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HeJ and HuJ simulation equivalence

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@@ -1114,7 +1114,7 @@ We have two more notions for coalgebraic bisimulation in $\Set$, that is to defi
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\begin{definition}[Hermida-Jacobs Simulation] \begin{definition}[Hermida-Jacobs Simulation]
Assuming that $\appr$ is a natural order structure on a functor $F\c\BC\to\BC$, an $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ of $\rel(\BC)$ is a \emph{Hermida-Jacobs simulation} from a coalgebra $(X,\alpha)$ to $(Y,\beta)$, whenever the following diagram commutes laxly: Assuming that $\appr$ is a natural order structure on a functor $F\c\BC\to\BC$, an $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ of $\rel(\BC)$ is a \emph{Hermida-Jacobs simulation} from a coalgebra $(X,\alpha)$ to $(Y,\beta)$, whenever the following diagram commutes laxly:
\begin{equation}\label{eq:diag-hj-sim} \begin{equation*}\label{eq:diag-hj-sim}
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
X \& R \& Y \\ X \& R \& Y \\
FX \& (FR)^\dagger \& FY FX \& (FR)^\dagger \& FY
@@ -1128,7 +1128,7 @@ We have two more notions for coalgebraic bisimulation in $\Set$, that is to defi
\arrow["{{(Fp_1)^\dagger}}", from=2-2, to=2-1] \arrow["{{(Fp_1)^\dagger}}", from=2-2, to=2-1]
\arrow["{{(Fp_2)^\dagger}}"', from=2-2, to=2-3] \arrow["{{(Fp_2)^\dagger}}"', from=2-2, to=2-3]
\end{tikzcd} \end{tikzcd}
\end{equation} \end{equation*}
\end{definition} \end{definition}
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\begin{prop} \begin{prop}
@@ -1194,22 +1194,24 @@ Traditionally, simulations in $\Set$ are defined using relators. In this section
\end{example} \end{example}
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\begin{example} \begin{example}
Given $F\c\Set\to\Set$ with an order structure $\appr$ on it the relator that sends $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ to $(FX \stackrel{q_1}{\leftarrow} \appr\comp(FR)^\dagger\comp\appr \stackrel{q_2}{\to}FY)$ is a relator. We call it \emph{bi-lax Barr Relator}. There are other variations of this: left-lax ($\appr\comp(FR)^\dagger$) and right-lax ($(FR)^\dagger\comp\appr$). Given $F\c\Set\to\Set$ with an order structure $\appr$ on it the relator that sends $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ to $(FX \stackrel{q_1}{\leftarrow} \appr\comp(FR)^\dagger\comp\appr \stackrel{q_2}{\to}FY)$ is a relator. We call it \emph{bi-lax Barr Relator}. There are other variations of this: left-lax ($\appr\comp(FR)^\dagger$) and right-lax ($(FR)^\dagger\comp\appr$). We denote this relator with $F^\leftrightarrow$.
\end{example} \end{example}
% %
\begin{prop}\label{prop:HeJ-HuJ} \begin{prop}\label{prop:HeJ-HuJ}
Every object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ in $\rel$: 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 $(Y,\beta)$:
\begin{itemize} \begin{enumerate}
\item is a Hughes-Jacobs simulation if it is a Hermida-Jacobs simulation. \item is a $F^\leftrightarrow$-simulation if it is a Hermida-Jacobs simulation.
\item is a Hermida-Jacobs simulation if it is a Hermida-Jacobs simulation, assuming the axiom of choice. \item is a Hermida-Jacobs simulation if it is a $F^\leftrightarrow$-simulation, assuming the axiom of choice.
\end{itemize} \end{enumerate}
\end{prop} \end{prop}
\begin{proof} \begin{proof}
\todo{Finish!} (1): $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ being a Hermida-Jacobs simulation means that for every $(x,y)\in R$, we have $\alpha\comp p_1(x,y)\appr(Fp_1)^\dagger\comp\sigma(x,y)$ and $(Fp_2)^\dagger\comp\sigma(x,y)\appr\beta\comp p_2(x,y)$. So, $x\mathrel{R}y$ gives that $\alpha(x)\mathrel{(\appr\comp(FR)^\dagger\comp\appr)}\beta(y)$ that means that $R$ is a $F^\leftrightarrow$-simulation.\\
(2): $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ being a $F^\leftrightarrow$-simulation means that $x\mathrel{R}y$ gives $\alpha(x)\mathrel{(\appr\comp(FR)^\dagger\comp\appr)}\beta(y)$ that means for every $(x,y)\in R$, there exist $(u,v)\in(FR)^\dagger$ such that $\alpha(x)\appr u$ and $v\appr\beta(y)$. We form a function $f\c R\to\powf(FR)^\dagger$ such that takes every $(x,y)$ to the set of the mentioned existing pairs $(u,v)$ in $(FR)^\dagger$. By the axiom of choice there exist a function $s\c\im_f\to(FR)^\dagger$. So, assuming that $f$ has the epi-mono factorization $(e,m)$, then we define $\sigma\c R\to(FR)^\dagger$ as $\sigma=s\comp e$. Now, the diagram~\eqref{eq:diag-hj-sim} commutes laxly for the defined $\sigma$.\qed
\end{proof} \end{proof}
\begin{rem} \begin{rem}
The proposition entails that Hermida-Jacobs simulation subsumes Hughes-Jacobs simulation. The proposition entails that Hermida-Jacobs simulation subsumes simulation relations defined with bi-lax Barr relators.
\end{rem} \end{rem}
\todo{Read section V of the LICS paper and see what is the non-symmetric relator. Then try to see if HJ-simulation subsumes it.}
\subsection{Simulations in Double Categories} \subsection{Simulations in Double Categories}
\todo{Try to define simulation in a given double category. Perhaps you need to order enrichment over the endofunctor on $\BC_0$. Then inspired by~\autoref{prop:HeJ-HuJ} you may be able to prove a general theorem for an arbitrary lifting!} \todo{Try to define simulation in a given double category. Perhaps you need to order enrichment over the endofunctor on $\BC_0$. Then inspired by~\autoref{prop:HeJ-HuJ} you may be able to prove a general theorem for an arbitrary lifting!}
%We show the category of partially ordered sets with monotone functions between them with $\poset$. %We show the category of partially ordered sets with monotone functions between them with $\poset$.