: ${\mathfrak{clop}}(\hom_{{\sf{SL}}}(M,2))$ \end{tabular}
&
\begin{tabular}{l} \\ \cite{KKV04} \\ Theorem~\ref{prop: H is a reflector} \end{tabular}
\\ \hline
\begin{tabular}{l}$\mathcal{U} : \caba \to {\sf{CSL}}$ \\ \phantom{algebraic} \\ \phantom{dual} \end{tabular}
&
\begin{tabular}{l}$\mathcal{L} : {\sf{CSL}} \to \caba$  \\ algebraic construction:  ${\sf Free}_{\caba}(M)/{\equiv}$ \\ dual construction: $\wp(\hom_{{\sf{CSL}}}(M,2)) \cong \wp(M)$ \end{tabular}
&
\begin{tabular}{l} \ \\ Theorem~\ref{prop: H is a reflector Venema def} \\ Theorem~\ref{thm:alternate L} \end{tabular}
\\ \hline
\end{tabular}
\end{center}

\medskip

In Section~\ref{sec:thomason} we define the endofunctor $\mathcal{H}:\caba\to\caba$ as the composition $\mathcal{H}=\mathcal{L}\mathcal{U}$. In Theorem~\ref{thm:Hwp=wpP} we prove that the following diagram commutes (up to natural isomorphism).

\begin{center}
\begin{tikzcd}[column sep = 8pc]
\caba \arrow[r, <->, "\textrm{Tarski duality}"] \arrow[d, "\mathcal{H}"'] &  {\sf{Set}} \arrow[d, "\mathcal P"] \\
\caba  \arrow[r, <->, "\textrm{Tarski duality}"'] & {\sf{Set}}
\end{tikzcd}
\end{center}
This paves the way towards proving that the category ${\sf{Alg}}(\mathcal{H})$ of algebras for $\mathcal{H}$ is dually equivalent to the category ${\sf{Coalg}}(\mathcal P)$ of coalgebras for $\mathcal P$ (see Theorem~\ref{cor: main}).

It is well known that
J\'onsson-Tarski and Thomason dualities are connected through the canonical extension and forgetful functors $(-)^\sigma:{\sf{MA}}\to{\sf{CAMA}}$ and $\mathcal{U}:{\sf{DFr}}\to{\sf{KFr}}$, making the following diagram commutative.
\begin{center}
\begin{tikzcd}[column sep = 8pc]
{\sf{MA}} \arrow[r, <->, "\textrm{J\'onsson-Tarski duality}"] \arrow[d, "(-)^\sigma"'] &  {\sf{DFr}} \arrow[d, "\mathcal{U}"] \\
{\sf{CAMA}}  \arrow[r, <->, "\textrm{Thomason duality}"'] & {\sf{KFr}}
\end{tikzcd}
\end{center}
We conclude the paper by Remark~\ref{rem: canonical ext on algebras} in which we show that there are analogous canonical extension and forgetful functors $(-)^\sigma:{\sf{Alg}}(\mathcal{K})\to{\sf{Alg}}(\mathcal{H})$ and $\mathcal{U}:{\sf{Coalg}}(\mathcal{V}) \to{\sf{Coalg}}(\mathcal{P})$ that make a similar diagram commutative.

\section{Coalgebraic approach to J\'onsson-Tarski duality}\label{sec:free}

In this section we give a brief account of J\'onsson-Tarski duality, and provide a construction of the left adjoint $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ of the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$ which is alternative to~\cite[Prop.~3.12]{KKV04}. This we do by utilizing Pontryagin duality for semilattices~\cite{HMS74}. We start by recalling the definition of a modal algebra.

\begin{defi}
A \emph{modal algebra} is a pair $(B,\Box)$ where $B$ is a boolean algebra and $\Box$ is a unary function on $B$ preserving finite meets. A \emph{modal algebra homomorphism} between modal algebras $(B_1,\Box_1)$ and $(B_2,\Box_2)$ is a boolean homomorphism $\alpha:B_1\to B_2$ such that $\alpha(\Box_1 a)=\Box_2\alpha(a)$ for each $a\in B_1$. Let ${\sf{MA}}$ be the category of modal algebras and modal algebra homomorphisms.
\end{defi}

A subset of a topological space $X$ is {\em clopen} if it is both closed and open, and $X$ is {\em zero-dimensional} if $X$ has a basis of clopen sets. A {\em Stone space} is a zero-dimensional compact Hausdorff space.

For a binary relation $R$ on $X$, we write
\[
R[x]:=\{y \in X \mid xRy\} \ \mbox{ and } \ R^{-1}[U]:=\{x\in X \mid \exists u\in U \mbox{ with } xRu\}
\]
for the $R$-image of $x\in X$ and $R$-inverse image of $U\subseteq X$.

\begin{defi}
A \emph{descriptive frame} is a pair $(X,R)$ where $X$ is a Stone space and $R$ is a binary relation on $X$ such that $R[x]$ is closed for each $x\in X$ and $R^{-1}[U]$ is clopen for each clopen $U \subseteq X$.
\end{defi}

Such relations are often called \emph{continuous relations} for the following reason. Let $\mathcal{V}(X)$ be the \emph{Vietoris space} of $X$. We recall (see, e.g.,~\cite[Sec.~III.4]{Joh82}) that $\mathcal{V}(X)$ is the set of closed subsets of $X$ topologized by the subbasis $\{\Box_U,\Diamond_V\mid U,V \mbox{ open in } X\}$ where
\[
\Box_U=\{ F\in\mathcal{V}(X) \mid F \subseteq U\} \ \mbox{ and } \ \Diamond_V=\{F\in\mathcal{V}(X) \mid F\cap V\ne\varnothing\}.
\]
Then $R$ is continuous iff the associated map $\rho_R:X\to\mathcal{V}(X)$, given by $\rho_R(x)=R[x]$, is a well-defined continuous map (that $\rho_R$ is well defined follows from (i), and that it is continuous from (ii)).

Let ${\sf{DFr}}$ be the category of descriptive frames and continuous p-morphisms, where a \textit{p-morphism} between $(X_1,R_1)$ and $(X_2,R_2)$ is a map $f:X_1 \to X_2$ satisfying $f[R_1[x]]=R_2[f(x)]$ for each $x\in X_1$.

\begin{thm} [J\'onsson-Tarski duality]
${\sf{MA}}$ is dually equivalent to ${\sf{DFr}}$.
\end{thm}

J\'onsson-Tarski duality generalizes Stone duality between the category ${\sf{BA}}$ of boolean algebras and boolean homomorphisms and the category ${\sf{Stone}}$ of Stone spaces and continuous maps. We recall that the contravariant functors ${\mathfrak{uf}}:{\sf{BA}}\to^{\sf{op}}{\sf{Stone}}$ and ${\mathfrak{clop}}:{\sf{Stone}}\to^{\sf{op}}{\sf{BA}}$ yielding Stone duality are constructed as follows.\footnote{To easily distinguish between covariant and contravariant functors, following the suggestion of one of the referees, we write $F:{\sf C}\to^{\sf{op}}{\sf D}$ for a contravariant functor $F$.}
The functor ${\mathfrak{uf}}$ assigns to each boolean algebra $A$ the set ${\mathfrak{uf}}(A)$ of ultrafilters of $A$ topologized by the basis $\{\beta_A(a)\mid a\in A\}$ where
\[
\beta_A(a)=\{x\in{\mathfrak{uf}}(A)\mid a\in x\}.
\]
To each boolean homomorphism $\alpha:A\to B$, the functor ${\mathfrak{uf}}$ assigns ${\mathfrak{uf}}(\alpha):=\alpha^{-1}:{\mathfrak{uf}}(B)\to{\mathfrak{uf}}(A)$.
The functor ${\mathfrak{clop}}$ assigns to each Stone space $X$ the boolean algebra ${\mathfrak{clop}}(X)$ of clopen subsets of $X$, and to each continuous map $f:X\to Y$ the boolean homomorphism ${\mathfrak{clop}}(f):=f^{-1}:{\mathfrak{clop}}(Y)\to{\mathfrak{clop}}(X)$.

One unit $\beta : 1_{{\sf{BA}}} \to {\mathfrak{clop}} \circ {\mathfrak{uf}}$ of this dual equivalence is given by the Stone maps $\beta_A:A\to{\mathfrak{clop}}({\mathfrak{uf}}(A))$ for $A\in{\sf{BA}}$, and the other unit $\eta : 1_{{\sf{Stone}}} \to {\mathfrak{uf}}\circ{\mathfrak{clop}}$ by the homeomorphisms $\eta_X:X\to{\mathfrak{uf}}({\mathfrak{clop}}(X))$ for $X\in{\sf{Stone}}$, which are given by
\[
\eta_X(x)=\{U\in{\mathfrak{clop}}(X)\mid x\in U\}.
\]

These functors naturally generalize to yield J\'onsson-Tarski duality. As we pointed out in the introduction, an alternative approach to J\'onsson-Tarski duality is by lifting Stone duality using algebra/coalgebra methods. We recall that the Vietoris construction extends to an endofunctor $\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ by sending a continuous map $f:X\to Y$ to $\mathcal{V}(f):\mathcal{V}(X)\to\mathcal{V}(Y)$ given by $\mathcal{V}(f)(G)=f[G]$ for each $G\in\mathcal{V}(X)$. We next consider the category ${\sf{Coalg}}(\mathcal{V})$ of coalgebras for $\mathcal{V}$. For this we recall the notion of a coalgebra for an endofunctor (see, e.g.,~\cite[Def.~9.1]{Ven07}).

\begin{defi}\label{def:coalgebra}
\begin{enumerate}
\item[]
\item A {\em coalgebra} for an endofunctor $\mathcal{T}:\sf{C} \to \sf{C}$ is a pair $(A,f)$ where $A$ is an object of the category $\sf{C}$ and $f:A \to \mathcal{T}(A)$ is a $\sf{C}$-morphism.
\item A {\em morphism} between two coalgebras $(A_1, f_1)$ and $(A_2,f_2)$ for $\mathcal{T}$ is a $\sf{C}$-morphism $\alpha : A_1 \to A_2$ such that the following square is commutative.
\[
\begin{tikzcd}[column sep = 5pc]
A_1 \arrow[d, "f_1"'] \arrow[r, "\alpha"] &  A_2 \arrow[d, "f_2"] \\
\mathcal{T}(A_1) \arrow[r, "\mathcal{T}(\alpha)"'] &\mathcal{T}(A_2)
\end{tikzcd}
\]
\item Let ${\sf{Coalg}}(\mathcal{T})$ be the category whose objects are coalgebras for $\mathcal{T}$ and whose morphisms are morphisms of coalgebras.
\end{enumerate}
\end{defi}

The dual endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ of the Vietoris endofunctor $\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ was described in~\cite{KKV04}. Let ${\sf{SL}}$ be the category of meet-semilattices with top and meet-homomorphisms preserving top. Then $\mathcal{K}$ is the composition $\mathcal{L}\mathcal{U}$, where $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ is the left adjoint of the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$. In~\cite[Prop.~3.12]{KKV04} the left adjoint is constructed algebraically, by taking the free boolean algebra $F$ over the underlying set of $M\in{\sf{SL}}$ and then taking the quotient of $F$ by the relations remembering that $M$ is a meet-semilattice with top. We give an alternative description of $\mathcal{L}$, which utilizes Pontryagin duality for semilattices~\cite{HMS74}, which we briefly recall next.

