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@@ -289,7 +289,8 @@
\newcommand{\obj}{\mathbf{Obj}}
\newcommand{\relar}{\mathbf{R}}
\newcommand{\emre}{\mathbf{L}}
\newcommand{\rto}{\mathrel{\tikz{\draw[-{Stealth}] (0,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);}}}
\newcommand{\spto}{\mathrel{\tikz{\draw[-{Stealth}] (0.0,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);}}}
\newcommand{\rto}{\mathrel{\tikz{\draw[-{Stealth}] (0.04,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);\draw (0.04,0) -- (0,-0.07);\draw (0.04,0) -- (0,0.07);}}}
\newcommand{\powf}{\mathcal{P}}
\newcommand{\sappr}{\sqsupseteq}
@@ -387,30 +388,84 @@ Pouya Partow\inst{1}\orcidID{0009-0003-9652-9469}}
%
\section{Coalgebraic Bisimulation}%\label{sec:}
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.
%\begin{definition}[Relation Lifting]
% 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:
% \begin{equation*}
% \begin{tikzcd}[ampersand replacement=\&]
% \rel(\BC) \&\& \rel(\BC) \\
% {\BC\times\BC} \&\& {\BC\times\BC}
% \arrow["{\rel(F)}", from=1-1, to=1-3]
% \arrow[from=1-1, to=2-1]
% \arrow[from=1-3, to=2-3]
% \arrow["{F\times F}"', from=2-1, to=2-3]
% \end{tikzcd}
% \end{equation*}
%\end{definition}
%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.
\begin{definition}[$F$-Relator]
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.
\end{definition}
%Also, for the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ we mean $\rel(\BC)$.
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 category of relations 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. We show morphisms in $\spa(\BC)$ by $\spto$, and in $\rel(\BC)$ by $\rto$.
A morphism $f\c R\rto S$ ($f\c R\spto S$) is a morphism in $\rel(\BC)$ ($\spa(\BC)$), whenever $f$, $g_1$ and $g_2$ are morphisms in $\BC$ for which the following diagram commutes in $\BC$:
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
{X_1} \& R \& {X_2} \\
{Y_1} \& S \& {Y_2}
\arrow["{g_1}"', from=1-1, to=2-1]
\arrow["{{p_1}}"', from=1-2, to=1-1]
\arrow["{{p_2}}", from=1-2, to=1-3]
\arrow["f", from=1-2, to=2-2]
\arrow["{g_2}", from=1-3, to=2-3]
\arrow["{q_1}", from=2-2, to=2-1]
\arrow["{q_2}"', from=2-2, to=2-3]
\end{tikzcd}
\end{equation*}
\begin{definition}[Relation Lifting]
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:
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
\rel(\BC) \&\& \rel(\BC) \\
{\BC\times\BC} \&\& {\BC\times\BC}
\arrow["{\rel(F)}", from=1-1, to=1-3]
\arrow[from=1-1, to=2-1]
\arrow[from=1-3, to=2-3]
\arrow["{F\times F}"', from=2-1, to=2-3]
\end{tikzcd}
\end{equation*}
\end{definition}
We have notions of bisimulation that may involve relation lifting. An example of a relation lifting is to use the image factorization provided in regular categories.
%
%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 $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$.
%\begin{equation*}
% \begin{tikzcd}[ampersand replacement=\&]
% R \& {R^\dagger} \&\& {X\times X}
% \arrow["{e_R}"', two heads, from=1-1, to=1-2]
% \arrow["{\brks{p_1,p_2}}", bend left=20, from=1-1, to=1-4]
% \arrow["{\brks{p^\dagger_1,p^\dagger_2}}"', tail, from=1-2, to=1-4]
% \end{tikzcd}
%\end{equation*}
We define a functor of type $(-)^\dagger\c\spa(\BC)\to\rel(\BC)$. It takes every span $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ to the image of its legs:
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
R \& {R^\dagger} \&\& {X\times Y}
\arrow["{e_R}"', two heads, from=1-1, to=1-2]
\arrow["{\brks{p_1,p_2}}", bend left=20, from=1-1, to=1-4]
\arrow["{\brks{p^\dagger_1,p^\dagger_2}}"', tail, from=1-2, to=1-4]
\end{tikzcd}
\end{equation*}
So, for every functor $F\c\BC\to\BC$ we have $(F-)^\dagger\c\rel(\BC)\to\rel(\BC)$ that takes every relation $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ to the following relation:
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
\& {(FR)^\dagger} \& \\
FX \&\& FY \\
\& {FX\times FY}
\arrow["{{{(Fp_1)^\dagger}}}"', from=1-2, to=2-1]
\arrow["{{{(Fp_2)^\dagger}}}", from=1-2, to=2-3]
\arrow["{{\brks{{(Fp_1)^\dagger},{(Fp_2)^\dagger}}}}"{description}, dashed, tail, from=1-2, to=3-2]
\end{tikzcd}
\end{equation*}
Also, there exist a well-known notion named relator, instead of relation lifting. A relator does not need to be a functor, but it should be a map of a specific type format that is monotone with respect to inclusion. Also, relators are defined only on $\Set$ unlike relation liftings. %We discuss relators more in depth in the later chapters.
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.
