Merge branch 'master' of git.wlog.site:pouya/coalgebraic-simulation

ACV-abstract-2026/sym-sim.tex

@@ -241,7 +241,7 @@ We ask:
 \begin{enumerate}
   \item How to construct relators for established notions of simulation? 
   \item How to generally prove that ensuing symmetric simulations are bisimulations?
-  \item How alternative notions of simulations align with the relator-based one? 
+  \item How alternative notions of simulation align with the relator-based one? 
 \end{enumerate}
 
 \paragraph{Relaxing Barr Relators} %\label{sec:} 
@@ -253,7 +253,7 @@ we consider four ways of relaxing Barr relators, to model simulation:
   \item\label{it:ll-barr} left-lax Barr relator $\appr\comp\bar{F}$ 
   \item\label{it:rl-barr} right-lax Barr relator $\bar{F}\comp\appr$  
   \item\label{it:bl-barr} bi-lax Barr relator $\appr\comp\bar{F}\comp\appr$ 
-  \item\label{it:ml-barr} mid-lax Barr relator $R\mapsto F\pi_2\comp\sappr\comp(F\pi_1)^\op$, by 
+  \item\label{it:ml-barr} mid-lax Barr relator $R\mapsto F\pi_2\comp\appr\comp(F\pi_1)^\op$, by 
   viewing $R$ as a span via projections $\pi_1\c R\to X$, $\pi_2\c R\to Y$.
 \end{enumerate}
 %
@@ -360,7 +360,7 @@ Take $R=\{(1,2),(2,1),(1,3),(3,1)\}$, and $X=\{1,2,3\}$, and $ c(x)=X$ for every
 		R\setminus\{(1,2)\} & w=(1,3)
 	\end{cases}
 \end{gather*}
-Then $\sigma$ is a witnesses of simulation but not a bisimulation: $c(p_2(1,3))=c(3)=X\neq\mathcal{P}p_2(\sigma(1,3))=X\setminus\{2\}$.
+Then $\sigma$ is a witnesses of simulation on $(X,c)$ but not a bisimulation: $c(p_2(1,3))=c(3)=X\neq\mathcal{P}p_2(\sigma(1,3))=X\setminus\{2\}$.
 \end{example}
 
 %Inspired by Hermida-Jacobs bisimulation\cite{HermidaJacobs98} we modify the definition by setting $\BC$ to be a regular category, and changing the span $(FR,Fp_1,Fp_2)$ in~\eqref{eq:diag-lax-sim} with the span $((FR)^\dagger,(Fp_1)^\dagger,(Fp_2)^\dagger)$, where $(FR)^\dagger$ is the image of $\brks{Fp_1,Fp_2}$, and $\brks{(Fp_1)^\dagger,(Fp_2)^\dagger}\c(FR)^\dagger\to FX\times FX$ is monic, so $(FR)^\dagger$ is a relation. By the symmetry of $R$ there exists $s\c R\to R$ that we call \emph{swap}, where $p_1\comp s=p_2$ and $p_2\comp s=p_1$. The necessary and sufficient condition for $\sigma$ to be a witness for $R$ to be a bisimulation is that