Let ${\sf{StoneSL}}$ be the category whose objects are topological meet-semilattices, where the topology is a Stone topology, and whose morphisms are continuous meet-homomorphisms. Pontryagin duality for ${\sf{SL}}$ establishes a dual equivalence between ${\sf{SL}}$ and ${\sf{StoneSL}}$.
The contravariant functor $(-)^* : {\sf{SL}} \to^{\sf{op}} {\sf{StoneSL}}$ sends $M$ to its dual $M^* := \hom_{{\sf{SL}}}(M, 2)$, where $2 = \{0,1\}$ is the two-element chain and meet on $M^*$ is pointwise meet. If $2$ is given the discrete topology and $2^M$ the product topology, then $M^*$ is easily seen to be a closed subspace of $2^M$, and so the subspace topology is a Stone topology. Moreover, pointwise meet is continuous, and hence $M^*\in{\sf{StoneSL}}$. On morphisms, if $\sigma : M \to N$ is an ${\sf{SL}}$-morphism, then $\sigma^* : N^* \to M^*$ is defined by $\sigma^*(\gamma) = \gamma \circ \sigma$. The contravariant functor in the other direction sends $A \in {\sf{StoneSL}}$ to $A^*:=\hom_{{\sf{StoneSL}}}(A, 2)$ and $\sigma : A \to B$ to $\sigma^* : B^* \to A^*$, defined in the same way as the previous functor. Finally, one natural isomorphism sends each $M \in {\sf{SL}}$ to its double dual $M^{**}$ by sending $m$ to the map $\sigma \mapsto \sigma(m)$ for each $m \in M$ and $\sigma \in M^*$. The other natural isomorphism sends each $A \in {\sf{StoneSL}}$ to its double dual $A^{**}$ and is given by the same formula.

\begin{thm} \label{prop: H is a reflector}
Associating with each $M \in {\sf{SL}}$ the boolean algebra ${\mathfrak{clop}}(M^*)$ of clopen subsets of its dual $M^*$ yields an alternative description of the functor $\mathcal{L}:{\sf{SL}} \to {\sf{BA}}$ that is left adjoint to the forgetful functor $\mathcal{U}:{\sf{BA}} \to {\sf{SL}}$.
\end{thm}

\begin{proof}
Let $M\in{\sf{SL}}$. Define $i_M : M \to {\mathfrak{clop}}(M^*)$ by $i_M(m) = \{ \sigma \in M^* \mid \sigma(m) = 1\}$. It is straightforward to see that $i_M$ is a well-defined ${\sf{SL}}$-morphism. By~\cite[p.~89]{Mac71} it is enough to show that for each $A\in{\sf{BA}}$ and ${\sf{SL}}$-morphism $\gamma : M \to A$ there is a unique ${\sf{BA}}$-morphism $\tau : {\mathfrak{clop}}(M^*) \to A$ such that $\tau\circ i_M = \gamma$.
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "i_M"] \arrow[dr, "\gamma"'] & {\mathfrak{clop}}(M^*) \arrow[d, "\tau"] \\
& A
\end{tikzcd}
\]
The map $\gamma^* : \hom_{{\sf{SL}}}(A, 2) \to M^*$ is continuous, so its restriction $\gamma^* : \hom_{{\sf{BA}}}(A, 2) \to M^*$ is continuous. Therefore, ${\mathfrak{clop}}(\gamma^*) : {\mathfrak{clop}}(M^*) \to {\mathfrak{clop}}(\hom_{{\sf{BA}}}(A, 2))$ is a ${\sf{BA}}$-morphism. If $m \in M$, then
\begin{align*}
{\mathfrak{clop}}(\gamma^*)(i_M(m)) &= (\gamma^*)^{-1}(i_M(m)) = \{ \sigma \in M^* \mid \gamma^*(\sigma) \in i_M(m) \} \\
&= \{ \sigma \in M^* \mid \sigma(\gamma(m)) = 1 \}.
\end{align*}
Let $\tau : {\mathfrak{clop}}(M^*) \to A$ be the composition of ${\mathfrak{clop}}(\gamma^*)$ with the inverse of the natural isomorphism $\beta_A: A \to {\mathfrak{clop}}(\hom_{{\sf{BA}}}(A, 2))$ of Stone duality which sends $a\in A$ to $\{ \sigma\in~\hom_{{\sf{BA}}}(A, 2) \mid \sigma(a) = 1\}$.\footnote{Here we make the well-known identification of ${\mathfrak{uf}}(A)$ with $\hom_{{\sf{BA}}}(A, 2)$.}
Then
\[
\tau(i_M(m)) = \beta_A^{-1} {\mathfrak{clop}}(\gamma^*) (i_M(m))= \beta_A^{-1}(\{ \sigma \in M^* \mid \sigma(\gamma(m)) = 1 \})= \gamma(m),
\]
so $\tau \circ i_M = \gamma$. Finally, uniqueness of $\tau$ follows since $i_M[M]$ generates ${\mathfrak{clop}}(M^*)$ as a boolean algebra.
\end{proof}

\begin{rem}
To see how $\mathcal{L}$ acts on morphisms, if $\sigma : M \to N$ is an ${\sf{SL}}$-morphism, then $\sigma^* : N^* \to M^*$ is a continuous ${\sf{SL}}$-morphism between Stone spaces, so ${\mathfrak{clop}}(\sigma^*) : {\mathfrak{clop}}(M^*) \to {\mathfrak{clop}}(N^*)$ is a ${\sf{BA}}$-morphism by Stone duality.  We then set $\mathcal{L}(\sigma) = {\mathfrak{clop}}(\sigma^*) :\mathcal{L}(M) \to \mathcal{L}(N)$.
\end{rem}

Let $\mathcal{K}=\mathcal{L}\mathcal{U}$. Then $\mathcal{K}$ is an endofunctor on ${\sf{BA}}$.
\[
\begin{tikzcd}[column sep = 5pc]
{\sf{BA}} \arrow[rr, bend left = 15, "\mathcal{K}"] \arrow[r, "\mathcal{U}"'] & {\sf{SL}} \arrow[r, "\mathcal{L}"'] & {\sf{BA}}
\end{tikzcd}
\]
Let ${\sf{Alg}}(\mathcal{K})$ be the category of algebras for $\mathcal{K}$ (see~\cite[Def.~5.37]{AHS06}). We recall that for an endofunctor $\mathcal{T}$, algebras for $\mathcal{T}$ are defined by reversing the arrows in the definition of coalgebras for $\mathcal{T}$. Since $\mathcal{K}$ is dual to $\mathcal{V}$, we have that ${\sf{Alg}}(\mathcal{K})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{V})$. Because ${\sf{Alg}}(\mathcal{K})$ is isomorphic to ${\sf{MA}}$ and ${\sf{Coalg}}(\mathcal{V})$ is isomorphic to ${\sf{DFr}}$, this gives an alternate proof of J\'onsson-Tarski duality (see~\cite{KKV04}).

\section{Two constructions of the left adjoint \texorpdfstring{$\mathcal{L} : {\sf{CSL}} \to \caba$}{L : CSL --> CABA}}\label{sec:coalgebra}

Let ${\sf{KFr}}$ be the category of Kripke frames and p-morphisms. Forgetting the topology of a descriptive frame yields the forgetful functor $\mathcal{U}:{\sf{DFr}}\to{\sf{KFr}}$. To describe the modal algebras corresponding to Kripke frames, we recall the notion of a completely multiplicative modal operator.

\begin{defi}
A modal operator $\Box$ on a complete boolean algebra $B$ is \emph{completely multiplicative} if $\Box\left(\bigwedge S\right)=\bigwedge\{\Box s\mid s\in S\}$ for each $S\subseteq B$. Let ${\sf{CAMA}}$ be the category whose objects are complete atomic modal algebras with completely multiplicative $\Box$, and whose morphisms are complete modal algebra homomorphisms.
\end{defi}

\begin{thm} [Thomason duality]
${\sf{CAMA}}$ is dually equivalent to ${\sf{KFr}}$.
\end{thm}

Thomason duality generalizes Tarski duality between $\caba$ and ${\sf{Set}}$ the same way J\'onsson-Tarski duality generalizes Stone duality. We recall that $\caba$ is the category of complete atomic boolean algebras and complete boolean homomorphisms and ${\sf{Set}}$ is the category of sets and functions.  The contravariant functors of Tarski duality are $\wp:{\sf{Set}}\to^{\sf{op}}\caba$ and ${\mathfrak{at}}:\caba\to^{\sf{op}}{\sf{Set}}$. The functor $\wp$ assigns to each set $X$ the powerset $\wp(X)$ and to each function $f:X\to Y$ its inverse image $f^{-1}:\wp(Y)\to\wp(X)$. The functor ${\mathfrak{at}}$ assigns to each $A\in\caba$ its set of atoms. If $\alpha:A\to B$ is a complete boolean homomorphism, it has a left adjoint $\alpha^*:B\to A$, which sends atoms to atoms, and the functor ${\mathfrak{at}}$ assigns to $\alpha$ the function $\alpha^*:{\mathfrak{at}}(B)\to{\mathfrak{at}}(A)$. One unit $\varepsilon : 1_{{\sf{Set}}} \to {\mathfrak{at}}\circ\wp$ of this dual equivalence is given by $\varepsilon_X(x)=\{x\}$ for each $x \in X \in {\sf{Set}}$, and the other unit $\vartheta : 1_{\caba} \to \wp \circ {\mathfrak{at}}$ by $\vartheta_A(a)={\downarrow}a\cap {\mathfrak{at}}(A)$ for each $a\in A \in \caba$.

To derive Thomason duality from Tarski duality the same way J\'onsson-Tarski duality was derived from Stone duality, we need to replace the Vietoris endofunctor $\mathcal{V}$ on ${\sf{Stone}}$ with the powerset endofunctor $\mathcal P$ on ${\sf{Set}}$. We recall that the endofunctor $\mathcal{P}:{\sf{Set}} \to {\sf{Set}}$ associates to each set $X$ its powerset $\mathcal{P}(X)$ and to each function $f:X \to Y$ the function $\mathcal{P}(f):\mathcal{P}(X) \to \mathcal{P}(Y)$ that maps each subset $S \subseteq X$ to its direct image $f[S]$. We also need to replace the endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ with an appropriate endofunctor $\mathcal{H}:\caba\to\caba$.

To describe $\mathcal{H}$, we need to construct the left adjoint to $\mathcal{U}:\caba\to{\sf{CSL}}$, where ${\sf{CSL}}$ is the category of complete meet-semilattices and complete meet-homomorphisms. As in the previous section, this can be done purely algebraically or using duality. To construct $\mathcal{H}$ algebraically, we need that free objects exist in $\caba$. Care is needed here since
it is a well-known result of Gaifman~\cite{Gai64} and Hales~\cite{Hal64} that free objects do not exist in the category of complete boolean algebras and complete boolean homomorphisms. On the other hand, free objects do exist in $\caba$, and this can be seen by observing that the Eilenberg-Moore algebras of the double contravariant powerset monad are exactly the objects of $\caba$~\cite{Tay02}, and that categories of algebras for monads have free objects~\cite[Prop.~20.7(2)]{AHS06}.

A more concrete construction of free objects in $\caba$ can be given utilizing the theory of canonical extensions. It is well known that free objects over any set exist in the category of complete and completely distributive lattices (see Markowski~\cite{Mar79} and Dwinger~\cite[Thm.~4.2]{Dwi81}). By~\cite[Cor.~2.3]{BHJ20}, the free complete and completely distributive lattice over a set $X$ is the canonical extension of the free bounded distributive lattice over $X$. We show that the same is true in $\caba$.
For this we need to recall the definition of a canonical extension of a boolean algebra.

\begin{defi}~\cite{JT51,GH01}
A {\em canonical extension} of a boolean algebra $A$ is a complete boolean algebra $A^\sigma$ together with a boolean embedding $e:A\to A^\sigma$ satisfying:
\begin{enumerate}
\item (Density) Each $x\in A^\sigma$ is a join of meets (and hence also a meet of joins) of $e[A]$.
\item (Compactness) For $S,T\subseteq A$, from $\bigwedge e[S] \le \bigvee e[T]$ it follows that $\bigwedge S_0 \le \bigvee T_0$ for some finite $S_0\subseteq S$ and $T_0\subseteq T$.
\end{enumerate}
\end{defi}

It is well known that canonical extensions are unique up to isomorphism, and that the correspondence $A\mapsto A^\sigma$ extends to a covariant functor $(-)^\sigma:{\sf{BA}}\to\caba$. It can conveniently be described as the composition $\wp\circ\mathcal{U}\circ{\mathfrak{uf}}$, where $\mathcal{U}:{\sf{Stone}}\to{\sf{Set}}$ is the forgetful functor.
\[
\begin{tikzcd}
{\sf{BA}} \arrow[rrr, bend left = 20, "(-)^\sigma"] \arrow[r, "{\mathfrak{uf}}"'] & {\sf{Stone}} \arrow[r, "\mathcal{U}"'] & {\sf{Set}} \arrow[r, "\wp"'] & \caba
\end{tikzcd}
\]
Thus, we can think of $A^\sigma$ as $\wp({\mathfrak{uf}}(A))$ and of $e:A\to A^\sigma$ as the Stone map $\beta_A:A\to\wp({\mathfrak{uf}}(A))$.

\begin{thm} \label{prop: free caba}
Let $X$ be a set. The canonical extension of the free boolean algebra over $X$ is the free object in $\caba$ over $X$.
\end{thm}

\begin{proof}
Let $F$ be the free boolean algebra over $X$, $f : X \to F$ the associated map, and $e : F \to F^\sigma$ the boolean embedding into the canonical extension. We show that $(F^\sigma, e\circ f)$ has the universal mapping property in $\caba$. Let $A \in \caba$ and $g : X \to A$ be a function. Since $A$ is a boolean algebra, there is a unique boolean homomorphism $\varphi : F \to A$ with $\varphi \circ f = g$. This induces a map ${\mathfrak{uf}}(\varphi) : {\mathfrak{uf}}(A) \to {\mathfrak{uf}}(F)$
given by ${\mathfrak{uf}}(\varphi)(y)=\varphi^{-1}(y)$. Define $\varphi_+ : {\mathfrak{at}}(A)\to {\mathfrak{uf}}(F)$ by $\varphi_+(x)=\varphi^{-1}({\uparrow}x)$. If we identify atoms with the principal ultrafilters, we can think of $\varphi_+$ as the restriction of ${\mathfrak{uf}}(\varphi)$ to ${\mathfrak{at}}(A)$.

