From 48728470f3a6de392a4539daea7be0d8e0af6691 Mon Sep 17 00:00:00 2001 From: partowp Date: Sat, 23 May 2026 18:03:20 +0100 Subject: [PATCH 1/5] good --- draft/draft.tex | 32 ++++++++++++++++++++++++++------ 1 file changed, 26 insertions(+), 6 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index ab9b52b..f67805b 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2321,13 +2321,33 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{proof} \todo{Finish.} \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{definition}[Natural Order Structure]\label{def:nat-ord} + A \emph{natural order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ such that if $\alpha\appr\beta$ in $\Hom(X,FY)$, $f\c X\to X'$, $g\c Y\to Y'$, then: \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] - \item If $\alpha\appr\beta$ in $\Hom(X,FY)$, $g\c Y\to Y'$, then $Fg\comp\alpha\appr Fg\comp\beta$ 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} + \item $\alpha\comp f\appr\beta\comp f$ in $\Hom(X',Y)$. \label{item:nat-ord:I} + \item $Fg\comp\alpha\appr Fg\comp\beta$ in $\Hom(X,Y')$. \label{item:nat-ord:II} \end{enumerate} \end{definition} +\begin{definition}[Good Order Structure]\label{def:good-ord} + A \emph{good order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and if $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $h\appr Fg\comp k$ in $\Hom(X,FZ)$, then there is $k'\c X\to FY$ such that $k'\appr k$ in $\Hom(X,FY)$ and $h=Fg\comp k'$. +\end{definition} +\begin{definition}[Cogood Order Structure]\label{def:cogood-ord} + A \emph{cogood order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and 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 $h=Fg\comp k'$. +\end{definition} +\begin{lemma} + Assuming that a a functor $F$ has an order structure $\appr$ that is cogood, then for every $f\in\Hom(X,FY)$ we have: + \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] + \item $Ff\comp\appr\quad=\quad\appr\comp Ff$ + \item $(Ff)^\op\comp\sappr\quad=\quad\sappr\comp (Ff)^\op$ + \end{enumerate} +\end{lemma} +\begin{proof} + $(I)$Assuming $t\mathrel{Ff\comp\appr} x$, there exists $s$ such that $t\appr s$ and $Ff(s)=x$. Since $\appr$ is good, and thus natural, by~\autoref{def:nat-ord}, from $t\appr s$ we get $Ff(t)\appr x$ that is $t\mathrel{\appr\comp Ff} x$. + + Assuming $t\mathrel{\appr\comp Ff} x$, there exists $y$ such that $Ff(t)=y$ and $y\appr x$. By~\autoref{def:cogood-ord} since $Ff(t)\appr x$ there exists $s$ that $t\appr s$ and $Ff(s)=x$ that is $t\mathrel{Ff\comp\appr}s$. + + $(II)$\todo{Finish.} +\end{proof} \begin{prop} For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a cogood order structure $\appr$, the following propositions hold: \begin{enumerate} @@ -2344,7 +2364,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i 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 $x\mathrel{\appr\comp F\pi_j\comp(F\pi_i)^\op} y$. + Then from $z \mathrel{\appr} z'$ since $\appr$ is a cogood order structure by~\ref{item:nat-ord: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*} @@ -2352,7 +2372,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i 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 + Since $\appr$ is a cogood order structure by~\autoref{def:cogood-ord} 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} For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a cogood order structure $\appr$, the following propositions hold: From ae5636cb23dd25bcbeb66f8f84f201770a89ed88 Mon Sep 17 00:00:00 2001 From: partowp Date: Sun, 24 May 2026 22:07:25 +0100 Subject: [PATCH 2/5] the equality --- draft/draft.tex | 65 +++++++++++++++++++++++++++++-------------------- 1 file changed, 38 insertions(+), 27 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index f67805b..33f7263 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2334,7 +2334,26 