4 defs!
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@@ -387,45 +387,51 @@ Pouya Partow\inst{1}\orcidID{0009-0003-9652-9469}}
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\section{Coalgebraic Bisimulation}%\label{sec:}
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\begin{definition}[Relation Lifting]
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Assuming $F\c\Set\to\Set$ is a functor, then we call $\rel(F)\c\rel\to\rel$ a relation lifting of $F$, where the following diagram commutes:
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\begin{equation*}
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\begin{tikzcd}[ampersand replacement=\&]
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\rel \&\& \rel \\
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{\Set\times\Set} \&\& {\Set\times\Set}
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\arrow["{\rel(F)}", from=1-1, to=1-3]
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\arrow[from=1-1, to=2-1]
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\arrow[from=1-3, to=2-3]
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\arrow["{F\times F}"', from=2-1, to=2-3]
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\end{tikzcd}
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\end{equation*}
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In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at least has products, and by $\rel(\BC)$ we refer to the relatios in $\BC$, and by relation we are referring to a span $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ such that the morphism $\brks{p_1,p_2}$ is a mono. For the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ and $\spa$ we mean $\rel(\Set)$ and $\spa(\Set)$, accordingly.
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%\begin{definition}[Relation Lifting]
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% Assuming $F\c\BC\to\BC$ is a functor, then we call $\rel(F)\c\rel(\BC)\to\rel(\BC)$ a relation lifting of $F$, where the following diagram commutes:
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% \begin{equation*}
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% \begin{tikzcd}[ampersand replacement=\&]
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% \rel(\BC) \&\& \rel(\BC) \\
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% {\BC\times\BC} \&\& {\BC\times\BC}
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% \arrow["{\rel(F)}", from=1-1, to=1-3]
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% \arrow[from=1-1, to=2-1]
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% \arrow[from=1-3, to=2-3]
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% \arrow["{F\times F}"', from=2-1, to=2-3]
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% \end{tikzcd}
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% \end{equation*}
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%\end{definition}
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%We have notions of bisimulation that may involve relation lifting. An example of relation lifting is to use the image factorization provided in regular categories.
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\begin{definition}[$F$-Relator]
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For a set functor $F$, and for sets $X$ and $Y$, an $F$-relator $\relar$ is a map that takes every relation on $X\times Y$ to a relation on $FX\times FY$, and it is monotone with respect to inclusion.
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\end{definition}
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%Also, for the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ we mean $\rel(\BC)$.
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%
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We take $\rel(F)\c\rel\to\rel$ to be the functor that for an arbitrary functor $F$ takes a relation $R$, where $R\in\obj(\rel)$ and $R\subseteq X_1\times X_2$, and gives the relation that is the image of the function $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$.
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\begin{definition}[Bisimulation]
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For a functor $F\c\Set\to\Set$, a bisimulation is a $\rel(F)$-coalgebra in $\rel$.
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\end{definition}
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%We take $\rel(F)\c\rel(\BC)\to\rel(\BC)$ to be the functor that for an arbitrary functor $F$ takes a relation $R$, where $R\in\obj(\rel)$ and $R\subseteq X_1\times X_2$, and gives the relation that is the image of the function $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$.
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%\begin{definition}[Bisimulation]
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% For a functor $F\c\BC\to\BC$, a bisimulation is a $\rel(F)$-coalgebra in $\rel$.
