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


From ae1c456f6467a50427fc485ec5ae163495ea0e52 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 13 Jul 2022 23:09:49 +0200
Subject: Ortografia

---
 sections/fraisse_classes.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 993ca73..87647c6 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -15,7 +15,7 @@
   \end{definition}
 
   \begin{definition}
-	We say that a class $\cK$ of finitely generated strcutures 
+	We say that a class $\cK$ of finitely generated structures 
 	is \emph{essentially countable} if it has countably many isomorphism types
 	of finitely generated structures.
   \end{definition}
@@ -40,7 +40,7 @@
   \end{definition}
 
   In terms of category theory we may say that $\cK$ is a category of finitely
-  generated strcutures where morphims are embeddings of those strcutures.
+  generated structures where morphisms are embeddings of those structures.
   Then the above diagram is a \emph{span} diagram in category $\cK$. 
 
   Fraïssé has shown fundamental theorems regarding age of a structure, one of 
@@ -272,7 +272,7 @@
 
   \begin{definition}
 	\label{definition:free_amalgamation} 
-	Let $L$ be a relational language and $\cK$ a class of $L$-strucutres.
+	Let $L$ be a relational language and $\cK$ a class of $L$-structures.
 	$\cK$ has \emph{free amalgamation} if for every
 	$A, B, C\in\cK$ such that $C = A\cap B$ the following diagram commutes:
 	\begin{center}
@@ -283,7 +283,7 @@
 	  \end{tikzcd}
 	\end{center}
 
-	$A\sqcup_C B$ here is an $L$-strcuture with domain $A\cup B$ such that
+	$A\sqcup_C B$ here is an $L$-structure 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 $B\models R(\bar{a})$].
@@ -346,11 +346,11 @@
 	\end{itemize}
   \end{definition}
 
-  From now on in the paper, when $A$ is an $L$-strcuture and $\alpha$ is 
+  From now on in the paper, when $A$ is an $L$-structure and $\alpha$ is 
   an automorphism of
-  $A$, then by $(A, \alpha)$ we mean the strucutre $A$ expanded by the
-  unary function corresping to $\alpha$, and $A$ constantly denotes the
-  $L$-strucutre.
+  $A$, then by $(A, \alpha)$ we mean the structure $A$ expanded by the
+  unary function corresponding to $\alpha$, and $A$ constantly denotes the
+  $L$-structure.
 
   \begin{theorem}
 	\label{theorem:canonical_amalgamation_thm}
@@ -376,7 +376,7 @@
 	\end{center}
 	
 	Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
-	of the pushout diagram that looks exaclty like the diagram in the second
+	of the pushout diagram that looks exactly like the diagram in the second
 	point of the Definition \ref{definition:canonical_amalgamation}.
 	
 	This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$
-- 
cgit v1.2.3


From 2b830cb2d9c2237fcb7809bab7c64966098ea6fb Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 17 Jul 2022 14:16:41 +0200
Subject: Oby ostatnie poprawki

---
 sections/fraisse_classes.tex | 11 ++++++++---
 1 file changed, 8 insertions(+), 3 deletions(-)

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 87647c6..1126dee 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -283,13 +283,14 @@
 	  \end{tikzcd}
 	\end{center}
 
+	and $A\sqcup_C B\in\cC$.
 	$A\sqcup_C B$ here is an $L$-structure 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
+	we have that $A\sqcup_C B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and
 	$A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$].
   \end{definition}
 
-  Actually we did already implicitly worked with free amalgamation in the 
+  Actually we did already implicitly work with free amalgamation in the 
   Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that
   the class of finite graphs is indeed a Fraïssé class.
 
@@ -298,7 +299,7 @@
   Recall, $\Cospan(\cC)$, $\Pushout(\cC)$ are the categories of cospan
   and pushout diagrams of the category $\cC$. We have also denoted the notion
   of cospans and pushouts with a fixed base structure $C$ denoted
-  as $\Cospan_C(\cC)$ and $Pushout_C(\cC)$.
+  as $\Cospan_C(\cC)$ and $\Pushout_C(\cC)$.
 
   \begin{definition}
 	\label{definition:canonical_amalgamation}
@@ -346,6 +347,10 @@
 	\end{itemize}
   \end{definition}
 
+  \begin{remark}
+	Free amalgamation is canonical.
+  \end{remark}
+
   From now on in the paper, when $A$ is an $L$-structure and $\alpha$ is 
   an automorphism of
   $A$, then by $(A, \alpha)$ we mean the structure $A$ expanded by the
-- 
cgit v1.2.3