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{definition}[Cogood Order Structure]\label{def:cogood-ord} A \emph{cogood order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and 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 $h=Fg\comp k'$. \end{definition} -\begin{lemma} +\begin{lemma}\label{lem:good} + Assuming that a a functor $F$ has an order structure $\appr$ that is good, then for every $f\in\Hom(X,FY)$ we have: + \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] + \item $Ff\comp\sappr\quad=\quad\sappr\comp Ff$ + \item $(Ff)^\op\comp\appr\quad=\quad\appr\comp (Ff)^\op$ + \end{enumerate} +\end{lemma} +\begin{proof} + $(I)$ Assuming $t\mathrel{Ff\comp\sappr} x$, there exists $s$ such that $t\sappr s$ and $Ff(s)=x$. Since $\appr$ is good, and thus natural, by~\autoref{def:nat-ord}, from $s\appr t$ we get $x\appr Ff(t)$ that is $t\mathrel{\sappr\comp Ff} x$. + + Assuming $t\mathrel{\sappr\comp Ff} x$, there exists $y$ such that $Ff(t)=y$ and $y\sappr x$. By~\autoref{def:good-ord} since $Ff(t)\sappr x$ there exists $s$ that $t\sappr s$ and $Ff(s)=x$ that is $t\mathrel{Ff\comp\appr}s$. + + $(II)$ Basically, by definition of $\op$ and relation composition we have + \begin{gather*} + (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ + (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + \end{gather*} + So it follows directly from applying $\op$ on both sides of $(I)$.\qed +\end{proof} +\begin{lemma}\label{lem:cogood} Assuming that a a functor $F$ has an order structure $\appr$ that is cogood, then for every $f\in\Hom(X,FY)$ we have: \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] \item $Ff\comp\appr\quad=\quad\appr\comp Ff$ @@ -2342,11 +2361,16 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \end{enumerate} \end{lemma} \begin{proof} - $(I)$Assuming $t\mathrel{Ff\comp\appr} x$, there exists $s$ such that $t\appr s$ and $Ff(s)=x$. Since $\appr$ is good, and thus natural, by~\autoref{def:nat-ord}, from $t\appr s$ we get $Ff(t)\appr x$ that is $t\mathrel{\appr\comp Ff} x$. + $(I)$ Assuming $t\mathrel{Ff\comp\appr} x$, there exists $s$ such that $t\appr s$ and $Ff(s)=x$. Since $\appr$ is cogood, and thus natural, by~\autoref{def:nat-ord}, from $t\appr s$ we get $Ff(t)\appr x$ that is $t\mathrel{\appr\comp Ff} x$. Assuming $t\mathrel{\appr\comp Ff} x$, there exists $y$ such that $Ff(t)=y$ and $y\appr x$. By~\autoref{def:cogood-ord} since $Ff(t)\appr x$ there exists $s$ that $t\appr s$ and $Ff(s)=x$ that is $t\mathrel{Ff\comp\appr}s$. - $(II)$\todo{Finish.} + $(II)$ Basically, by definition of $\op$ and relation composition we have + \begin{gather*} + (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ + (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + \end{gather*} + So it follows directly from applying $\op$ on both sides of $(I)$.\qed \end{proof} \begin{prop} For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a cogood order structure $\appr$, the following propositions hold: @@ -2375,34 +2399,21 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i Since $\appr$ is a cogood order structure by~\autoref{def:cogood-ord} 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} - For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a cogood order structure $\appr$, the following propositions hold: + For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has an order structure $\appr$, the following propositions hold: \begin{enumerate} - \item $F\pi_2\comp\sappr\comp(F\pi_1)^\op\quad=\quad F\pi_2\comp(F\pi_1)^\op\comp\appr$ - \item $F\pi_1\comp\sappr\comp(F\pi_2)^\op\quad=\quad F\pi_1\comp(F\pi_2)^\op\comp\appr$ + \item If $\appr$ is good, we have $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad=\quad F\pi_2\comp(F\pi_1)^\op\comp\appr$ + \item If $\appr$ is cogood, we have $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad=\quad \appr\comp F\pi_2\comp(F\pi_1)^\op$ + \item If $\appr$ is both good and cogood, all the following are equal: + \begin{itemize} + \item $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad$ + \item $F\pi_2\comp(F\pi_1)^\op\comp\appr$ + \item $\appr\comp F\pi_2\comp(F\pi_1)^\op$ + \item $\appr\comp F\pi_2\comp(F\pi_1)^\op\comp\appr$ + \end{itemize} \end{enumerate} \end{prop} \begin{proof} - \todo{Finish.