We identify $F^\sigma$ with $\wp({\mathfrak{uf}}(F))$. Then $e:F\to F^\sigma$ becomes the Stone map $\beta_F$.
The map $\varphi_+:{\mathfrak{at}}(A)\to {\mathfrak{uf}}(F)$ yields a $\caba$-morphism $\wp(\varphi_+) : F^\sigma \to \wp({\mathfrak{at}}(A))$. Since $A \in \caba$, the map $\vartheta_A : A \to \wp({\mathfrak{at}}(A))$ is an isomorphism. We set $\psi = \vartheta_A^{-1} \circ \wp(\varphi_+)$. Clearly $\psi  : F^\sigma \to A$ is a $\caba$-morphism.
We show that $\vartheta_A \circ \varphi = \wp(\varphi_+) \circ e$.
\[
\begin{tikzcd}[column sep = 5pc]
& F \arrow[r, "e"] \arrow[d, "\varphi"] & F^\sigma \arrow[d, "\wp(\varphi_+)"] \arrow[dl, shift right = .3pc, "\psi"] \\
X \arrow[ur, "f"] \arrow[r, "g"'] &A \arrow[r, "\vartheta_A"'] & \wp({\mathfrak{at}}(A))
\end{tikzcd}
\]
Let $a \in F$. Since $\vartheta_A\varphi(a) = \{ x \in {\mathfrak{at}}(A) \mid x \le \varphi(a) \}$ and $e(a) = \beta_F(a) = \{ y \in {\mathfrak{uf}}(F) \mid a \in y\}$, we have
\begin{align*}
(\wp(\varphi_+)\circ e)(a) &= \varphi_+^{-1}e(a) = \{ x \in {\mathfrak{at}}(A) \mid \varphi_+(x) \in e(a) \} \\
&= \{ x \in {\mathfrak{at}}(A) \mid a \in \varphi_+(x) \} = \{ x \in {\mathfrak{at}}(A) \mid a \in \varphi^{-1}({\uparrow}x) \} \\
&= \{ x \in {\mathfrak{at}}(A) \mid x \le \varphi(a)\} = \vartheta_A \varphi(a).
\end{align*}
This shows that $\vartheta_A \circ \varphi = \wp(\varphi_+) \circ e$, so
\[
\psi \circ (e \circ f) =  \vartheta_A^{-1} \circ \wp(\varphi_+) \circ e \circ f =  \vartheta_A^{-1} \circ \vartheta_A \circ \varphi \circ f =  \varphi \circ f = g.
\]

It is left to show uniqueness. Suppose that $\mu : F^\sigma \to A$ is a $\caba$-morphism satisfying $\mu \circ (e \circ f) = g$. Then $(\mu \circ e) \circ f = (\psi \circ e) \circ f = \varphi \circ f$. By uniqueness of $\varphi$, we have $ \mu \circ e = \varphi = \psi \circ e$. Therefore, $\mu$ and $\psi$ agree on $e[F]$. Since $e[F]$ is dense in $F^\sigma$ and $\mu, \psi$ are $\caba$-morphisms, we conclude that $\mu = \psi$.
\end{proof}

We next show that the forgetful functor $\mathcal{U}:\caba\to{\sf{CSL}}$ has a left adjoint $\mathcal{L}:{\sf{CSL}}\to\caba$.
Let $A\in\caba$. We recall that a boolean congruence $\equiv$ on $A$ is a \emph{complete congruence} if $a_i\equiv b_i$ for each $i\in I$ imply $\bigwedge\{ a_i\mid i\in I\} \equiv \bigwedge\{ b_i \mid i\in I\}$. It is well known that the quotient algebra $A/{\equiv}$ is also an object in $\caba$. As usual, for $a\in A$ we write $[a]$ for the equivalence class of $a$. Then the quotient map $\pi:A\to A/{\equiv}$, given by $a \mapsto [a]$, is a $\caba$-morphism.

\begin{rem} \label{rem:comp cong}
There is a well-known one-to-one correspondence between congruences and ideals of a boolean algebra $A$, which associates to each boolean congruence $\equiv$ on $A$ the equivalence class of $0$. If $A \in \caba$, this correspondence restricts to a one-to-one correspondence between complete congruences and principal ideals. In this case, the equivalence class of $0$ is generated by the element $x=\bigvee\{a\vartriangle b\mid a\equiv b\}$, where $\vartriangle$ denotes symmetric difference in $A$.
\end{rem}

If $M \in {\sf{CSL}}$, let $F(M)$ be the free object in $\caba$ over the underlying set of $M$, and let $f_M : M  \to F(M)$ be the associated map. We let $\equiv$ be the complete congruence on $F(M)$ generated by the relations:
\[
f_M\left(\bigwedge S\right) \equiv \bigwedge\{ f_M(s) \mid s\in S \}, \mbox{ where } S\subseteq M.
\]
We then set $\mathcal{L}(M)$ to be the quotient algebra $F(M)/{\equiv}$. Since $F(M)\in\caba$ and $\equiv$ is a complete congruence, $\mathcal{L}(M)\in\caba$. For $a\in M$, let $\Box_a = [f_M(a)] \in \mathcal{L}(M)$. Let $\alpha_M : M \to \mathcal{L}(M)$ be the composition of the quotient map $\pi : F(M) \to \mathcal{L}(M)$ and $f_M$. Then $\alpha_M(a) = \Box_a$ for each $a \in M$.
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "f_M"] \arrow[dr, "\alpha_M"'] & F(M) \arrow[d, "\pi"] \\
& \mathcal{L}(M)
\end{tikzcd}
\]
By the definition of $\equiv$ we see that $\Box_{\bigwedge S} = \bigwedge \{ \Box_s  \mid s \in S\}$ in $\mathcal{L}(M)$ for each $S \subseteq M$. Thus, $\alpha_M$ is a ${\sf{CSL}}$-morphism.

\begin{rem} \label{rem: kernel}
In view of Remark~\ref{rem:comp cong}, the equivalence class $[0]\in\mathcal{L}(M)$ is the principal ideal generated by the element
\[
\bigvee \left\{ f_M\left({\bigwedge S}\right) \vartriangle \bigwedge \{ f_M(s) \mid s \in S\} \;\middle|\; S \subseteq M\right\}.
\]
\end{rem}

\begin{thm} \label{prop: H is a reflector Venema def}
The correspondence $M \mapsto \mathcal{L}(M)$ defines a functor $\mathcal{L} : {\sf{CSL}} \to \caba$ that is left adjoint to the forgetful functor $\mathcal{U}:\caba \to {\sf{CSL}}$.
\end{thm}

\begin{proof}
By~\cite[p.~89]{Mac71} it is enough to show that for each $M \in {\sf{CSL}}$, $A\in\caba$, and ${\sf{CSL}}$-morphism $\gamma : M \to A$ there is a unique $\caba$-morphism $\tau : \mathcal{L}(M) \to A$ such that $\tau\circ\alpha_M = \gamma$. There is a unique $\caba$-morphism $\varphi : F(M) \to A$ with $\varphi \circ f_M = \gamma$. To see that $\varphi$ factors through $\equiv$, let $S \subseteq M$. Since $\gamma$ is a ${\sf{CSL}}$-morphism, $\gamma(\bigwedge S) = \bigwedge \{ \gamma(s) \mid s \in S\}$. Therefore,
\[
\varphi f_M\left(\bigwedge S \right) = \gamma \left(\bigwedge S  \right) = \bigwedge \gamma[S]
\]
and
\[
\varphi\left(\bigwedge \{ f_M(s) \mid s \in S\}\right) = \bigwedge \{ \varphi f_M(s) \mid s \in S\} = \bigwedge \{ \gamma(s) \mid s \in S\} = \bigwedge \gamma[S].
\]
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "f_M"'] \arrow[dr, "\gamma"'] \arrow[rr, bend left = 20, "\alpha_M"] & F(M) \arrow[r, "\pi"'] \arrow[d, "\varphi"] & \mathcal{L}(M) \arrow[dl, "\tau"] \\
& A&
\end{tikzcd}
\]
Thus, $\varphi f_M\left(\bigwedge S \right) = \varphi\left(\bigwedge \{ f_M(s) \mid s \in S\}\right)$.
This implies that $\equiv$ is contained in $\ker(\varphi)$, and hence $\varphi$ induces a $\caba$-morphism $\tau : \mathcal{L}(M) \to A$ with $\tau \circ \alpha_M = \gamma$. Since $\mathcal{L}(M)$ is generated by $\alpha_M[M]$ and $\tau$ is a $\caba$-morphism, $\tau$ is uniquely determined by the equation $\tau \circ \alpha_M = \gamma$.
\end{proof}

\begin{rem} \label{rem: H on maps}
To describe how $\mathcal{L}$ acts on morphisms, let $\gamma : M \to N$ be a ${\sf{CSL}}$-morphism. Then $\alpha_N \circ \gamma : M \to \mathcal{L}(N)$ is a ${\sf{CSL}}$-morphism, so there is a unique $\caba$-morphism $\mathcal{L}(\gamma) : \mathcal{L}(M) \to \mathcal{L}(N)$ such that $\mathcal{L}(\gamma) \circ \alpha_M = \alpha_N \circ \gamma$.
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "\gamma"] \arrow[d, "\alpha_M"'] & N \arrow[d, "\alpha_N"] \\
\mathcal{L}(M) \arrow[r, "\mathcal{L}(\gamma)"'] & \mathcal{L}(N)
\end{tikzcd}
\]
Therefore, if $a \in M$, then $\mathcal{L}(\gamma)(\Box_a) = \mathcal{L}(\gamma)\alpha_M(a) = \alpha_N\gamma(a) = \Box_{\gamma(a)}$.
\end{rem}

We conclude this section by giving an alternative construction of the left adjoint of the forgetful functor $\caba\to{\sf{CSL}}$.
In parallel to Theorem~\ref{prop: H is a reflector}, we can replace ${\sf{SL}}$ by ${\sf{CSL}}$ and Stone duality by Tarski duality. Then for $M \in {\sf{CSL}}$ we can consider $\mathcal{L}(M)$ to be $\wp(\hom_{{\sf{CSL}}}(M, 2))$. If $\sigma \in \hom_{{\sf{CSL}}}(M, 2)$, then $\sigma^{-1}(1)$ is a filter of $M$. Since $\sigma$ preserves arbitrary meets, letting $a = \bigwedge \sigma^{-1}(1)$
yields $\sigma^{-1}(1) = {\uparrow}a$. Conversely, if $a \in M$, then defining $\sigma_a$ by
\[
\sigma_a(m) = \left\{\begin{array}{ll}1 & \textrm{if }a \le m\\ 0 & \textrm{otherwise}\end{array}\right.
\]
yields $\sigma_a \in \hom_{{\sf{CSL}}}(M, 2)$.
Because this correspondence reverses the order, there is an order-reversing bijection $f  : M \to \hom_{{\sf{CSL}}}(M, 2)$ sending $a$ to $\sigma_a$.
Thus, $\wp(f) : \wp(\hom_{{\sf{CSL}}}(M, 2)) \to~\wp(M)$
is a $\caba$-isomorphism. If $i_M : M \to \wp(\hom_{{\sf{CSL}}}(M,2))$ is given by $i_M(m) = \{ \sigma \in \hom_{{\sf{CSL}}}(M,2) \mid \sigma(m) = 1\}$, then we have the function $\iota_M = \wp(f) \circ i_M : M \to \wp(M)$ given by
\begin{align*}
\iota_M(m) &= \wp(f)(i_M(m)) = \{ a \in M \mid f(a) \in i_M(m) \} = \{ a \in M \mid f(a)(m) = 1 \} \\
&= \{ a \in M \mid a \le m\} = {\downarrow}m .
\end{align*}
Thus, we may set $\mathcal{L}(M) = \wp(M)$ and define $\iota_M : M \to \mathcal{L}(M)$ by $\iota(m) = {\downarrow}m$.

\begin{thm}\label{thm:alternate L}
Associating with each $M \in {\sf{CSL}}$ its powerset yields an alternative description of the functor $\mathcal{L}:{\sf{CSL}} \to \caba$ that is left adjoint to the forgetful functor $\mathcal{U}:\caba \to {\sf{CSL}}$.
\end{thm}

\begin{proof}
Let $M\in{\sf{CSL}}$. Clearly the powerset of $M$ is an object in $\caba$. Define $\iota_M:M\to\mathcal{L}(M)$ by $\iota_M(a)={\downarrow} a$ for each $a \in M$. For each $S \subseteq M$, we have
\[
  \bigwedge \{ \iota_M(s) \mid s \in S \} = \bigcap \{ {\downarrow} s \mid s \in S \} = \big\downarrow\! \left(\bigwedge S\right) = \iota_M \left( \bigwedge S \right).
\]
Therefore, $\iota_M$ is a ${\sf{CSL}}$-morphism. It is enough to show that for each
$A\in\caba$ and ${\sf{CSL}}$-morphism $\gamma : M \to A$ there is a unique $\caba$-morphism $\tau : \mathcal{L}(M) \to A$ such that $\tau\circ\iota_M = \gamma$.
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "\iota_M"] \arrow[dr, "\gamma"'] & \mathcal{L}(M) \arrow[d, "\tau"] \\
& A
\end{tikzcd}
\]

