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draft/draft.tex
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@ -1025,12 +1025,12 @@ We show this relation with $F_\rel(R,X)$.
X \& R \& X \\
FX \& (FR)^\dagger \& FX
\arrow["\alpha"', from=1-1, to=2-1]
\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=1-1, to=2-2]
\arrow["\appr"{marking, allow upside down}, draw=none, from=1-1, to=2-2]
\arrow["{p_1}"', from=1-2, to=1-1]
\arrow["{p_2}", from=1-2, to=1-3]
\arrow["\sigma", from=1-2, to=2-2]
\arrow["\alpha", from=1-3, to=2-3]
\arrow["\sqsubseteq"{marking, allow upside down}, draw=none, from=2-2, to=1-3]
\arrow["\appr"{marking, allow upside down}, draw=none, from=2-2, to=1-3]
\arrow["{{(Fp_1)^\dagger}}", from=2-2, to=2-1]
\arrow["{{(Fp_2)^\dagger}}"', from=2-2, to=2-3]
\end{tikzcd}
@ -1123,7 +1123,7 @@ We show this relation with $F_\rel(R,X)$.
\end{proof}
%
Now, we give a counter example of a symmetric relation on $\Set$ that is a simulation according to~\autoref{def:sim}, i.e, exists the morphism $\sigma$ that commutes laxly in~\eqref{eq:diag-lax-sim}, but $\sigma$ is not a coalgebraic bisimulation, although the relation that we give is clearly a bisimulation in the classic sense.
We set $R=\{(A,B),(B,A),(C_1,C_2),(C_2,C_1),(C'_2,C_2),(C_2,C'_2),(C_2,C_2)\}$, $F=\mathbf{Id}$, $\sqsubseteq=\Delta\cup\{(C_1,C_2),(C_2,C'_2)\}$, and the coalgebra $\alpha$ is defined with the following set of reductions:
We set $R=\{(A,B),(B,A),(C_1,C_2),(C_2,C_1),(C'_2,C_2),(C_2,C'_2),(C_2,C_2)\}$, $F=\mathbf{Id}$, $\appr=\Delta\cup\{(C_1,C_2),(C_2,C'_2)\}$, and the coalgebra $\alpha$ is defined with the following set of reductions:
\begin{gather*}
A\to C_1\qquad B\to C_2\qquad C_1\to C_1\qquad C_2\to C_2\qquad C'_2\to C_2
\end{gather*}
@ -2214,7 +2214,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i
=&F\pi_2\comp(F\pi_1)^\op\cap F\pi_2\comp (F\pi_1)^\op\\
=&F\pi_2\comp(F\pi_1)^\op\\
=&\bar{F}r
\end{align*}
\end{align*}\qed
\end{proof}
\begin{prop}
$\hat{L}$ is a Barr relator.
@ -2224,10 +2224,10 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i
\end{proof}
\begin{definition}[One-sided Barr relator]
Given a relation $r$, and take a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$. Assuming that $\sqsubseteq$ is a partial order over a functor $F$, then the relator over $F$ and shown with $\overrightarrow{F}$ is a \emph{one-sided Barr relator} iff we have:
Given a relation $r$, and take a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$. Assuming that $\appr$ is a partial order over a functor $F$, then the relator over $F$ and shown with $\overrightarrow{F}$ is a \emph{one-sided Barr relator} iff we have:
% A relator over a functor $F$ is a one-sided Barr relator, shown by $\overrightarrow{F}$, iff for a partial order $\appr$ over $F$, a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have:
\begin{gather*}
\overrightarrow{F}r=F\pi_2\comp\sappr\comp(F\pi_1)^\op
\overrightarrow{F}r=F\pi_2\comp\appr\comp(F\pi_1)^\op
\end{gather*}
\end{definition}
@ -2239,17 +2239,17 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i
\begin{align*}
s \;\hat{\overrightarrow{F}}r\; t&\\
&\iff s\;\overrightarrow{F}r\;t\qquad\&\qquad s \;(\overrightarrow{F}r^\op)^\op\; t\\
&\iff s\;F\pi_2\comp\sappr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;(F\pi_1\comp\sappr\comp(F\pi_2)^\op)^\op\;t\\
&\iff s\;F\pi_2\comp\sappr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;F\pi_2\comp\appr\comp(F\pi_1)^\op\;t