In $\Set$, there is another way to define the morphisms that is to say that $f\c R\rto S$ ($f\c R\spto S$) is a morphism in $\rel$ ($\spa$), whenever there are functions $g_1$ and $g_2$ that $x\mathrel{R}y\Rightarrow g_1(x)\mathrel{(FR)}g_2(y)$.
Now, considering the two types of morphisms that we can define for $\rel$, and having lifting over the relation or not, we give four notions of bisimulation:
%We take $\rel(F)\c\rel(\BC)\to\rel(\BC)$ to be the functor that for an arbitrary functor $F\c\BC\to\BC$ 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 $\brks{Fp_1,Fp_2}\c FR\to FX\times FY$.
%\begin{definition}[Bisimulation]
% For a functor $F\c\BC\to\BC$, a bisimulation is a $\rel(F)$-coalgebra in $\rel$.
%\end{definition}
%\begin{definition}[$F$-Relator]
% 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.
%\end{definition}
%Also, for the time being, we limit the discussion to the case $\BC=\Set$. For simplicity, by $\rel$ we mean $\rel(\BC)$.
%
%\begin{prop}
% 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:
@@ -428,7 +483,6 @@ In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at le
% \end{tikzcd}
% \end{equation*}
%\end{prop}
\begin{definition}[Aczel-Mendler Bisimulation]
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:
\begin{equation*}\label{eq:acz-mend-diag}
@@ -445,9 +499,10 @@ In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at le
\end{tikzcd}
\end{equation*}
\end{definition}
Aczel-Mendler bisimulation can be defined for an arbitrary category $\BC$ instead of $\Set$. Worth noting that with this definition,if $R$ is a bisimulation, it does not necassirily mean that $FR$ is a relation as well, for example $FR$ can have elements that are not pairs.
\begin{definition}[Witnessless $\relar$-Bisimulation]
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 $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(\relar R)}\beta(y)$.
\begin{definition}[Witnessless Bisimulation]
A relation $R\subseteq X\times Y$ is a \emph{witnessless 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 $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(FR)^\dagger}\beta(y)$.
% \begin{equation*}
% \begin{tikzcd}[ampersand replacement=\&]
% {X} \& R \& {Y} \\
@@ -462,23 +517,24 @@ In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at le
% \end{equation*}
\end{definition}
\begin{definition}[Witnessful $\relar$-Bisimulation]
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:
\begin{definition}[Hermida-Jacobs Bisimulation]
A relation $R\subseteq X\times Y$ is a \emph{Hermida-Jacobs bisimulation} from an $F$-coalgebra $(X,\alpha)$ to an $F$-coalgebra $(Y,\beta)$ whenever there is a morphism $\gamma\c R\to (FR)^\dagger$ called witness that commutes in the following diagram:
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
{X} \& R \& {Y} \\
{FX} \& \relar R \& {FY}
{FX} \& (FR)^\dagger \& {FY}
\arrow["{\alpha}"', from=1-1, to=2-1]
\arrow["{p_1}"', from=1-2, to=1-1]
\arrow["{p_2}", from=1-2, to=1-3]
\arrow["\gamma", from=1-2, to=2-2]
\arrow["{\beta}", from=1-3, to=2-3]
\arrow["{q_1}", from=2-2, to=2-1]
\arrow["{q_2}"', from=2-2, to=2-3]
\arrow["{(Fp_1)^\dagger}", from=2-2, to=2-1]
\arrow["{(Fp_2)^\dagger}"', from=2-2, to=2-3]
\end{tikzcd}
\end{equation*}
\end{definition}
\begin{definition}[Vanilla Bisimulation]
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 $x\mathrel{R} y\Rightarrow \alpha(x)\mathrel{(FR)}\beta(y)$.
% \begin{equation*}
@@ -499,10 +555,12 @@ In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at le
\begin{enumerate}[label=(\Roman*), ref=(\Roman*)]
\item Every Aczel-Mendler bisimulation is a vanilla bisimulation.
\item Every vanilla bisimulation is an Aczel-Mendler bisimulation.
\item Every Aczel-Mendler bisimulation is a witnessful bisimulation.
\item Assuming the axiom of choice, every witnessful bisimulation is an Aczel-Mendler bisimulation.
\item Every witnessless bisimulation is a witnessful bisimulation.
\item Assuming the axiom of choice, every witnessless bisimulation is a witnessful bisimulation.
\item Every Aczel-Mendler bisimulation is a Hermida-Jacobs bisimulation.
\item Assuming the axiom of choice, every Hermida-Jacobs bisimulation is an Aczel-Mendler bisimulation.
\item Every witnessless bisimulation is a Hermida-Jacobs bisimulation.
\item Assuming the axiom of choice, every witnessless bisimulation is a Hermida-Jacobs bisimulation.
\item Every vanilla bisimulation is a witnessless bisimulation.
\item Every witnessless bisimulation is a vanilla bisimulation.
\end{enumerate}
\end{prop}
\begin{proof}
@@ -510,7 +568,13 @@ In this section, by $\spa(\BC)$ we refer to spans in a category $\BC$ that at le
(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}.
(III):
(III): It is trivial.
(IV): Given by Staton.
(V): Similar to (I) we can define $\gamma(x,y)=(\alpha(x),\beta(y))$ as the witness for $R$ to be a Hermida-Jacobs bisimulation.
(VI):
\end{proof}
\todo{Discuss 4 versions of bisimulation (with witness/without witness, for relations/for spans). Which are equivalent? Which do not make sense?}