Let $\gamma^*$ be the left adjoint of $\gamma$, and consider its restriction $\gamma^*:{\mathfrak{at}}(A) \to M$. Also recall that $\vartheta_A:A\to\wp({\mathfrak{at}}(A))$ is a $\caba$-isomorphism, hence so is $\vartheta_A^{-1}:\wp({\mathfrak{at}}(A))\to A$ which is given by $\vartheta_A^{-1} (S)= \bigvee S$ for $S \subseteq {\mathfrak{at}}(A)$.
We
set $\tau=\vartheta_A^{-1} \circ \wp(\gamma^*): \mathcal{L}(M) \to A$.
\[
\begin{tikzcd}
\mathcal{L}(M)=\wp(M) \arrow[rr, bend left = 20, "\tau"] \arrow[r, "\wp(\gamma^*)"'] & \wp({\mathfrak{at}}(A)) \arrow[r, "\vartheta_A^{-1}"'] & A
\end{tikzcd}
\]
Then $\tau$ is the composition of two $\caba$-morphisms, so is a $\caba$-morphism. Moreover, for $S\subseteq M$, we have
\[
\tau(S)= \vartheta_A^{-1}\wp(\gamma^*)(S) = \vartheta_A^{-1}(\{ x \in {\mathfrak{at}}(A) \mid \gamma^*(x) \in S \}) = \bigvee \{ x \in {\mathfrak{at}}(A) \mid \gamma^*(x) \in S \}.
\]
Thus, for $a \in M$, we have
\[
\tau(\iota_M(a))=\tau({\downarrow} a) = \bigvee \{ x \in {\mathfrak{at}}(A) \mid \gamma^*(x) \le a \} = \bigvee \{ x \in {\mathfrak{at}}(A) \mid x \le \gamma(a) \} = \gamma(a)
\]
since $A$ is atomic.
To show that $\tau$ is uniquely determined by the equation $\tau \circ \iota_M=\gamma$, it is enough to show that $\mathcal{L}(M)$ is generated as a complete boolean algebra by $\iota_M[M]$. Since each $S \subseteq M$ is the union of singletons, this follows from the equation
\[
\{ a \} = {\downarrow} a \setminus \{ b \mid b < a \} = {\downarrow} a \setminus \bigcup \{ {\downarrow} b \mid b < a \} = \iota_M(a) \wedge \neg \bigvee \{ \iota_M(b) \mid b < a \}.
\qedhere
\]
\end{proof}

\begin{rem}\label{rem:H on morphisms}
It is worth mentioning that the equation $\{ a \} = \iota_M(a) \wedge \neg \bigvee \{ \iota_M(b) \mid b < a \}$ above
allows an alternate description of $\tau:\mathcal{L}(M)\to A$ that does not involve atoms.
Indeed, since $\tau$ is a $\caba$-morphism, we have
\begin{align*}
\tau(\{ a \}) &= \tau \left(\iota_M(a) \wedge \neg \bigvee \{ \iota_M(b) \mid b < a \} \right) = \tau \iota_M(a) \wedge \neg \bigvee \{ \tau \iota_M(b) \mid b < a \}\\
&= \gamma(a) \wedge \neg \bigvee \{ \gamma(b) \mid b < a \}.
\end{align*}
Thus, for each $S \subseteq M$, we have
\[
  \tau(S) = \bigvee \{ \tau(\{ a \}) \mid a \in S \} = \bigvee \left\lbrace \gamma(a) \wedge \neg \bigvee \{\gamma(b) \mid b < a \} \;\middle|\; a \in S \right\rbrace.
\]
\end{rem}

\begin{rem} \label{rem:morphisms}
To describe how $\mathcal{L}$ acts on morphisms, let $\gamma : M \to N$ be a ${\sf{CSL}}$-morphism. Then $\iota_N \circ \gamma : M \to \mathcal{L}(N)$ is a ${\sf{CSL}}$-morphism, so there is a unique $\caba$-morphism $\mathcal{L}(\gamma) : \mathcal{L}(M) \to \mathcal{L}(N)$ such that $\mathcal{L}(\gamma) \circ \iota_M = \iota_N \circ \gamma$.
\[
\begin{tikzcd}[column sep = 5pc]
M \arrow[r, "\gamma"] \arrow[d, "\iota_M"'] & N \arrow[d, "\iota_N"] \\
\mathcal{L}(M) \arrow[r, "\mathcal{L}(\gamma)"'] & \mathcal{L}(N)
\end{tikzcd}
\]
Therefore, if $a \in M$, then $\mathcal{L}(\gamma)({\downarrow} a) = \mathcal{L}(\gamma)\iota_M(a) = \iota_N\gamma(a) = {\downarrow} \gamma(a)$.
\end{rem}

\section{Coalgebraic approach to Thomason duality}\label{sec:thomason}

\begin{defi}
Let $\mathcal{H}:\caba\to\caba$ be the composition $\mathcal{H}=\mathcal{L}\mathcal{U}$.
\[
\begin{tikzcd}[column sep = 5pc]
\caba \arrow[rr, bend left = 15, "\mathcal{H}"] \arrow[r, "\mathcal{U}"'] & {\sf{CSL}} \arrow[r, "\mathcal{L}"'] & \caba
\end{tikzcd}
\]
\end{defi}

\begin{rem}
In Theorems~\ref{prop: H is a reflector Venema def} and~\ref{thm:alternate L} we have given two alternative constructions of $\mathcal{L}:{\sf{CSL}}\to\caba$. Thus, we have two alternative descriptions of $\mathcal{H}:\caba\to\caba$. For $A \in \caba$ we can think of $\mathcal{H}(A)$ as the powerset of $A$ (Theorem~\ref{thm:alternate L}) or as the quotient of the free object in $\caba$ over $A$ (Theorem~\ref{prop: H is a reflector Venema def}). The resulting two functors are naturally isomorphic. In this section we will always assume that $\mathcal{H}(A)$ is the powerset of $A$, but will indicate how the corresponding result can be proved if we think of $\mathcal{H}(A)$ as the quotient of the free object in $\caba$ over $A$.
\end{rem}

We show that the diagram in Figure~\ref{diagram} is commutative up to natural isomorphism. The horizontal arrows in the diagram represent the contravariant functors ${\mathfrak{at}}:{\sf{Set}} \to^{\sf{op}} \caba$ and $\wp:{\sf{Set}}\to^{\sf{op}}\caba$ of Tarski duality, whereas the vertical arrows represent the endofunctors $\mathcal{H}$ and $\mathcal{P}$ on $\caba$ and ${\sf{Set}}$, respectively. This together with standard algebra/coalgebra machinery then allows us to prove that ${\sf{Alg}}(\mathcal{H})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{P})$, thus yielding an alternate proof of Thomason duality.
\begin{figure}[H]
\begin{tikzcd}[column sep = 5pc]
\caba \arrow[r, shift left = .5ex, "{\mathfrak{at}}"] \arrow[d, "\mathcal{H}"'] &  {\sf{Set}} \arrow[d, "\mathcal{P}"] \arrow[l, shift left, "\wp"] \\
\caba  \arrow[r, shift left = .5ex, "{\mathfrak{at}}"] & {\sf{Set}} \arrow[l, shift left, "\wp"]
\end{tikzcd}
\caption{}\label{diagram}
\end{figure}

\begin{thm}\label{thm:Hwp=wpP}
  \hspace{1em}
\begin{enumerate}[$(1)$]
\item $\mathcal{H} \circ \wp = \wp\circ\mathcal{P}$.
\item ${\mathfrak{at}}\circ\mathcal{H}$ is naturally isomorphic to $\mathcal{P}\circ{\mathfrak{at}}$.
\end{enumerate}
\end{thm}

\begin{proof}
(1) If $X \in {\sf{Set}}$, then $\wp \mathcal{P}(X)$ and $\mathcal{H} \wp (X)$ are both objects in $\caba$ obtained by taking the double powerset of $X$ ordered by inclusion. We show that the two compositions also agree on morphisms.
Let $f:X \to Y$ be a map. It is sufficient to show that $\wp \mathcal{P}(f)({\downarrow} S)=\mathcal{H} \wp(f)({\downarrow} S)$ for each $S \subseteq Y$. By Remark~\ref{rem:morphisms}, we have
\[
\mathcal{H} \wp(f)({\downarrow} S) = {\downarrow} \wp(f)(S)={\downarrow} f^{-1}(S).
\]
On the other hand,
\begin{align*}
\wp \mathcal{P}(f)({\downarrow} S) &= \mathcal{P}(f)^{-1}({\downarrow} S)=\{ T \in \wp(X) \mid \mathcal{P}(f)(T) \in {\downarrow} S \}\\
&=\{ T \in \wp(X) \mid f[T] \subseteq S \}=\{ T \in \wp(X) \mid T \subseteq f^{-1}(S) \}={\downarrow} f^{-1}(S).
\end{align*}
Thus, $\wp \mathcal{P}(f)({\downarrow} S)=\mathcal{H} \wp(f)({\downarrow} S)$, completing the proof.

(2) follows from (1) since the horizontal arrows in the diagram in Figure~\ref{diagram} form a dual equivalence.
\end{proof}

\begin{rem}\label{rem:box vs down}
If we use the alternative description of $\mathcal{H}$, then Theorem~\ref{thm:Hwp=wpP}(1) should be phrased as $\mathcal{H} \circ \wp$ is naturally isomorphic to $\wp\circ\mathcal{P}$. The natural isomorphism $\xi:\mathcal{H} \circ \wp\to\wp\circ\mathcal{P}$ is given on the generators of $\mathcal{H}\wp(X)$ by $\xi_X(\Box_S)={\downarrow} S$ for each $S\subseteq X\in{\sf{Set}}$.
\end{rem}

In the next remark we give an explicit description of the natural isomorphism $\zeta : {\mathfrak{at}}\circ\mathcal{H} \to \mathcal{P}\circ{\mathfrak{at}}$. This will be used in Remark~\ref{rem: 4.11}.

\begin{rem}\label{cor:AtH and PAt naturally iso}
For $A \in \caba$ define $\zeta_A : {\mathfrak{at}}\mathcal{H}(A) \to \mathcal{P}{\mathfrak{at}}(A)$ by
\[
\zeta_A(\{ a \})= \{ x \in {\mathfrak{at}}(A) \mid x \le a \}
\]
for each $a \in A$.
Since $\mathcal{H} \circ \wp = \wp\circ\mathcal{P}$ and $\varepsilon,\vartheta$ are natural isomorphisms of Tarski duality, we have that the composition ${\mathfrak{at}} \mathcal{H}(\vartheta_A) \circ \varepsilon_{\mathcal{P} {\mathfrak{at}}(A)}$ is a bijection.
\[
\begin{tikzcd}[column sep = 5pc]
\mathcal{P} {\mathfrak{at}}(A) \arrow[r, "\varepsilon_{\mathcal{P} {\mathfrak{at}}(A)}"] & {\mathfrak{at}} \wp \mathcal{P} {\mathfrak{at}}(A) = {\mathfrak{at}} \mathcal{H} \wp {\mathfrak{at}}(A) \arrow[r, "{\mathfrak{at}} \mathcal{H}(\vartheta_A)"] & {\mathfrak{at}} \mathcal{H}(A)
\end{tikzcd}
\]
We show that for each $a \in A$ we have
\[
({\mathfrak{at}} \mathcal{H}(\vartheta_A) \circ \varepsilon_{\mathcal{P} {\mathfrak{at}}(A)})(\{ x \in {\mathfrak{at}}(A) \mid x \le a \})=\{a \}.
\]
Since $\varepsilon_{\mathcal{P} {\mathfrak{at}}(A)}(\{ x \in {\mathfrak{at}}(A) \mid x \le a \})=\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \}$, it is sufficient to prove that
\[
{\mathfrak{at}} \mathcal{H}(\vartheta_A)(\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \})=\{a \}.
\]
It follows from Remark~\ref{rem:H on morphisms} that
\begin{align*}
\mathcal{H}(\vartheta_A)(\{a \}) &={\downarrow} \vartheta_A(a) \wedge \neg \bigvee \{ {\downarrow} \vartheta_A(b) \mid b < a \} \\
&={\downarrow} \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \setminus \bigcup \{ {\downarrow} \{ x \in {\mathfrak{at}}(A) \mid x \le b \} \mid b < a \} \\
&=\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \}.
\end{align*}
In particular, $\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \} \le \mathcal{H}(\vartheta_A)(\{ a \})$, and so
\[
{\mathfrak{at}} \mathcal{H}(\vartheta_A) (\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \}) \le \{ a \}
\]
 because ${\mathfrak{at}}\mathcal{H}(\vartheta_A)$ is left adjoint to $\mathcal{H}(\vartheta_A)$. Therefore, ${\mathfrak{at}} \mathcal{H}(\vartheta_A)(\{ \{ x \in {\mathfrak{at}}(A) \mid x \le a \} \})=\{a \}$ since both sides of the last inequality are atoms. Thus, $\zeta_A =({\mathfrak{at}} \mathcal{H}(\vartheta_A) \circ \varepsilon_{\mathcal{P} {\mathfrak{at}}(A)})^{-1}$, and hence $\zeta$ is a natural isomorphism.
\end{rem}

