pc sync

draft/draft.tex

@@ -82,6 +82,8 @@
 \usetikzlibrary{arrows.meta}
 \usetikzlibrary{decorations}           % Required for all decorations
 \usetikzlibrary{decorations.pathmorphing} % Specifically for 'zigzag'
+\usetikzlibrary{matrix,positioning,arrows.meta,shapes.geometric}
+
 
 \tikzset{
   commutative diagrams/.cd,
@@ -1403,6 +1405,44 @@ We have two more notions for coalgebraic bisimulation in $\Set$, that is to defi
 \end{proof}
 %
 \subsection{Simulations in Set}
+
+%\begin{figure}[ht]
+%	\centering
+%	\begin{tabular}{|c|c|c|c|c|}
+%		\hline
+%		 & \textbf{Hughes-Jacobs} & \textbf{Hermida-Jacobs} & \textbf{Span-based} & \textbf{Aczel-Mendler} \\
+%		\hline
+%		$\appr\cdot-$ & ? & ? & ? & ? \\
+%		\hline
+%		$-\cdot\appr$ & ? & ? & ? & ? \\
+%		\hline
+%		$\appr\cdot-\cdot\appr$ & ? & ? & ? & ? \\
+%		\hline
+%	\end{tabular}
+%	\caption{Comparison of anonymous and onymous settings for relations and spans in $\Set$.}
+%	\label{fig:anonymous_onymous_transposed}
+%\end{figure}
+
+
+\begin{tikzpicture}[
+    scale=0.5, every node/.style={transform shape}
+]
+\matrix (m) [matrix of nodes,
+             nodes={draw, ellipse, minimum width=2.6cm, minimum height=1cm,
+                    align=center, font=\scriptsize, inner sep=2pt},
+             row sep=6mm, column sep=6mm]
+{
+  $\appr\cdot-$-Hughes-Jacobs   & $\appr\cdot-$-Hermida-Jacobs   & $\appr\cdot-$-Span-based   & $\appr\cdot-$-Aczel-Mendler   \\
+  $-\cdot\appr$-Hughes-Jacobs   & $-\cdot\appr$-Hermida-Jacobs   & $-\cdot\appr$-Span-based   & $-\cdot\appr$-Aczel-Mendler   \\
+  $\appr\cdot-\cdot\appr$-Hughes-Jacobs & $\appr\cdot-\cdot\appr$-Hermida-Jacobs & $\appr\cdot-\cdot\appr$-Span-based & $\appr\cdot-\cdot\appr$-Aczel-Mendler \\
+};
+
+% Example arrows (uncomment / edit as needed):
+\draw[-{Latex[length=2mm]}] (m-3-1) to[bend right=30] node[midway, above] {AC} (m-3-2);
+\draw[-{Latex[length=2mm]}] (m-3-2) to[bend right=30] (m-3-1);
+
+\end{tikzpicture}
+
 Traditionally, simulations in $\Set$ are defined using relators. In this section we make a comparison on this concept with the other notions of simulation that we mentioned.
 %\begin{notation}
 %	We show the category of sets and binary relations with $\rels$, and we show a morphism in this category as $R\c X\rto Y$ that is a relation $R$.
@@ -1417,7 +1457,7 @@ Traditionally, simulations in $\Set$ are defined using relators. In this section
 %\end{proof}
 %The above translation seems to be true in a more general case, where $\spa$ and $\rel$ are defined on an arbitrary category (the latter is called an allegory then).
 \begin{definition}[Relator]\label{def:relator}
-	Assuming $F$ is a functor on $\Set$, a $F$-relator or simply a relator $\relar$ is a monotone map that sends an object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ of $\Rel$ to $(FX \stackrel{q_1}{\leftarrow} \relar R \stackrel{q_2}{\to}FY)$. 
+	Assuming $F$ is a functor on $\Set$, an $F$-relator or simply a relator $\relar$ is a monotone map that sends an object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ of $\Rel$ to $(FX \stackrel{q_1}{\leftarrow} \relar R \stackrel{q_2}{\to}FY)$. 
 \end{definition}
 %
 %\begin{definition}[Hermida-Jacobs Simulation]\label{def:hej-sim-rela}
@@ -1458,10 +1498,10 @@ Traditionally, simulations in $\Set$ are defined using relators. In this section
 \end{example}
 %
 \begin{prop}\label{prop:HeJ-HuJ}
-	For a functor $F\c\Set\to\Set$, every object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ in $\rel$ from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$:
+Given $F\c\Set\to\Set$ and an object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ in $\rel$,
 	\begin{enumerate}
-		\item is a $\tilde{F}$-simulation if it is a Hermida-Jacobs simulation.
+		\item $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ is a $\tilde{F}$-simulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if it is a Hermida-Jacobs simulation.
-		\item is a Hermida-Jacobs simulation if it is a $\tilde{F}$-simulation, assuming the axiom of choice.
+		\item $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ is a Hermida-Jacobs simulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if it is a $\tilde{F}$-simulation, assuming the axiom of choice.
 	\end{enumerate}
 \end{prop}
 \begin{proof}