From 3285b1fdf61deabd498a3d4527d884542735700e Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 11 Aug 2026 16:58:42 +0100 Subject: [PATCH] the homework is under progress --- draft/draft.tex | 65 ++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 62 insertions(+), 3 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index b8c2049..e07d774 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -936,9 +936,68 @@ We do not know if this definition exists anywhere. \end{rem} \todo{Discuss 4 versions of bisimulation (with witness/without witness, for relations/for spans). Which are equivalent? Which do not make sense?} -\todo{In next section run a similar analysis for simulation: relator-based vs. Aczel-Mendler.} - - +\todo{In next section run a similar analysis for simulation: relator-based vs. Aczel-Mendler.}\\ +-------------------------------------------------------------- +\begin{definition}[Aczel-Mendler Bisimulation] + In an arbitrary category $\BC$, a span $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ is an \emph{Aczel-Mendler bisimulation} over $F$-coalgebras $(X,\alpha)$ and $(Y,\beta)$, if there exists a morphism in $\spa(\BC)$ of the type $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)\to(FX \stackrel{Fp_1}{\leftarrow} FR \stackrel{Fp_2}{\to}FY)$. +\end{definition} +% +\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["{U}"',from=1-1, to=2-1] + \arrow["{U}",from=1-3, to=2-3] + \arrow["{F\times F}"', from=2-1, to=2-3] + \end{tikzcd} + \end{equation*} +\end{definition} +\begin{definition}[Abstract Relational Bisimulation] + In an arbitrary category $\BC$, a relation $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ is an \emph{abstract relational bisimulation} over $F$-coalgebras $(X,\alpha)$ and $(Y,\beta)$, if there exists a morphism in $\rel(\BC)$ of the type $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)\to(FX \stackrel{\rel(F)p_1}{\leftarrow} \rel(F)R \stackrel{\rel(F)p_2}{\to}FY)$. +\end{definition} +The given definition is highly abstract. There is a relation lifting that abstracts Barr-relators that are known to be well-behaved relators, and it also gives an interesting notion of bisimulation, called \emph{Hermida-Jacobs bisimulation}. +% + The lifting is using the image factorization in regular categories. %\ppnote{Initially, I wanted to give the definitions for an arbitrary relation lifting. I think it can be doable, but for simplicity I preferred to stick to this one.} + % + %\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*} + For a regular category $\BC$, we define a functor of type $(-)^\clubsuit\c\spa(\BC)\to\rel(\BC)$. 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*} + Also, for every functor $F\c\BC\to\BC$ we have a trivial lifting to $\spa(\BC)$ that takes every object $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ to $(FX \stackrel{Fp_1}{\leftarrow} FR \stackrel{Fp_2}{\to}FY)$, and every morphism $(f,g,w)$ to $(Ff,Fg,Fw)$, and we show it with $\spa(F)$. Since $\rel(\BC)$ is a subcategory of $\spa(\BC)$, we have an inclusion functor $I\c\rel(\BC)\to\spa(\BC)$ as well. + So, given a functor $F\c\BC\to\BC$ we define its lifting $(F-)^\dagger\c\rel(\BC)\to\rel(\BC)$ as $(F-)^\dagger=(\spa(F)I-)^\clubsuit$. $(F-)^\dagger$ 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*} +% +\begin{definition}[Hermida-Jacobs Bisimulation] + In an arbitrary category $\BC$, a relation $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ is a \emph{Hermida-Jacobs bisimulation} over $F$-coalgebras $(X,\alpha)$ and $(Y,\beta)$, if there exists a morphism in $\rel(\BC)$ of the type $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)\to(FX \stackrel{(Fp_1)^\dagger}{\leftarrow} (FR)^\dagger \stackrel{(Fp_2)^\dagger}{\to}FY)$. +\end{definition} +\subsection{Coalgebraic Bisimulation in Set} +The have two more notions for coalgebraic bisimulation in $\Set$, that is to define them in $\spa_a$ and $\rel_a$. % \section{Coalgebraic Simulation} We show the category of preorders with monotone functions between them with $\preord$. In the diagrams, any arrow that shows a functor, but does not have a label is showing a forgetful functor. Also, we use $\rel$ to refer to the category of binary relations. Assuming $R\in\obj(\rel)$ and $R\subseteq X_1\times X_2$, and $S\in\obj(\rel)$ and $S\subseteq Y_1\times Y_2$, then a morphism $f\c R\to S$ in this category is the pair $(f_1,f_2)$ of morhpisms in $\Set$, where, $f_1\c X_1\to Y_1$ and $f_2\c X_2\to Y_2$, and for each $(x_1,x_2)\in R$ we have $(f_1(x_1),f_2(x_2))\in S$. Also, we show projections of $R\in\obj(\rel)$ with $p_1$ and $p_2$ that are morphisms in $\Set$.