We next utilize Theorem~\ref{thm:Hwp=wpP} and standard algebra/coalgebra machinery to show that Tarski duality lifts to a dual equivalence between ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal P)$.
We start with the following well-known result (see, e.g.,~\cite[Sec.~9]{Ven07}). Since we will be using the functors establishing the isomorphism of Theorem~\ref{thm: KFr} in Remark~\ref{rem: canonical ext on algebras}, we sketch the proof.

\begin{thm}\label{thm: KFr}
${\sf{KFr}}$ is isomorphic to ${\sf{Coalg}}(\mathcal{P})$.
\end{thm}

\begin{proof}[Proof (Sketch)]
To each Kripke frame $\mathfrak F=(X,R)$ we associate the coalgebra $\rho_R:X \to \mathcal{P}(X)$ defined by $\rho_R(x)=R[x]$. If $f:X_1 \to X_2$ is a p-morphism between Kripke frames $(X_1, R_1)$ and $(X_2, R_2)$, then $f$ is also a morphism between the coalgebras $(X_1,\rho_{R_1})$ and $(X_2, \rho_{R_2})$. This defines a covariant functor $\mathcal{C} : {\sf{KFr}} \to {\sf{Coalg}}(\mathcal{P})$. To each coalgebra $(X,\rho)$ for $\mathcal{P}$, we associate the Kripke frame $(X,R_\rho)$ where $xR_\rho y$ iff $y \in \rho(x)$. If $f$ is a morphism between two coalgebras $(X_1,\rho_1)$ and $(X_2,\rho_2)$, then $f$ is also a p-morphism between the Kripke frames $(X_1,R_{\rho_1})$ and $(X_2, R_{\rho_2})$. This defines a covariant functor $\mathcal{F} : {\sf{Coalg}}(\mathcal{P}) \to {\sf{KFr}}$. It is straightforward to see that $R=R_{\rho_R}$ for each $(X,R) \in {\sf{KFr}}$ and $\rho=\rho_{R_\rho}$ for each $(X,\rho) \in {\sf{Coalg}}(\mathcal{P})$. Thus, the functors $\mathcal{C}$ and $\mathcal{F}$ yield an isomorphism of ${\sf{KFr}}$ and ${\sf{Coalg}}(\mathcal{P})$.
\end{proof}

We next show that ${\sf{CAMA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{H})$. This is parallel to the well-known fact that ${\sf{MA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{K})$ (see, e.g.,~\cite[Cor.~3.11]{KKV04}).

\begin{thm}\label{thm: cama}
${\sf{CAMA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{H})$.
\end{thm}

\begin{proof}
Let $(A, \Box) \in {\sf{CAMA}}$.
Since $\Box : A \to A$ is a ${\sf{CSL}}$-morphism, by Theorem~\ref{thm:alternate L}, there is a unique $\caba$-morphism $\tau_\Box : \mathcal{H}(A) \to A$ such that $\tau_\Box({\downarrow} a) = \Box a$ for each $a \in A$. Therefore, $(A, \tau_\Box) \in {\sf{Alg}}(\mathcal{H})$. Let $\alpha : A_1 \to A_2$ be a ${\sf{CAMA}}$-morphism and $a \in A_1$. Since $\alpha(\Box_1 a)=\Box_2 \alpha(a)$, by Remark~\ref{rem:morphisms},
\[
\tau_{\Box_2} \mathcal{H}(\alpha)({\downarrow} a) = \tau_{\Box_2}({\downarrow} {\alpha(a)}) = \Box_2 {\alpha(a)} = \alpha(\Box_1 a) = \alpha\tau_{\Box_1}({\downarrow} a).
\]
Because $\mathcal{H}(A)$ is generated by $\{ {\downarrow} a \mid a \in A\}$, we obtain that $\tau_{\Box_2} \circ \mathcal{H}(\alpha) = \alpha \circ \tau_{\Box_1}$. Therefore, $\alpha$ is also a morphism in ${\sf{Alg}}(\mathcal{H})$. This defines a covariant functor $\mathcal{A} : {\sf{CAMA}} \to {\sf{Alg}}(\mathcal{H})$.

Conversely, let $(A,\tau) \in {\sf{Alg}}(\mathcal{H})$ so $A \in \caba$ and $\tau:\mathcal{H}(A) \to A$ is a $\caba$-morphism. If we define $\Box_\tau$ on $A$ by $\Box_\tau a = \tau({\downarrow} a)$, it is easy to see that $\Box_\tau$ is completely multiplicative, so $(A, \Box_\tau) \in {\sf{CAMA}}$. Let $\alpha : A_1 \to A_2$ be a morphism in ${\sf{Alg}}(\mathcal{H})$ and $a \in A_1$. By Remark~\ref{rem:morphisms},
\[
\Box_{\tau_2} \alpha(a) = \tau_2({\downarrow} {\alpha(a)}) = \tau_2 \mathcal{H}(\alpha)({\downarrow} a) = \alpha\tau_1({\downarrow} a) = \alpha(\Box_{\tau_1} a).
\]
Therefore, $\alpha$ is also a ${\sf{CAMA}}$-morphism. This defines a covariant functor $\mathcal{M} :  {\sf{Alg}}(\mathcal{H}) \to {\sf{CAMA}}$.

Let $(A, \Box) \in {\sf{CAMA}}$. For $a \in A$, we have $\Box_{\tau_\Box}a = \tau_\Box({\downarrow} a) = \Box a$. Therefore, $\Box_{\tau_\Box} = \Box$.  Next, let $(A, \tau) \in {\sf{Alg}}(\mathcal{H})$. For $a \in A$, we have $\tau_{\Box_\tau}({\downarrow} a) = \Box_\tau a = \tau({\downarrow} a)$. Since $\mathcal{H}(A)$ is generated by $\{ {\downarrow} a \mid a \in A \}$, we obtain that $\tau_{\Box_\tau}  = \tau$. Thus, the functors $\mathcal{A}$ and $\mathcal{M}$ yield an isomorphism of ${\sf{CAMA}}$ and ${\sf{Alg}}(\mathcal{H})$.
\end{proof}

\begin{rem}
If we use the alternative description of $\mathcal{H}$, then the previous theorem can be proved using Theorem~\ref{prop: H is a reflector Venema def} and Remark~\ref{rem: H on maps}. The advantage of using this description of $\mathcal{H}$ lies in the suggestive definitions $\tau_\Box(\Box_a) = \Box a$ and $\Box_\tau a = \tau(\Box_a)$.
\end{rem}

We are ready to lift Tarski duality to a dual equivalence between ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$. For this we utilize
\cite[Thm.~2.5.9]{Jac17} which states that, under certain conditions, adjunctions lift to adjunctions between categories of algebras. For our purposes, we require the following reformulation of
\cite[Thm.~2.5.9]{Jac17} for dual equivalences.

\begin{lem}\label{lem: Jacobs}
Let $\mathbb{S}:\sf{C} \to \sf{C}$, $\mathbb{T}:\sf{D} \to \sf{D}$ be two endofunctors and $\mathbb{Q}:\sf{C} \to^{\sf{op}} \sf{D}$, $\mathbb{R}:\sf{D} \to^{\sf{op}} \sf{C}$ two contravariant functors forming a dual equivalence. Suppose that $\mathbb{S}\mathbb{R}$ and $\mathbb{R}\mathbb{T}$ are naturally isomorphic \emph{(}and hence so are $\mathbb{T}\mathbb{Q}$ and $\mathbb{Q}\mathbb{S}$\emph{)}. Then $\mathbb{Q}$ and $\mathbb{R}$ lift to contravariant functors
$\widehat{\mathbb{Q}}:{\sf{Alg}}(\mathbb{S}) \to^{\sf{op}} {\sf{Coalg}}(\mathbb{T})$ and $\widehat{\mathbb{R}}:{\sf{Coalg}}(\mathbb{T}) \to^{\sf{op}} {\sf{Alg}}(\mathbb{S})$
which yield a dual equivalence between ${\sf{Alg}}(\mathbb{S})$ and ${\sf{Coalg}}(\mathbb{T})$.
\[
\begin{tikzcd}[column sep = 5pc]
\sf{C} \arrow[r,shift left = .5ex, "\mathbb{Q}"] \arrow[d, "\mathbb{S}"'] & \sf{D} \arrow[d, "\mathbb{T}"] \arrow[l, shift left = .5ex, "\mathbb{R}"] \\
\sf{C} \arrow[r,shift left = .5ex, "\mathbb{Q}"] & \sf{D} \arrow[l, shift left = .5ex, "\mathbb{R}"]
\end{tikzcd}
\]
\end{lem}

Theorem~\ref{thm:Hwp=wpP} and Lemma~\ref{lem: Jacobs} then immediately give:

\begin{thm}\label{cor: main}
Tarski duality between $\caba$ and ${\sf{Set}}$ lifts to a dual equivalence between ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$.
\end{thm}

\begin{rem} \label{rem: 4.11}
By adapting the proof of~\cite[Thm.~2.5.9]{Jac17}, the contravariant functors $\widehat{{\mathfrak{at}}} : {\sf{Alg}}(\mathcal{H}) \to^{\sf{op}} {\sf{Coalg}}(\mathcal{P})$ and $\widehat{\wp} :{\sf{Coalg}}(\mathcal{P}) \to^{\sf{op}} {\sf{Alg}}(\mathcal{H})$ lifting ${\mathfrak{at}}:\caba \to^{\sf{op}} {\sf{Set}}$ and $\wp:{\sf{Set}} \to^{\sf{op}} \caba$ can be defined as follows:

If $(A, f) \in {\sf{Alg}}(\mathcal{H})$, then
$({\mathfrak{at}}(A), \zeta_A \circ {\mathfrak{at}}(f))\in {\sf{Coalg}}(\mathcal{P})$ where $\zeta_A$ is defined in Remark~\ref{cor:AtH and PAt naturally iso}. Moreover, if $\alpha : A_1 \to A_2$ is a morphism in ${\sf{Alg}}(\mathcal{H})$, then ${\mathfrak{at}}(\alpha)$ is a morphism of the corresponding coalgebras. This defines the functor $\widehat{{\mathfrak{at}}} : {\sf{Alg}}(\mathcal{H}) \to^{\sf{op}} {\sf{Coalg}}(\mathcal{P})$.
\[
\begin{tikzcd}[column sep = 5pc]
{\mathfrak{at}}(A_2) \arrow[d, "{\mathfrak{at}}(\alpha)"'] \arrow[r, "\zeta_{A_2} \circ {\mathfrak{at}}(f_2)"] & \mathcal{P}{\mathfrak{at}}(A_2) \arrow[d, "\mathcal{P}{\mathfrak{at}}(\alpha)"]\\
{\mathfrak{at}}(A_1) \arrow[r, "\zeta_{A_1} \circ {\mathfrak{at}}(f_1)"'] & \mathcal{P}{\mathfrak{at}}(A_1)
\end{tikzcd}
\]