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%\end{definition}
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%
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\begin{prop}
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Assuming that $(R,\alpha)$ is a $\rel(F)$-coalgebra, where $\alpha=\beta_1\times\beta_2$ in $\Set\times\Set$, then the following diagram commutes, and vice-versa:
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\begin{equation*}
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\begin{tikzcd}[ampersand replacement=\&]
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{X_1} \& R \& {X_2} \\
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{FX_1} \& FR \& {FX_2}
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\arrow["{\beta_1}"', from=1-1, to=2-1]
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\arrow["{p_1}"', from=1-2, to=1-1]
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\arrow["{p_2}", from=1-2, to=1-3]
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\arrow["\beta", from=1-2, to=2-2]
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\arrow["{\beta_2}", from=1-3, to=2-3]
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\arrow["{Fp_1}", from=2-2, to=2-1]
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\arrow["{Fp_2}"', from=2-2, to=2-3]
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\end{tikzcd}
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\end{equation*}
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\end{prop}
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%\begin{prop}
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% Assuming that $(R,\alpha)$ is a $\rel(F)$-coalgebra, where $\alpha=\beta_1\times\beta_2$ in $\BC\times\BC$, then the following diagram commutes, and vice-versa:
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% \begin{equation*}
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% \begin{tikzcd}[ampersand replacement=\&]
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% {X_1} \& R \& {X_2} \\
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% {FX_1} \& FR \& {FX_2}
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% \arrow["{\beta_1}"', from=1-1, to=2-1]
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% \arrow["{p_1}"', from=1-2, to=1-1]
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% \arrow["{p_2}", from=1-2, to=1-3]
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% \arrow["\beta", from=1-2, to=2-2]
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% \arrow["{\beta_2}", from=1-3, to=2-3]
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% \arrow["{Fp_1}", from=2-2, to=2-1]
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% \arrow["{Fp_2}"', from=2-2, to=2-3]
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% \end{tikzcd}
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% \end{equation*}
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%\end{prop}
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\begin{definition}[Aczel-Mendler Bisimulation]
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\begin{equation*}
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A relation $R\subseteq X\times Y$ is an \emph{Aczel-Mendler bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever there is a morphism $\gamma\c R\to FR$ called witness that commutes in the following diagram:
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\begin{equation*}\label{eq:acz-mend-diag}
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\begin{tikzcd}[ampersand replacement=\&]
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{X} \& R \& {Y} \\
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{FX} \& FR \& {FY}
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@@ -440,51 +446,72 @@ We take $\rel(F)\c\rel\to\rel$ to be the functor that for an arbitrary functor $
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\end{equation*}
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\end{definition}
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\begin{definition}[Hermida-Jacobs Bisimulation]
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\begin{definition}[Witnessless $\relar$-Bisimulation]
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For a relator $\relar$, a relation $R\subseteq X\times Y$ is a \emph{witnessless $\relar$-bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever for every $x\in X$ and $y\in Y$, we have $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(\relar R)}\beta(y)$.
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% \begin{equation*}
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% \begin{tikzcd}[ampersand replacement=\&]
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% {X} \& R \& {Y} \\
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% {FX} \& \rel(F)R \& {FY}
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% \arrow["{\alpha}"', from=1-1, to=2-1]
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% \arrow["{p_1}"', from=1-2, to=1-1]
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% \arrow["{p_2}", from=1-2, to=1-3]
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% \arrow["{\beta}", from=1-3, to=2-3]
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% \arrow["{q_1}", from=2-2, to=2-1]
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% \arrow["{q_2}"', from=2-2, to=2-3]
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% \end{tikzcd}
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% \end{equation*}
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\end{definition}
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\begin{definition}[Witnessful $\relar$-Bisimulation]
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For a relator $\relar$, a relation $R\subseteq X\times Y$ is a \emph{witnessful $\relar$-bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever there is a morphism $\gamma\c R\to \relar R$ called witness that commutes in the following diagram:
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\begin{equation*}
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\begin{tikzcd}[ampersand replacement=\&]
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{X} \& R \& {Y} \\
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{FX} \& (FR)^\dagger \& {FY}
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{FX} \& \relar R \& {FY}
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\arrow["{\alpha}"', from=1-1, to=2-1]
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\arrow["{p_1}"', from=1-2, to=1-1]
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\arrow["{p_2}", from=1-2, to=1-3]