&\iff s\;F\pi_2\comp\appr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;(F\pi_1\comp\appr\comp(F\pi_2)^\op)^\op\;t\\
&\iff s\;F\pi_2\comp\appr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;F\pi_2\comp\appr\comp(F\pi_1)^\op\;t
\end{align*}
Since $F\pi_1$ is a surjective function, then exists at least one $w\in FA$ such that $(F\pi_1)^\op(s)=w$, and:
\begin{gather*}
w\;F\pi_2\comp\sappr\; t \qquad\&\qquad w\;F\pi_2\comp\appr\; t
w\;F\pi_2\comp\appr\; t \qquad\&\qquad w\;F\pi_2\comp\appr\; t
\end{gather*}
And similarly, since $F\pi_2$ is also a surjective function we have at least one $v\in FA$ such that $(F\pi_2^\op)(t)=v$, and:
\begin{align*}
&w\;\sappr\; v \qquad\&\qquad w\;\appr\; v\\
\iff&(F\pi_1)^\op(s)\;\sappr\; (F\pi_2^\op)(t) \qquad\&\qquad (F\pi_1)^\op(s)\;\appr\; (F\pi_2^\op)(t)\\
&w\;\appr\; v \qquad\&\qquad w\;\appr\; v\\
\iff&(F\pi_1)^\op(s)\;\appr\; (F\pi_2^\op)(t) \qquad\&\qquad (F\pi_1)^\op(s)\;\appr\; (F\pi_2^\op)(t)\\
\iff&(F\pi_1)^\op(s)\;=\; (F\pi_2^\op)(t)\\
\iff&s\; F\pi_2\comp(F\pi_1)^\op\;t\\
\iff&s\;\bar{F}r\;t
@ -2258,31 +2258,31 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i
Now, we are left to show that $\bar{F}\leq\hat{\overrightarrow{F}}$. For that, reading the given proof from the end to the starting point is sufficient.
\end{proof}
\begin{lemma}
The following propositions hold:
\begin{prop}
For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ the following propositions hold:
\begin{enumerate}
\item $\powf\pi_2\comp\supseteq\comp(\powf\pi_1)^\op\quad=\quad\supseteq\comp \powf\pi_2\comp(\powf\pi_1)^\op$
\item $\powf\pi_1\comp\supseteq\comp(\powf\pi_2)^\op\quad=\quad\supseteq\comp \powf\pi_1\comp(\powf\pi_2)^\op$
\item $\powf\pi_2\comp\subseteq\comp(\powf\pi_1)^\op\quad=\quad\subseteq\comp \powf\pi_2\comp(\powf\pi_1)^\op$
\item $\powf\pi_1\comp\subseteq\comp(\powf\pi_2)^\op\quad=\quad\subseteq\comp \powf\pi_1\comp(\powf\pi_2)^\op$
\end{enumerate}
\end{lemma}
\end{prop}
\begin{proof}
Without loss of generality, we assume $i,j\in\{1,2\}$, and $i\neq j$, and prove $\powf\pi_j\comp\supseteq\comp(\powf\pi_i)^\op\quad=\quad\supseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op$.
Without loss of generality, we assume $i,j\in\{1,2\}$, and $i\neq j$, and prove $\powf\pi_j\comp\subseteq\comp(\powf\pi_i)^\op\quad=\quad\subseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op$.
Assuming $x \mathrel{\powf\pi_j\comp\supseteq\comp(\powf\pi_i)^\op} y$, then exist $z$ and $z'$ such that
Assuming $x \mathrel{\powf\pi_j\comp\subseteq\comp(\powf\pi_i)^\op} y$, then exist $z$ and $z'$ such that
\begin{gather*}
z \mathrel{(\powf\pi_i)} x,\\
z \mathrel{\subseteq} z',\\
z'\mathrel{(\powf\pi_j)} y.
\end{gather*}
Then from $z \mathrel{\subseteq} z'$ we get $\powf\pi_j(z)\mathrel{\subseteq}y$. So, we have $z\mathrel{\supseteq\comp \powf\pi_j} y$, thus $x\mathrel{\supseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op} y$.
Then from $z \mathrel{\subseteq} z'$ we get $\powf\pi_j(z)\mathrel{\subseteq}y$. So, we have $z\mathrel{\subseteq\comp \powf\pi_j} y$, thus $x\mathrel{\subseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op} y$.
Now, assuming $x \mathrel{\supseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op} y$, then there exist $z$ and $y'$ such that
Now, assuming $x \mathrel{\subseteq\comp \powf\pi_j\comp(\powf\pi_i)^\op} y$, then there exist $z$ and $y'$ such that
\begin{gather*}
z\mathrel{(\powf\pi_i)} x,\\
z \mathrel{(\powf\pi_j)} y',\\
y' \mathrel{\subseteq} y.