Let $(X,g) \in {\sf{Coalg}}(\mathcal{P})$. Then
$\wp(g): \wp\mathcal{P}(X) \to \wp(X)$ is a $\caba$-morphism. Since $\mathcal{H} \circ \wp = \wp\circ\mathcal{P}$ (see Theorem~\ref{thm:Hwp=wpP}(1)), we have that $(\wp(X), \wp(g)) \in {\sf{Alg}}(\mathcal{H})$. Moreover, if $h:(X_1,g_1) \to (X_2,g_2)$ is a ${\sf{Coalg}}(\mathcal{P})$-morphism, then $\wp(h)$ is an ${\sf{Alg}}(\mathcal{H})$-morphism. This defines the functor $\widehat{\wp}:{\sf{Coalg}}(\mathcal{P}) \to^{\sf{op}} {\sf{Alg}}(\mathcal{H})$.
\[
\begin{tikzcd}[column sep = 5pc]
\mathcal{H}\wp(X_2)=\wp \mathcal{P}(X_2) \arrow[r, "\wp(g_2)"] \arrow[d, shift left=12, "\wp \mathcal{P}(h)"] \arrow[d, shift right=7, "\mathcal{H} \wp(h)"'] & \wp(X_2) \arrow[d, "\wp(h)"] \\
\mathcal{H}\wp(X_1)=\wp \mathcal{P}(X_1) \arrow[r, "\wp(g_1)"'] & \wp(X_1)
\end{tikzcd}
\]
\end{rem}

\begin{rem}
Putting Theorems~\ref{thm: KFr}, \ref{thm: cama}, and \ref{cor: main} together yields an alternate proof of Thomason duality.
We recall that the contravariant functors establishing Thomason duality extend the contravariant functors of Tarski duality. Namely, the functor $\wp:{\sf{KFr}}\to^{\sf{op}}{\sf{CAMA}}$ associates to each $(X,R) \in {\sf{KFr}}$ the algebra $(\wp(X), \Box_R) \in {\sf{CAMA}}$ where $\Box_R$ is defined by
\[
  \Box_R(S)=\{ x \in X \mid R[x] \subseteq S \}.\footnote{Thus, $\Box_R$ is the predicate lifting for $\Box$ composed with the coalgebra map $\rho_R : X \to \mathcal{P}(X)$.}
\]
Also, $\wp$ associates to each ${\sf{KFr}}$-morphism $f:X\to Y$ the ${\sf{CAMA}}$-morphism $\wp(f):\wp(Y)\to\wp(X)$ given by $\wp(f)=f^{-1}$.
The functor ${\mathfrak{at}}:{\sf{CAMA}}\to^{\sf{op}}{\sf{KFr}}$ associates to each $(A,\Box) \in {\sf{CAMA}}$ the Kripke frame $({\mathfrak{at}}(A),R_\Box)$ where
\[
xR_\Box y \mbox{ iff } x \wedge \Box\neg y=0 \mbox{ iff } (\forall a \in A) (x \le \Box a \Rightarrow y \le a).
\]
Also, ${\mathfrak{at}}$ associates to each ${\sf{CAMA}}$-morphism $\alpha:A\to B$ the p-morphism ${\mathfrak{at}}(\alpha) : {\mathfrak{at}}(B)\to{\mathfrak{at}}(A)$ given by ${\mathfrak{at}}(\alpha)=\alpha^*$.

We show that $\wp : {\sf{KFr}} \to^{\sf{op}} {\sf{CAMA}}$ is the composition
\[
\begin{tikzcd}
{\sf{KFr}} \arrow[r, "\mathcal{C}"] & {\sf{Coalg}}(\mathcal{P}) \arrow[r, "\widehat{\wp}"] & {\sf{Alg}}(\mathcal{H}) \arrow[r, "\mathcal{M}"] & {\sf{CAMA}}
\end{tikzcd}
\]
and that ${\mathfrak{at}} : {\sf{CAMA}} \to^{\sf{op}} {\sf{KFr}}$ is the composition
\[
 \begin{tikzcd}
 {\sf{CAMA}} \arrow[r, "\mathcal{A}"] & {\sf{Alg}}(\mathcal{H}) \arrow[r, "\widehat{{\mathfrak{at}}}"] & {\sf{Coalg}}(\mathcal{P}) \arrow[r, "\mathcal{F}"] & {\sf{KFr}}.
 \end{tikzcd}
 \]

Let $(X,R) \in {\sf{KFr}}$. Then
\[
\mathcal{M} \, \widehat{\wp} \, \mathcal{C}(X,R) = \mathcal{M} \, \widehat{\wp} \, (X, \rho_R) =\mathcal{M} (\wp(X), \wp(\rho_R))=(\wp(X), \Box_{\wp(\rho_R)}).
\]
For $S \subseteq X$, we have
\[
\Box_{\wp(\rho_R)} S = \wp(\rho_R)({\downarrow} S)=\rho_R^{-1}({\downarrow} S)=\{ x \in X \mid \rho_R(x) \subseteq S \} = \{ x \in X \mid R[x] \subseteq S \} = \Box_R S.
\]
Thus, $\Box_{\wp(\rho_R)}=\Box_R$, so $\mathcal{M} \, \widehat{\wp} \, \mathcal{C}(X,R) = (\wp(X), \Box_R)$, and hence $\mathcal{M} \, \widehat{\wp} \, \mathcal{C} = \wp$.

Let $(A,\Box)\in{\sf{CAMA}}$ . Then
\[
\mathcal{F} \, \widehat{{\mathfrak{at}}} \, \mathcal{A}(A, \Box) = \mathcal{F} \, \widehat{{\mathfrak{at}}} \, (A, \tau_\Box)= \mathcal{F} ({\mathfrak{at}}(A), \zeta_{A} \circ {\mathfrak{at}}(\tau_\Box)) =({\mathfrak{at}}(A), R_{\zeta_{A} \circ {\mathfrak{at}}(\tau_\Box)})
\]
For $x,y \in {\mathfrak{at}}(A)$, we have
\[
x R_{\zeta_{A} \circ {\mathfrak{at}}(\tau_\Box)} y \ \mbox{ iff } \ y \in \zeta_{A}{\mathfrak{at}}(\tau_\Box)(x) \ \mbox{ iff } \ y \in \zeta_A((\tau_\Box)^*(x)).
\]
By Remark~\ref{rem:H on morphisms}, for $a \in A$, we have $\tau_\Box(\{ a \})= \Box a \wedge \neg \bigvee \{ \Box b \mid b < a \}$.
Therefore, for each $x \in {\mathfrak{at}}(A)$, we have
\[
x \le \tau_\Box(\{ a \}) \mbox{ iff } x \le \Box a \mbox{ and } x \nleq \Box b \mbox{ for each } b < a.
\]
Since $\Box$ is completely multiplicative, $c:=\bigwedge \{ a \in A \mid x \le \Box a \}$ is the least element satisfying $x \le \Box c$. Thus, $x \le \tau_\Box(\{ c \})$. Consequently, $(\tau_\Box)^*(x)=\{ c \}$
since $(\tau_\Box)^*$ is left adjoint to $\tau_\Box$ and both $(\tau_\Box)^*(x)$ and
$\{ c \}$
 are atoms of $\mathcal{H}(A)$.
It follows that
\[
y \in \zeta_A((\tau_\Box)^*(x)) \ \mbox{iff} \ y \le \bigwedge \{ a \in A \mid x \le \Box a \} \ \mbox{iff} \ (\forall a \in A) (x \le \Box a \Rightarrow y \le a).
\]
Thus, $x R_{\zeta_{A} \circ {\mathfrak{at}}(\tau_\Box)} y$ iff $x R_\Box y$, so $\mathcal{F} \, \widehat{{\mathfrak{at}}} \, \mathcal{A}(A, \Box) = {\mathfrak{at}}(A,\Box)$, and hence $\mathcal{F} \, \widehat{{\mathfrak{at}}} \, \mathcal{A} = {\mathfrak{at}}$.
\end{rem}

\begin{rem}\label{rem: canonical ext on algebras}
We conclude the paper by connecting the coalgebraic approaches to J\'onsson-Tarski and Thomason dualities. As follows from~\cite{KKV04}, ${\sf{Alg}}(\mathcal{K})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{V})$, from which J\'onsson-Tarski duality follows. By Theorem~\ref{cor: main}, ${\sf{Alg}}(\mathcal{H})$ is dually equivalent to ${\sf{Coalg}}(\mathcal P)$, from which Thomason duality follows.

Let $\mathcal{U}:{\sf{Stone}} \to {\sf{Set}}$ be the forgetful functor. For each $X \in {\sf{Stone}}$, viewing the underlying set of the Vietoris space $\mathcal{V}(X)$ as a subset of $\mathcal P(X)$, we have an inclusion map $i : \mathcal{U}\mathcal{V}(X) \to \mathcal{P}\mathcal{U}(X)$. We extend $\mathcal{U}$ to a forgetful functor on the level of coalgebras. Let $(X,g) \in {\sf{Coalg}}(\mathcal{V})$, so $g:X \to \mathcal{V}(X)$ is a continuous map. Set $\mathcal{U}(X,g):=(\mathcal{U}(X), g')$ where $g' : \mathcal{U}(X) \to \mathcal{PU}(X)$ is given by $g'=i \circ \mathcal{U}(g)$.
\[
\begin{tikzcd}
\mathcal{U}(X) \arrow[rr, bend right = 18, "g'"'] \arrow[r, "\mathcal{U}(g)"] & \mathcal{U}\mathcal{V}(X) \arrow[r, "i"] &  \mathcal{P}\mathcal{U}(X)
\end{tikzcd}
\]
Then $(\mathcal{U}(X), g') \in {\sf{Coalg}}(\mathcal{P})$. If $\alpha:(X_1,g_1)\to(X_2,g_2)$ is a ${\sf{Coalg}}(\mathcal{V})$-morphism, it is straightforward to see that $\mathcal{U}(\alpha):(\mathcal{U}(X_1),g_1')\to(\mathcal{U}(X_2),g_2')$ is a ${\sf{Coalg}}(\mathcal{P})$-morphism. This yields a functor $\mathcal{U} : {\sf{Coalg}}(\mathcal{V}) \to {\sf{Coalg}}(\mathcal{P})$. If we identify $X$ with $\mathcal{U}(X)$ and $g : X \to \mathcal{V}(X)$ with $g' : \mathcal{U}(X) \to \wp\mathcal{U}(X)$, then $\mathcal{U}(X,g) = (X,g)$. We have thus forgotten the topological structure of $X$ and the continuity of $g$. This justifies thinking about $\mathcal{U} : {\sf{Coalg}}(\mathcal{V}) \to {\sf{Coalg}}(\mathcal{P})$ as a forgetful functor.

We next define a functor $(-)^\sigma : {\sf{Alg}}(\mathcal{K}) \to {\sf{Alg}}(\mathcal{H})$ which can be thought of as the canonical extension functor for algebras for $\mathcal{K}$. Use Lemma~\ref{lem: Jacobs} to lift the contravariant functors of Stone duality to $\widehat{{\mathfrak{uf}}},\widehat{{\mathfrak{clop}}}$ and those of Tarski duality to $\widehat{{\mathfrak{at}}},\widehat{\wp}$ and set $(-)^\sigma = \widehat{\wp} \circ \mathcal{U} \circ \widehat{{\mathfrak{uf}}}$.
Thus, $(-)^\sigma \circ \widehat{{\mathfrak{clop}}}$ and $\widehat{\wp}\circ \mathcal{U}$ are naturally isomorphic, and hence so are $\mathcal{U}\circ \widehat{{\mathfrak{uf}}}$ and $\widehat{{\mathfrak{at}}} \circ (-)^\sigma$.

\[
\begin{tikzcd}[column sep = 5pc]
{\sf{Alg}}(\mathcal{K}) \arrow[r, shift right = .5ex, "\widehat{{\mathfrak{uf}}}"'] \arrow[d, "(-)^\sigma"'] & {\sf{Coalg}}(\mathcal{V}) \arrow[d, "\mathcal{U}"] \arrow[l, shift right = .5ex, "\widehat{{\mathfrak{clop}}}"'] \\
{\sf{Alg}}(\mathcal{H}) \arrow[r, shift right = .5ex, "\widehat{{\mathfrak{at}}}"'] & {\sf{Coalg}}(\mathcal{P}) \arrow[l, shift right = .5ex, "\widehat{\wp}"']
\end{tikzcd}
\]

The functor $\mathcal{K}$ satisfies ${\mathfrak{uf}}\mathcal{K}(A) \cong \mathcal{V}{\mathfrak{uf}}(A)$~\cite[Cor.~3.11]{KKV04}. By identifying these two Stone spaces, for $(A, \alpha) \in {\sf{Alg}}(\mathcal{K})$ we describe $(A, \alpha)^\sigma$ directly. Since $\alpha : \mathcal{K}(A) \to A$ is a ${\sf{BA}}$-morphism, ${\mathfrak{uf}}(\alpha) : {\mathfrak{uf}}(A) \to {\mathfrak{uf}}\mathcal{K}(A) = \mathcal{V}{\mathfrak{uf}}(A)$ is a continuous map. Therefore, $({\mathfrak{uf}}(A), {\mathfrak{uf}}(\alpha)) \in {\sf{Coalg}}(\mathcal{V})$. The forgetful functor (after appropriate identifications) sends this to $({\mathfrak{uf}}(A), {\mathfrak{uf}}(\alpha)) \in {\sf{Coalg}}(\mathcal{P})$. Finally, $\widehat{\wp}$ sends $({\mathfrak{uf}}(A), {\mathfrak{uf}}(\alpha))$ to $(\wp{\mathfrak{uf}}(A), \wp{\mathfrak{uf}}(\alpha)) = (A^\sigma, \alpha^\sigma)$. Because of these calculations, we can view the functor ${\sf{Alg}}(\mathcal{K}) \to {\sf{Alg}}(\mathcal{H})$ as the canonical extension functor for algebras for $\mathcal{K}$.
\end{rem}

\section*{Acknowledgements}

We would like to thank the referees for their comments. One of the referees suggested to us an alternative description of $\mathcal{L}$ given in Theorem~\ref{thm:alternate L}. We also thank Nick Bezhanishvili, Jim de Groot, Sebastian Enqvist, Clemens Kupke, Phil Scott, and Yde Venema for their comments on an earlier version of the paper.