} -\end{proof} -\begin{prop} - For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a cogood order structure $\appr$, the following propositions hold: - \begin{enumerate} - \item $F\pi_2\comp\appr\comp\sappr\comp(F\pi_1)^\op\quad=\quad \appr\comp F\pi_2\comp(F\pi_1)^\op\comp\appr$ - \item $F\pi_1\comp\appr\comp\sappr\comp(F\pi_2)^\op\quad=\quad \appr\comp F\pi_1\comp(F\pi_2)^\op\comp\appr$ - \end{enumerate} -\end{prop} -\begin{proof} - \todo{Finish.} -\end{proof} -\begin{prop} - For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has a good order structure $\appr$, the following propositions hold: - \begin{enumerate} - \item $F\pi_2\comp\sappr\comp(F\pi_1)^\op\quad=\quad\appr\comp F\pi_2\comp(F\pi_1)^\op$ - \item $F\pi_1\comp\sappr\comp(F\pi_2)^\op\quad=\quad\appr\comp F\pi_1\comp(F\pi_2)^\op$ - \end{enumerate} -\end{prop} -\begin{proof} - \todo{Finish.} + They all follow in an obvious way from~\autoref{lem:good} and~\autoref{lem:cogood}. The last one needs $\appr\comp\appr=\appr$ that comes from transitivity of $\appr$. \qed \end{proof} \begin{prop} Assuming that $\relar$ is a difunctionally functorial relator, then the symmetrization of the relator that takes $r\c X\rto Y$ to $\relar r\comp\appr$ is a sound relator. From 1a3e7ec1a940bbb6ba5a0f818ba1d72407525a4c Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 26 May 2026 16:25:28 +0100 Subject: [PATCH 3/5] proof and counter-example --- draft/draft.tex | 89 ++++++++++++++++++++++++++++++++++--------------- 1 file changed, 62 insertions(+), 27 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 33f7263..b7cc2c4 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2334,6 +2334,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{definition}[Cogood Order Structure]\label{def:cogood-ord} A \emph{cogood order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and 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 $h=Fg\comp k'$. \end{definition} + \begin{lemma}\label{lem:good} Assuming that a a functor $F$ has an order structure $\appr$ that is good, then for every $f\in\Hom(X,FY)$ we have: \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] @@ -2372,32 +2373,32 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \end{gather*} So it follows directly from applying $\op$ on both sides of $(I)$.\qed \end{proof} -\begin{prop} - For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ 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~\ref{item:nat-ord: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-ord} 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} +% For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ 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~\ref{item:nat-ord: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-ord} 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} For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that $F$ has an order structure $\appr$, the following propositions hold: \begin{enumerate} @@ -2415,6 +2416,41 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{proof} They all follow in an obvious way from~\autoref{lem:good} and~\autoref{lem:cogood}. The last one needs $\appr\comp\appr=\appr$ that comes from transitivity of $\appr$. \qed \end{proof} +\begin{example} + In the category of sets, subset over the powerset functor is an example of a good order. For every $h\c X\to\powf Z$, $k\c X\to \powf Y$, and $g\c Y\to Z$, such that $h\subseteq\powf