From c4b7a43bce4793684788209a02546fe249be2e20 Mon Sep 17 00:00:00 2001 From: partowp Date: Wed, 20 May 2026 12:38:25 +0100 Subject: [PATCH] minor --- draft/draft.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index c13055f..9cfac8c 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2244,7 +2244,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \end{proof} \begin{lemma} - Then the following propositions hold: + 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 $\powf\pi_1\comp\supseteq\comp(\powf\pi_2)^\op\quad=\quad\supseteq\comp \powf\pi_1\comp(\powf\pi_2)^\op$ @@ -2267,7 +2267,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i 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 $x \mathrel{\powf\pi_j\comp\supseteq\comp(\powf\pi_i)^\op} y$.\qed \end{proof} %\begin{lemma}