\bibliographystyle{alphaurl}

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Home News India Larger conspiracy, PM must answer: Cong
Larger conspiracy, PM must answer: Cong
on: January 15, 2020 In: India, Latest News, News, Top StoriesTags: No Comments
New Delhi: There is a larger conspiracy behind Jammu and Kashmir Police officer Davinder Singh's arrest, the Congress alleged on Tuesday, asking the prime minister and the home minister to order a thorough probe.
The opposition claimed that Singh "cannot be a standalone carrier", with the Congress' leader in Lok Sabha, Adhir Ranjan Chowdhury, saying the question will now arise who were the real culprits behind the last year Pulwama attack. Taking a swipe at the Rashtriya Swayamsevak Sangh (RSS), the BJP's ideological mentor, Chowdhury said had police officer Davinder Singh been "Davinder Khan", the reaction of the "troll regiment of the RSS" would have been different.
Congress chief spokesperson Randeep Surjewala said, "There is a larger conspiracy. At whose instance was Davinder Singh acting and bringing terrorists to Delhi, as reported."
"Are there any people who are in the echelons of power who are involved with him (Singh). Is there a larger conspiracy? Is he only a pawn or a carrier in the larger conspiracy that is unleashed?" he asked.
Police on Saturday had arrested Singh, a deputy superintendent of police, at Mir Bazar in Jammu and Kashmir's Kulgam district, along with Hizbul Mujahideen terrorists Naveed Baba and Altaf, besides an unidentified lawyer who was working as an overground worker for terror outfits.
Surjewala said these are questions that prime minister and the home minister need to answer, as Davinder Singh "cannot be a stand alone carrier".
The Congress spokesperson said Modi and Shah should take note of this entire issue and carry out a thorough investigation into the entire episode.
He also asked who Singh was, who was he linked to and what connection he has with the Hizbul Mujahideen and other terrorist groups.
"I find this story preposterous and I find his explanation a hogwash," he said.
"For how long has he been acting as a conduit for JK terrorists and other terror factories. Was Davinder Singh connected to the 2001 Parliament attack. What is Davinder Singh's connection with the Pulwama attack, where he was posted as DSP, and 42 of our soldiers lost their lives," the Congress leader asked.
"What is this conspiracy is what the nation wants to know," Surjewala asked, alleging that Singh was the same officer who was posted in Pulwama when the attack happened.
He also asked as to where did the RDX in Pulwama attack come from.
"Had Davindar Singh by default been Davindar Khan, the reaction of troll regiment of RSS would have been more strident and vociferous. Enemies of our country ought to be condemned irrespective of colour, creed and religion," Chowdhury said on Twitter.
The Congress leader also said now, the question will arise who were the real culprits behind the Pulwama attack.
Searches continue at Dy SP's residence, documents seized
BSF opens fire at 'drone-like objects' near IB

redpajama_set_name: RedPajamaCommonCrawl


SQL database login for 'SharePoint_ConfigDB' on instance DATABASENAME failed. Additional error information from SQL Server is included below.
Error message repeated once a minute.
After rebooting the server, the error went away.
I'd love to be able to explain this, or say the discovery of the solution was a result of my genius. Sadly, no. After several days deep-diving the logs and exploring the breadth and depth of the blogosphere to no avail, temporarily disabling various timer jobs that appeared in the logs just prior to the error message, and chasing dead ends, it was a power outage and a failure of the backup power supply which forced the server to be restarted and made the error disappear.
When bulk uploading a large number of documents into SharePoint 2010, a user ran into an unexpected error.


TAPAS partners organized a successful stakeholder workshop at the end of September as part of the Aquaculture Europe 2016 conference, which is one of the major events of the European aquaculture industry. The main aim of the TAPAS workshop was to inform aquaculture operators, organizations, regulators, policy makers and scientists about the project in general, discuss any issues and potential "bottlenecks" in European aquaculture legislation and deliberate about ecosystem services provided by aquaculture and its public perception. The event was an integral part of the TAPAS stakeholder involvement process to collaborate on issues that limit the development of Europe's aquaculture sector. Presentations and small discussion groups allowed the participants to discuss key issues regarding aquaculture's role in food security, what is considered as acceptable aquaculture impacts, and bottleneck for aquaculture. Small discussion, or "break-out" groups, allowed the TAPAS consortium members to record findings, mediate discussions and prepare a summary of the key opinions and issues raised.
There was no unanimous agreement about the role of aquaculture in European food security as it was acknowledged to be on a global scale, rather than regional. There was also no apparent agreement on what are the acceptable impacts of aquaculture. The opinions raised varied locally and regionally, as public perception of aquaculture impacts, environmental status and highlighted media trends differed. The key to this discussion was the public perception and derived benefits of aquaculture and their ecosystem impacts. Attendees raised that there is often a lack of structure for decision-making processes across Europe, rather than the actual duration of the timelines themselves. In addition, key bottlenecks hindering aquaculture development included lack of consistency in licensing, finance and innovation. License processing time, often due to limited resources at regulatory agencies, were also outlined as a main limiting factor.
Ecosystem services provided by aquaculture were not fully understood by stakeholders and the benefits of aquaculture to the wider ecosystem remained unclear. These issues, as well as aquaculture policy, regulation and social carrying capacity were addressed, summarized and will later influence the TAPAS plan to consolidate the environmental sustainability of European aquaculture.
The workshop participants strongly agreed that stakeholder engagement is a key component to the success of the project. The views and requirements of the stakeholders will also be integrated into the final research plans of the TAPAS case studies developed in the coming months.


There are two types of admission of children into school – the 'Main Round' and 'In-Year' admissions.
This is when you wish to change schools during the school year, due to a house move for example. You may apply for an In-Year place using the Hampshire County Council (HCC) online application. If you do not wish to apply online a paper form can be obtained from any HCC school.
Documents relating to our school admission policies, FAQ's about starting school, and our new parent information are shown below.
You can find out more about school admissions and catchment areas by clicking on the links below.
Admissions – Click on this link to take you to the Admissions to Hampshire Schools website.


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LSU Football Team Getting White House Visit in Quickly
President Donald Trump will welcome the Louisiana State University football team to the White House on Friday after the Tigers defeated Clemson on Monday night in the college playoff final. Before visiting the White House, the LSU team will tour the nearby National Museum of African American History and Culture, according to The Advocate newspaper in Baton Rouge. Trump attended the national championship game in New Orleans, which LSU won 42-25, and he received an enthusiastic welcome from the crowd. It's unclear what could be on the menu for the team, but the president took much delight in serving last year's champions hamburgers and pizza during the partial government shutdown.
Filed Under: clemson, college playoff final, football team, Louisiana State University, President Donald Trump, tigers, white house
Categories: Featured, Football, Sports


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Career tips for public health professionals
Industry experts give their top tips for public health job seekers
Photo by Jason Goodman on Unsplash
By Hanna Knowles
Looking for a job in public health? Do you want to boost your public health career with an online master of public health degree?
Before taking the plunge, review what these public health experts — and Oregon State's online MPH instructors — have to say about the field.
Why do you love public health?
Paula Kaiser | Director, Health Outcomes Research and Evaluation, Samaritan Health Services
I love being able to help people at the population level, being able to do good in terms of systems and having an impact broader than just working with people one-on-one.
I love that public health is an applied field, but that it includes elements of social science. And that it requires really thoughtful critical efforts to think through what we're trying to do and really understand the kind of challenges and resources that are available to be as efficient as possible in terms of optimizing population health.
Alice Gandleman | Former Director, CA Prevention Training Center, UC San Francisco (retired)
The realization I had about 15 years ago is that when public health is working at its best nobody knows it exists. When there are no disease outbreaks, people don't even know what public health is.
When things are going well — working to keep the public healthy — the whole field seems invisible. I think that people become familiar with the critical role of public health when something is going on.
What skills and traits are important for public health professionals?
Paula Kaiser
I think the most important things really are being curious and being collaborative.
Public health is not a solo sport. Any sort of public health position involves working with lots of people who know lots of different types of information. Being able to approach people and be humble and respect the knowledge that they're bringing to the table and knowing how to how to work with lots of different people is one of the most essential skills and traits.
Within public health, my training is in epidemiology, and one of the things that I think is most important for epidemiologists in particular are the skills of critical thinking and really being very thoughtful and intentional about our goals and how we are going to accomplish them.
One of the things that I most appreciate about my training in epidemiology is that it really helps me formalize and think through that process.
Alice Gandleman
You have to believe in prevention.
There are many different levels of prevention — primary, secondary and tertiary. Primary prevention is preventing infections from occurring, and most public health conditions, such as chronic diseases and communicable diseases, for the most part are preventable. But just because they're preventable doesn't mean it's easy.
You also have to recognize that behaviors contribute to our health status, but also the circumstances in which people are born, live and grow contribute quite a bit.
Those include economic and social injustices and these bigger structural factors that people often have no control over, but they can also impact health.
I think public health practitioners today must take a multi-level approach to address these factors, which mean understanding biological, genetic and behavioral factors, and also the social, economic and political factors that really impact health.
What public health jobs are in demand?
At Samaritan, at a health care system, I see a lot of demand for people with training and background in population health sciences.
The medical system in the U.S. is going through this very slow and torturous transition from a fee-for-service model to value-based payment, including this whole idea of incorporating population health and how health systems address social determinants of health, and the kind of obligation and responsibilities that health systems have to improve population health outcomes.
It's a tough thing, and health systems really need people who understand the population health perspective. I think public health training is so good for that.
I personally would love to see more people with public health training coming into health systems and working on population in the health care setting.
Many health departments received quite a bit of money from the CDC.
I know that many public health departments are hiring communicable disease specialists. They're called different things in different areas, but they're really the backbone of public health departments.
Sometimes they're called disease intervention specialists or public health investigators, but they are the ones who really help to investigate health problems. They are at the core of the contact tracing case investigation work we saw during the pandemic.
They're also hiring a lot of health educators, epidemiologists and other health care professionals to work with and coordinate a lot of these public health issues, and to work with communities that are most affected by them.
So, again, it's really understanding not only who's becoming affected by these conditions, but why, and really having the skills to work with them and not place the burden on the individual, but to really recognize these larger factors.
I would also say that there's a lot more funding coming out around health equity and trying to end health disparities.
Can you describe the online public health courses you teach?
I work at Samaritan Health Services, which is not a traditional public health setting, but that being said, the class I teach is health care epidemiology, and so I think it's directly related to my work at Samaritan.
My job at Samaritan, as director for research and evaluation, involves a lot of supporting Samaritan investigators doing research projects and evaluation projects.
We also do a lot of collaborative research, for example, with other faculty at Oregon State University. The work that I do at Samaritan is actually directly related to the class that I teach, and I try really hard to keep the class relevant and reflect the kind of challenges that I'm working through and things I have learned from the job.
In my graduate program, I used almost every class that I took in my career, and I want to create that experience for students.
So, you know, there's assessing implementation, evaluation, quality assurance. There's a variety of connected activities that students are always going to use in their public health careers.
The thing that they also learn in the first class, the evidence-based public health one, is how to construct a logic model, and it's really basically just a one-page roadmap of a public health program.
The second course is a grant writing course, and throughout any public health practitioner's career they're going to be either responsible for, or significantly included in the grant writing process, because that's how public health operates. I make the assignments as real-world as possible.
There's a lot of group work in both of these classes. That's another reality of public health —you're always working with individuals and groups. I think it's really important to hone those skills in graduate school.
How do you build a connection with your online MPH students?
One of the assignments in the first week of classes is for all students to record an introductory video saying hi, talking a little bit about their background and what they're hoping to get out of the class.
I watch all of those, and I really appreciate the opportunity to see the students and hear their own words. I think it's good for the students to be able to meet the other students and get to know who else is in the class.
The other thing that I do, that I think has been really valuable, is that every week I have students submit a reflection on the coursework. It just has become a really helpful way for me get informal feedback. Sometimes it's related to the content, something that's still confusing or something that was particularly interesting.
But sometimes it's also a student's opportunity to share personal experiences or professional experiences, or just something else that they heard about or read about somewhere that connects.
From a didactic perspective, I think it's helpful to help students make those connections to what everything else is happening in their life.
From a personal perspective, it's been a really great way for me to build a connection.
Above all, I want my students to feel comfortable. I want them to ask questions if something isn't clear; I want them to let me know.
I don't want to provide superfluous information that they don't need, but I provide a lot of extensive feedback on their assignments because these are almost all writing assignments, and so I really want to help them be as good as they can be. I will provide suggestions and comments and ask them to consider things, as if I was talking to them.
It's a good time to be in public health
Within the field of public health, there is something for anyone who cares about population health and prevention.
And there is demand for skilled public health professionals, making it a worthwhile career choice and a good time to be pursuing a job in public health.
Learn more about Oregon State's Online MPH-Public Health Practice program.
Tags MPH Online, Public Health
← When it comes to changing your metabolism, small choices mean big results → Physical education requirements continue to decline at colleges and universities


