From 006c57fdeb81d97b1ee222d14346e3844df343f5 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 13 Jul 2022 23:06:05 +0200
Subject: Chyba poprawiony dowod 4.4

---
 sections/fraisse_classes.tex | 21 +++++----------------
 1 file changed, 5 insertions(+), 16 deletions(-)

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 0254280..993ca73 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -277,13 +277,13 @@
 	$A, B, C\in\cK$ such that $C = A\cap B$ the following diagram commutes:
 	\begin{center}
 	  \begin{tikzcd}
-		& A\sqcup B & \\
+		& A\sqcup_C B & \\
 		A \ar[ur, hook] & & B \ar[ul, hook'] \\
 		& C \ar[ur, hook] \ar[ul, hook'] & 
 	  \end{tikzcd}
 	\end{center}
 
-	$A\sqcup B$ here is an $L$-strcuture with domain $A\cup B$ such that
+	$A\sqcup_C B$ here is an $L$-strcuture with domain $A\cup B$ such that
 	for every $n$-ary symbol $R$ from $L$, $n$-tuple $\bar{a}\subseteq A\cup B$,
 	we have that $A\sqcup B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and
 	$A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$].
@@ -291,8 +291,7 @@
 
   Actually we did already implicitly worked with free amalgamation in the 
   Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that
-  the class of finite strcuture is indeed a Fraïssé class.
-
+  the class of finite graphs is indeed a Fraïssé class.
 
   \subsection{Canonical amalgamation}
 
@@ -377,19 +376,9 @@
 	\end{center}
 	
 	Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
-	of the pushout diagram:
+	of the pushout diagram that looks exaclty like the diagram in the second
+	point of the Definition \ref{definition:canonical_amalgamation}.
 	
-	\begin{center}
-	  \begin{tikzcd}
-		& A\otimes_C B \ar[loop above, "\delta"] & \\
-		A \ar[ur] \ar[loop left, "\alpha"]& & B \ar[ul] \ar[loop right, "\beta"]\\
-		& C \ar[ur] \ar[ul] \ar[loop below, "\gamma"] & 
-	  \end{tikzcd}
-	\end{center}
-	
-	TODO: ten diagram nie jest do końca taki jak trzeba, trzeba w zasadzie skopiować
-	ten z definicji kanonicznej amalgamcji. Czy to nie będzie wyglądać źle?
-
 	This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$
 	has to be automorphism. Thus, by the fact that the diagram commutes, 
 	we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ 
-- 
cgit v1.2.3