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\arrow["\gamma", from=1-2, to=2-2]
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\arrow["{\beta}", from=1-3, to=2-3]
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\arrow["{(Fp_1)^\dagger}", from=2-2, to=2-1]
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\arrow["{(Fp_2)^\dagger}"', from=2-2, to=2-3]
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\end{tikzcd}
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\end{equation*}
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\end{definition}
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\begin{definition}[Hughes-Jacobs Bisimulation]
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\begin{equation*}
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\begin{tikzcd}[ampersand replacement=\&]
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{X} \& R \& {Y} \\
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{FX} \& (FR)^\dagger \& {FY}
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\arrow["{\alpha}"', from=1-1, to=2-1]
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\arrow["{p_1}"', from=1-2, to=1-1]
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\arrow["{p_2}", from=1-2, to=1-3]
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\arrow["{\beta}", from=1-3, to=2-3]
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\arrow["{(Fp_1)^\dagger}", from=2-2, to=2-1]
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\arrow["{(Fp_2)^\dagger}"', from=2-2, to=2-3]
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\arrow["{q_1}", from=2-2, to=2-1]
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\arrow["{q_2}"', from=2-2, to=2-3]
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\end{tikzcd}
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\end{equation*}
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\end{definition}
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\begin{definition}[Vanilla Bisimulation]
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\begin{equation*}
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\begin{tikzcd}[ampersand replacement=\&]
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{X} \& R \& {Y} \\
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{FX} \& FR \& {FY}
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\arrow["{\alpha}"', from=1-1, to=2-1]
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\arrow["{p_1}"', from=1-2, to=1-1]
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\arrow["{p_2}", from=1-2, to=1-3]
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\arrow["{\beta}", from=1-3, to=2-3]
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\arrow["{Fp_1}", from=2-2, to=2-1]
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\arrow["{Fp_2}"', from=2-2, to=2-3]
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\end{tikzcd}
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\end{equation*}
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A relation $R\subseteq X\times Y$ is a \emph{vanilla bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever for every $x\in X$ and $y\in Y$, we have $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(FR)}\beta(y)$.
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% \begin{equation*}
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% \begin{tikzcd}[ampersand replacement=\&]
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% {X} \& R \& {Y} \\
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% {FX} \& FR \& {FY}
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% \arrow["{\alpha}"', from=1-1, to=2-1]
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% \arrow["{p_1}"', from=1-2, to=1-1]
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% \arrow["{p_2}", from=1-2, to=1-3]
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% \arrow["{\beta}", from=1-3, to=2-3]
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% \arrow["{Fp_1}", from=2-2, to=2-1]
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% \arrow["{Fp_2}"', from=2-2, to=2-3]
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% \end{tikzcd}
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% \end{equation*}
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\end{definition}
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\begin{prop}
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The following propositions hold:
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\begin{enumerate}[label=(\Roman*), ref=(\Roman*)]
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\item Every Aczel-Mendler bisimulation is a vanilla bisimulation.
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\item Every vanilla bisimulation is an Aczel-Mendler bisimulation.
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\item Every Aczel-Mendler bisimulation is a witnessful bisimulation.
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\item Assuming the axiom of choice, every witnessful bisimulation is an Aczel-Mendler bisimulation.
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\item Every witnessless bisimulation is a witnessful bisimulation.
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\item Assuming the axiom of choice, every witnessless bisimulation is a witnessful bisimulation.
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\end{enumerate}
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\end{prop}
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\begin{proof}
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(I): Assuming $x\mathrel{R}y$, then given by~\eqref{eq:acz-mend-diag} we have $\gamma(x,y)\in FR$, $Fp_1\comp\gamma(x,y)=\alpha(x)$, and $Fp_2\comp\gamma(x,y)=\beta(y)$ that means $\alpha(x)\mathrel{(FR)}\beta(y)$.
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(II): Since $R$ is a vanilla bisimulation, for every $(x,y)\in R$ we have $\alpha(x)\mathrel{(FR)}\beta(y)$, so we can define $\gamma\c R\to FR$ as $\gamma(x,y)=(\alpha(x),\beta(y))$, and then $\gamma$ commutes in~\eqref{eq:acz-mend-diag}.
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(III):
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\end{proof}
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\todo{Discuss 4 versions of bisimulation (with witness/without witness, for relations/for spans). Which are equivalent? Which do not make sense?}
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\todo{In next section run a similar analysis for simulation: relator-based vs. Aczel-Mendler.}
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