text:
Sri Lanka / Players / Chandika Hathurusingha
Domestic Competitions
PREMIER LEAGUE TOURNAMENT TIER A
Avishka Fernando
Angelo Mathews
Lasith Malinga
Kusal Mendis
Kusal Perera
Dimuth Karunaratne
Dhananjaya de Silva
Jeevan Mendis
Isuru Udana
Chandika Hathurusingha
Full name Upul Chandika Hathurusingha
Born September 13, 1968, Colombo
Major teams Sri Lanka, Moors Sports Club, Tamil Union Cricket and Athletic Club
Bowling style Right-arm medium-fast
Other Coach
Relation Brother - CM Hathurusingha
Tests 26 44 1 1274 83 29.62 3573 35.65 0 8 154 0 7 0
ODIs 35 33 1 669 66 20.90 1318 50.75 0 4 59 0 6 0
First-class 207 326 28 10861 191 36.44 20 61 161 0
List A 96 86 7 2203 93 27.88 0 16 26 0
T20s 5 4 1 137 57* 45.66 95 144.21 0 1 11 6 2 0
Tests 26 32 1962 789 17 4/66 5/145 46.41 2.41 115.4 1 0 0
ODIs 35 25 954 709 14 4/57 4/57 50.64 4.45 68.1 1 0 0
First-class 207 22359 9381 425 8/29 22.07 2.51 52.6 12 2
List A 96 2663 1752 59 4/18 4/18 29.69 3.94 45.1 3 0 0
T20s 5 3 72 85 3 1/23 1/23 28.33 7.08 24.0 0 0 0
Test debut New Zealand v Sri Lanka at Hamilton, Feb 22-26, 1991 scorecard
Last Test Pakistan v Sri Lanka at Lahore, Mar 4-8, 1999 scorecard
ODI debut Pakistan v Sri Lanka at Sargodha, Jan 10, 1992 scorecard
Last ODI Pakistan v Sri Lanka at Jamshedpur, Mar 19, 1999 scorecard
First-class debut 1988/89
Last First-class Tamil Union Cricket and Athletic Club v Burgher Recreation Club at Colombo (PSS), Nov 19-21, 2004 scorecard
List A debut 1990/91
Last List A Galle Cricket Club v Tamil Union Cricket and Athletic Club at Colombo (RPS), Dec 14, 2004 scorecard
T20s debut Tamil Union Cricket and Athletic Club v Singha Sports Club at Colombo (Bloomfield), Oct 9, 2005 scorecard
Last T20s Sinhalese Sports Club v Tamil Union Cricket and Athletic Club at Colombo (Burgher), Nov 5, 2005 scorecard
44 Tamil Union v Sinhalese Colombo (Burgher) 5 Nov 2005 T20
- Tamil Union v Police SC Colombo (Colts) 30 Oct 2005 T20
1/24, 21 Tamil Union v Burgher Colombo (SSC) 26 Oct 2005 T20
1/23, 57* Tamil Union v Sebast CAC Colombo (CCC) 16 Oct 2005 T20
1/38, 15 Tamil Union v Singha SC Colombo (Bloomfield) 9 Oct 2005 T20
An opening batsman with a compact style and steadfast nature, Hathurusingha forged a steady opening partnership with Roshan Mahanama. Capable of playing lengthy innings when the situation warranted ,Hathurusingha possessed a variety of strokes. A handy medium-pace bowler, capable of genuine movement through the air. Called into the side against New Zealand at Hamilton as a replacement for the injured Mahanama, he responded by making three half-centuries in his first three Tests. Despite some handy contributions in following series, after an arduous time against the pace of South Africans he found himself left out. After a period on the sidelines, during which no other opener could establish a permanent position, he was selected to tour Pakistan and responded with an excellent series. After the emergence of Sanath Jayasuriya on the proceeding tour of Australia, Hathurusingha was again left out. Sri Lanka's failure to develop a competent allrounder saw Hathrurusingha reincarnated as a middle-order batsman and medium-pace bowler during the tour of England and The Asian Test Championship, though this experiment was shortlived. Chosen for the 1999 World Cup, Hathurusinghe was not called upon to play a single match, and the selectors' policy of picking younger players singled the end to his international career. He quit first-class cricket in 2005 to focus on coaching and has since coached UAE and Sri Lanka A.
Hathurusingha expects Sri Lanka's batsmen to step up (Jun 27, 2019)
Dysfunction, hope, more dysfunction; Sri Lanka's World Cup rolls on (Jun 15, 2019)
Past success could return to haunt Hathurusingha (Jun 11, 2019)
Hathurusingha, de Mel 'thrash things out' to work together for World Cup (May 2, 2019)
Hathurusingha ordered to return to Sri Lanka after South Africa ODIs (Mar 14, 2019)
Chandika Hathurusingha news and articles »
Chandika Hathurusingha attends a training session
© AFP/Getty
Chandika Hathurusingha helps Avishka Fernando work out a technical kink at training
© Peter Della Penna
Sri Lanka coach Chandika Hathurusingha keeps a watchful eye at training
Chandika Hathurusingha cricket photos
Player of the Premier Tournament 2001-02, 2002-03, 2003-04


In the wake of 9/11, newspapers around the world carried some version of the headline, "We're All Americans Now." The idea was that the destruction of the Twin Towers wasn't just an assault on the United States but on civilization itself, and everyone had a stake in defending it.
That solidarity dissipated fairly quickly amid political debates over post-9/11 foreign policy, but for a brief, shining moment the world was one.
In similar fashion, perhaps today's headline, in a particular way for the 1.3 billion Catholics of the world, ought to be, "We're All Parisians Now." This time, one hopes the possibilities won't slip away quite so quickly.
Yesterday's stunning fire at the Cathédrale Notre-Dame de Paris, which has left the magnificent Gothic structure a smoldering hulk of its former self, represents a devastating blow not merely to French Catholics but to Catholicism everywhere, since it's one of those sites that transcends its nationality and culture.
Dominating the Paris skyline for eight centuries, there's almost no chapter of Church history the cathedral doesn't somehow capture and reflect.
While its role in such upheavals as the French Revolution as the Napoleonic era are well-known, Notre Dame's history also includes less familiar nuggets. For instance, the late Cardinal Jean-Marie Lustiger, a convert from Judaism, liked to point out that the façade of the cathedral displays statues of the 28 kings of Israel, which was a way for designers to express gratitude to Paris's Jewish community for its financial support of the original construction.
In that sense, Lustiger would say, Notre Dame was also a unique symbol of Jewish/Catholic friendship.
Investigations were still underway as of this writing as to the causes of the fire, though officials in Paris seem to have excluded either arson or terrorism.
In all honesty, though, it doesn't really matter right now. The point is that it's happened, and it's left a gash right at the heart not only of France and Europe but Catholicism everywhere, because Notre-Dame de Paris is among that handful of special places around the world that symbolize the beauty and depth of the faith at its most compelling.
Late last night in Paris, French President Emanuel Macron stood before the Cathedral of Notre Dame and vowed to the nation that it would be rebuilt.
"Notre-Dame is burning, and I know the sadness, and this tremor felt by so many fellow French people. But tonight, I'd like to speak of hope too," he said. Macron announced the launch of a global fundraising campaign, promising that the "greatest talents" will be recruited to assist.
"I solemnly say tonight: we will rebuild it together," Macron said.
That spirit, many observers believe, should facilitate cooperation between the French government and the Church in the work ahead.
Though Macron spoke of rebuilding Notre Dame as the destiny of the French, it's arguably every bit as much a call to the global Church. While last night's fire struck Paris, it could just as easily have happened anywhere.
We know, for instance, that a fire laid waste to Rome's famed Basilica of St. Paul Outside the Walls in 1823, so what one sees today is essentially an exact replica. As many of Europe's famed cathedrals fall increasingly into disuse as places of worship amid growing secularism, maintenance becomes more difficult and the risk of similar incidents elsewhere grows.
Wherever a cornerstone of the Church's heritage is at risk, all Catholics stand to lose.
While that can be a noble sentiment, given the reputation of his right-wing Law and Justice party, some may hear echoes of the nationalist push that characterizes much of European political life. Leaders will need to show restraint about any rhetoric that could suggest rebuilding the cathedral is partisan.
Though many have commented on the horror that this tragedy struck at the beginning of Holy Week, it's also the case that the Church's holiest period of the year creates a special moment of opportunity to galvanize Catholic support worldwide.
Imagine, for instance, a special collection for Cathédrale Notre-Dame de Paris during the Saturday Easter vigil later this week, when Catholics all around the world enter their local church to find it darkened, later broken by a flame - the lighting of the Paschal candle. More powerful symbolism is difficult to imagine.
Whatever form it takes, one hopes the global Church will respond energetically to save not only its past but also its present, since Notre Dame is also a living place of faith in the heart of one of the world's most thoroughly secular environments.
Much is at stake - because, as things stand, we're all Parisians now.


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Colorway in The Brattleboro Reformer
Big thanks to Dave Madeloni at The Brattleboro Reformer for his great piece on the band in advance of our show at Metropolis on May 20, 2016.
Originally published 5/19/16 in the Brattleboro Reformer.
BRATTLEBORO >> When I asked Colorway's F. Alex Johnson about his band's approach to playing live, his answer exuded a rock attitude: "Well, you're only as good as your last show. And our last show was pretty unbeatable."
That is good news for area rock fans since the trio is coming to Brattleboro for the first time tomorrow night on the heels of its most important and impressive gig. According to Johnson, their guitarist/singer/songwriter, Colorway rose to the occasion, opening a sold out show for Foreigner at the Calvin Theater in Northampton back on May 6. "My fiancé said she was so nervous for me and so were the people around the theater who helped me land the gig. But I was so sure of the band and the material and what I've worked for these past three years not to mention my eight years of sobriety, that I just went out there in front of 1,200 people who didn't come to see us and put on 33 minutes of a rock show that felt like it belonged absolutely."
For Johnson-a veteran performer with about 30 years of experience-reading an audience is old hat and he knew that he and his rhythm section connected with the Foreigner fanatics. "Believe me, I've opened shows where you can sense an unease and restlessness by the crowd who just want to see the headliner but these folks just understood that here are three guys who just love music—that they've played it all their lives and they probably will forever. And they got it. They actually got it. And I couldn't have been prouder, happier or more thankful."
"I know in the rock world smiling isn't "cool" but I was ear-to-ear and so were my guys (bassist Matt Clegg and drummer Riley Godleski). It was like a surprise celebration of songs very few people knew . . . but they got it by the middle of the first song and that's why I do this thing we're talking about here."
Johnson, who has several years of experience performing in front of large audiences as the guitar player for the renowned Young@Heart Chorus, has fond memories of rocking the region's small clubs. "As for Brattleboro, I rocked those streets in my 20s (read: the 1990s). Drunk Stuntmen (actually Soup for a time) played the Common Ground and The Mole's Eye and a few other places I can vaguely remember. It was an awesome town to play and to hang out in.. But it's been so long and I know places have closed or had fires or whatever and the scene has changed. But music is music and if you make good sounds people will come to see you. That's what I'm hoping for: make good sounds and have people hear them. That's pretty much my life's mission."
Johnson's mission was almost scuttled by the demons that derailed many a rocker: drugs and alcohol. "I went through some major life changes about ten years ago. I lost my mom and aunt to cancer within 18 months of each other. I had been in a successful local band (Drunk Stuntmen) for 15 years and was touring non-stop with the Young@Heart Chorus. But the booze and the drugs caught up to me and I had to give up the Stuntmen. The lifestyle involved was just too much for me and I needed to just step away from it."
It took about four years of soul searching and abstinence for Johnson to feel like he could start making original music again. "I met the love of my life and one day the songs started pouring out. I contacted two of my oldest and most musical friends, Dave Hayes and JJ O'Connell, and we put the band together. But, Dave had to step away for personal reason and JJ was just too in-demand so last summer I had to regroup. Thankfully I found a rhythm section that already played together in Shokazoba and knew each other's quirks."
By finding love, sobriety, and a band, F Alex Johnson is back to doing what he does best: making tasteful rock music. And Colorway is on its way, with a new line-up touting "Black Sky Sequined" a strong new CD. "I'm doing what I love." added Johnson. "I hope it can sustain me for the long term and that I can somehow continue to create and to have my art support me. It's been my goal in life since I got my first round of applause for a soft shoe routine I did at age four or five at Durgin Park in Boston. I heard those claps and knew they were for me. I looked at my mom and she was beaming. I know she's still beaming and, well, I never really stopped dancing, I guess."
Colorway will be at Metropolis on Friday, from 10 p.m. to 1 a.m., 55 Elliot St. Brattleboro. There is no charge, but must be 21 or over.
For information call 802-254-8500 or visit www.colorwaymusic.com.
Dave Madeloni writes music reviews for the entertainment section of the Brattleboro Reformer.