From 3d36dc32935237bebe6ee7652d1a5e8cd34af3e3 Mon Sep 17 00:00:00 2001 From: Sergey Goncharov Date: Mon, 20 Jul 2026 14:05:33 +0100 Subject: [PATCH 1/2] pc sync --- draft/draft.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/draft/draft.tex b/draft/draft.tex index 003d259..19894dd 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -3077,7 +3077,7 @@ The following example justifies why the $g$ in~\autoref{def:coliftable-ord} shou \end{align*}\qed \end{proof} \begin{cor} - Since the symmetrization of left-lax Barr relator that is laxed with a lifatble order structure is natural, and normal, it is a normal relational connector. So, it is a sound and complete relator. + Since the symmetrization of left-lax Barr relator with a lifatble order structure is natural, and normal, it is a normal relational connector. So, it is a sound and complete relator. \end{cor} \begin{cor} By~\autoref{prop:all-rel-compa}.(1), if $\appr$ is liftable, then the mid-lax Barr relator is a normal relation connector, and thus a sound and complete relator as well. From 9da15b278cd6b7cf8499d546610b247c9734a061 Mon Sep 17 00:00:00 2001 From: Sergey Goncharov Date: Mon, 20 Jul 2026 15:17:54 +0100 Subject: [PATCH 2/2] pc sync --- draft/draft.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 9f03924..8367017 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2163,8 +2163,7 @@ We recall that in the above diagram $\sigma_3$ is a bisimulation, and the rest a % We have $(Fp_1)^\dagger\comp\sigma$. %\end{proof} \begin{example} -And another counter-example!!! -Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. We show the functor with $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ +Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. Let us call this functor $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined as below: \begin{gather*} \alpha(x)=