\end{gather*}
We take the set $w=z\cup (\powf\pi_i(z)\times y)$ for which we have $z\subseteq w$ and $\powf\pi_j(w)=y$. So, we have $w \mathrel{(\powf\pi_j)} y$, $z\mathrel{\subseteq} w$, and $z\mathrel{(\powf\pi_i)} x$ that gives $x \mathrel{\powf\pi_j\comp\supseteq\comp(\powf\pi_i)^\op} y$.\qed
We take the set $w=z\cup (\powf\pi_i(z)\times y)$ for which we have $z\subseteq w$ and $\powf\pi_j(w)=y$. So, we have $w \mathrel{(\powf\pi_j)} y$, $z\mathrel{\subseteq} w$, and $z\mathrel{(\powf\pi_i)} x$ that gives $x \mathrel{\powf\pi_j\comp\subseteq\comp(\powf\pi_i)^\op} y$.\qed
\end{proof}
%\begin{lemma}
@ -2310,7 +2310,39 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i
% y' \mathrel{\sqsubseteq} y.
% \end{gather*}
%\end{proof}
\begin{definition}[Cogood Order Structure]\label{def:cogood}
A \emph{cogood order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ such that:
\begin{enumerate}[label=(\Roman*), ref=(\Roman*)]
\item If $\alpha\appr\beta$ in $\Hom(X,FY)$, $f\c X'\to X$ and $g\c Y\to Y'$, then $Fg\comp\alpha\comp f\appr Fg\comp\beta\comp f$ in $\Hom(X',Y')$.\label{item:cogood:I}
\item If $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $Fg\comp k\appr h $ in $\Hom(X,FZ)$, then there is $k'\c X\to FY$ such that $k\appr k'$ in $\Hom(X,FY)$ and $h=Fg\comp k'$.\label{item:cogood:II}
\end{enumerate}
\end{definition}
\begin{prop}
For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ is has a cogood order structure $\appr$, the following propositions hold:
\begin{enumerate}
\item $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad=\quad\appr\comp F\pi_2\comp(F\pi_1)^\op$
\item $F\pi_1\comp\appr\comp(F\pi_2)^\op\quad=\quad\appr\comp F\pi_1\comp(F\pi_2)^\op$
\end{enumerate}
\end{prop}
\begin{proof}
Without loss of generality, we assume $i,j\in\{1,2\}$, and $i\neq j$, and prove $F\pi_j\comp\appr\comp(F\pi_i)^\op\quad=\quad\appr\comp F\pi_j\comp(F\pi_i)^\op$.
Assuming $x \mathrel{F\pi_j\comp\appr\comp(F\pi_i)^\op} y$, then exist $z$ and $z'$ such that
\begin{gather*}
z \mathrel{(F\pi_i)} x,\\
z \mathrel{\appr} z',\\
z'\mathrel{(F\pi_j)} y.
\end{gather*}
Then from $z \mathrel{\appr} z'$ since $\appr$ is a cogood order structure by~\autoref{def:cogood}.\ref{item:cogood:I} we get $F\pi_j(z)\mathrel{\appr}y$. So, we have $z\mathrel{\appr\comp F\pi_j} y$, thus $x\mathrel{\appr\comp F\pi_j\comp(F\pi_i)^\op} y$.
Now, assuming $x \mathrel{\appr\comp F\pi_j\comp(F\pi_i)^\op} y$, then there exist $z$ and $y'$ such that
\begin{gather*}
z\mathrel{(F\pi_i)} x,\\
z \mathrel{(F\pi_j)} y',\\
y' \mathrel{\appr} y.
\end{gather*}
Since $\appr$ is a cogood order structure by~\autoref{def:cogood}.\ref{item:cogood:II} there exists a $w$ such that $z\appr w$ and $F\pi_j(w)=y$. So, we have $w \mathrel{(F\pi_j)} y$, $z\mathrel{\appr} w$, and $z\mathrel{(F\pi_i)} x$ that gives $x \mathrel{F\pi_j\comp\appr\comp(F\pi_i)^\op} y$.\qed
\end{proof}
\begin{prop}
Assuming that $\relar$ is a relator over $F\c\Set\to\Set$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then the relator that takes $r\c X\rto Y$ to $\appr_{X};\relar r;\appr_{Y}$ is a Barr relator.
\end{prop}