g\comp k$, there exists $k'\c X\to\powf Y$ that for every $x\in X$, $k'(x)=k(x)\setminus\{y\mid g(y)\notin h(x)\}$. We show that for every $x\in X$, we have $\powf g\comp k'(x)=h(x)$. + + Assuming $z\in\powf g(k'(x))$, then there exist $y'\in k(x)\setminus \{y\mid g(y)\notin h(x)\}$ that $g(y')=z$, so $y'\in k(x)$, $y'\notin\{y\mid g(y)\notin h(x)\}$. $y'\notin\{y\mid g(y)\notin h(x)\}$ means that $g(y')\in h(x)$, thus $\powf g\comp k'\subseteq h$. + + Assuming $z\in h(x)$, since $h\subseteq \powf g\comp k$, then $z\in\powf g\comp k(x)$. So, there exists $y'$ such that $y'\in k(x)$ and $g(y')=z$. Since $g(y')\in h(x)$ then $y'\notin\{y\mid g(y)\notin h(x)\}$ meaning that $y'\in k'(x)$ that means $z\in \powf g(k'(x))$, thus $h\subseteq \powf g\comp k'$.\qed +\end{example} +\begin{example} + In the category of sets, subset over the powerset functor is NOT an example of a cogood order! For some set $X$ we take $h\c X\to\powf\mathbb{Z}$, for every $x\in X$, $h(x)=\mathbb{Z}$, $g\c\mathbb{Z}\to\mathbb{Z}$, and for every $z\in\mathbb{Z}$, + \begin{gather*} + g(z)= + \begin{cases} + 0 & z\in\mathbb{Z}^+ \\ + 1 & \mathsf{otherwise} + \end{cases} + \end{gather*} + then + \begin{gather*} + \powf g(A)= + \begin{cases} + \emptyset & A=\emptyset \\ + \{0\} & A\subseteq \mathbb{Z}^+,A\neq\emptyset\\ + \{1\} & A\subseteq \mathbb{Z}^-\cup\{0\},A\neq\emptyset\\ + \{0,1\} & \mathsf{otherwise} + \end{cases} + \end{gather*} + then no matter what $k\c X\to\powf\mathbb{Z}$ is there will be no $k'\c X\to\powf\mathbb{Z}$, for every $x\in X$, $\powf g(k'(x))=h(x)$, as $\powf g(k'(x))\subseteq\{0,1\}$, while $h(x)=\mathbb{Z}$, so $\powf g(k'(x))\subset h(x)$. +\end{example} +The previous example suggests that cogoodness is a strong condition. Perhaps it can be limited to be meaningful. Maybe the morphism $g$ can be limited to legs of a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, where for a relation $r$, $r=\pi_2\comp\pi_1^\op$. +\begin{prop} + For a natural order structure $\appr$ on a set-functor $F$, if for every $f\in\Hom(X,FY)$ we have $Ff\comp\appr=\appr\comp Ff$, then $\appr$ is cogood. +\end{prop} +\begin{proof} + \todo{Investigate if it's true! If it is, is naturality necessary? It is really weird! If having a good order structure forces every witness for AM-simulation to have the left cell of the lax diagram as equality, and subset on powerset functor is a good order structure, then what is the counter-example number 4 that you have?!} +\end{proof} \begin{prop} Assuming that $\relar$ is a difunctionally functorial relator, then the symmetrization of the relator that takes $r\c X\rto Y$ to $\relar r\comp\appr$ is a sound relator. \end{prop} @@ -2427,7 +2463,6 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{proof} \todo{Finish.} \end{proof} - \end{document} From c812597e18e057355745cc62b5b421d0aa0cb487 Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 26 May 2026 17:33:22 +0100 Subject: [PATCH 4/5] minor --- draft/draft.pdf | Bin 624241 -> 642100 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/draft/draft.pdf b/draft/draft.pdf index 1ad9f3f4013c57d3a75fa25b02072ec42c4cb68e..fd1f6fe0af50d044f8c710e502283aa0695dde0d 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b937b17..dbf302f 100644 --- a/ACV-abstract-2026/sym-sim.tex +++ b/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:} diff --git a/draft/draft.pdf b/draft/draft.pdf index 1ad9f3f4013c57d3a75fa25b02072ec42c4cb68e..9f085de785cd7bf0693a714a0e15c1cc05221b4e 100644 GIT binary patch delta 177447 zcmV)kK%l?zj3%_0CxEm8G_(UZH#nEEc>ySY?OR)K+c*|}_pi{S3{^4YC5q&^f9LSH%+9hhdog-8 z89hEyEHlI^C9=tBM!6tb$xK;BXeqPFyX;NQ%W=V!wD~a^7hJIX#cFvnW+p$Mu2*A! zX7f*v+3UNQxElX9`RAw%%hsQdk2K9J)@I7BU1kN7#FpHz`f@#% zG+$N!PDwPu<0GzHFLFv)Y4N0b?>}S1^Vzr%jONSP?47^DKB~|6ZpSA78V!(dBlX>GEA-7FH4_Xg%gE 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