merge

7 files changed, 101 insertions, 106 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 9f6ec58..417923d 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/lic_malinka.tex b/lic_malinka.tex
index e76d33b..738ff71 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -83,6 +83,7 @@
 \newtheorem{axiom}[theorem]{Axiom}

 \newtheorem{question}[theorem]{Question}

 \newtheorem{corollary}[theorem]{Corollary} 

+\newtheorem{remark}[theorem]{Remark}

 \newtheorem*{theorem2}{Theorem}

 \newtheorem*{claim2}{Claim} 

 \newtheorem*{corollary2}{Corollary}

@@ -97,7 +98,6 @@
 \newtheorem{example}[theorem]{Example}

 

 \theoremstyle{remark} 

-\newtheorem{remark}[theorem]{Remark}

 \newtheorem*{remark2}{Remark}

 

 

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 9ec4b0c..d37afd5 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -90,19 +90,11 @@
 	automorphism group of the $\Flim(\cC)$.
   \end{theorem}
 
-  Before we get to the proof, let us establish some notions. If 
-  $g\colon\Gamma\to\Gamma$ is a finite injective function, then we say that
-  $g$ is \emph{good} if it gives (in a natural way) an isomorphism between
-  $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures 
-  generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
-  case, $g$ is good
-  if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
-  of $\Gamma$).
-
-  Also it is important to mention that an isomorphism between two finitely
+  Before we get to the proof, it is important to mention that an isomorphism 
+  between two finitely
   generated structures is uniquely given by a map from generators of one structure
   to the other. This allow us to treat a finite function as an isomorphism
-  of finitely generated structures.
+  of finitely generated structures (if it yields one) and \textit{vice versa}. 
 
   \begin{proof}
 	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
@@ -115,16 +107,17 @@
 	By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove 
 	that this class is comeagre.
 
-	Recall, $G$ has a basis consisting of open
+	Recall, $G$ has a basis consisting of 
 	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some
 	finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other
-	words, a basic open set is a set of all extensions of some finite partial
-	automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
-	open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$
+	words, a basic open set is a set of all extensions of some partial
+	automorphism $g_0$ of finitely generated substructures of $\Gamma$. 
+	By $B_{g}\subseteq G$ we denote a basic 
+	open subset given by a partial isomorphism $g$. Again, Note that $B_g$
 	is nonempty because of ultrahomogeneity of $\Gamma$.
 
 	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
-	only games where both players choose finite partial isomorphisms. Namely,
+	only games where both players choose partial isomorphisms. Namely,
 	player \textit{I} picks functions $f_0, f_1,\ldots$ and player \textit{II}
 	chooses $g_0, g_1,\ldots$ such that
 	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify
@@ -144,20 +137,18 @@
 	For technical reasons, let $g_{-1} = \emptyset$ and
 	$X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit 
 	$\Gamma = \{v_0, v_1, \ldots\}$.
-	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
-	automorphism $f_n$. We will construct a finite partial automorphism 
+	Suppose that player \textit{I} in the $n$-th move chooses a partial
+	automorphism $f_n$. We will construct a partial automorphism 
 	$g_n\supseteq f_n$ together with a finitely generated substructure
 	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
 	such that the following properties hold:
 
 	\begin{enumerate}[label=(\roman*)]
-	  \item $g_n$ is good and 
-		$\dom(g_n)\cup\rng(g_n)\subseteq\langle\dom(g_n)\rangle = \Gamma_n$,
-		i.e. $g_n$ gives an automorphism of a finitely generated 
-		substructure $\Gamma_n$
-	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined,
-
+	  \item $g_n$ is is a partial automorphism of $\Gamma$ and an automorphism of
+		finitely generated substructure $\Gamma_n$,
+	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined.
 	  \end{enumerate}
+
 	  Before we give the third point, suppose recursively that $g_{n-1}$ already
 	  satisfy all those properties. Let us enumerate 
 	  $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
@@ -166,14 +157,14 @@
 	  $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$
 	  is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $(\FrGr_{n-1}, g_{n-1})$.
 	  We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function,
-	  we may treat it as an automorphism as we have said before.
+	  we may treat it as a partial automorphism as we have said before.
 
 	  \begin{enumerate}[resume, label=(\roman*)]
 	  \item 
 		Let 
-		$(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the
+		$(i, j) = \min\{(\{0, 1, \ldots, n\} \times \bN) \setminus X_{n-1}\}$ (with the
 		order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and
-		$(B_{n,k}, \beta_{n,k})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
+		$(B_{i,j}, \beta_{i,j})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
 		commutes:
 
 		\begin{center}
@@ -216,7 +207,7 @@
 	Take any $(B, \beta)\in\cD$. Then, there are
 	$i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$.
 	By the bookkeeping there was $n$ such that 
-	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
+	$(i, j) = \min\{\{0,1,\ldots n\}\times\bN\setminus X_{n}\}$.
 	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
 	into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$.
 	The other inclusion is obvious.	Hence, the age of $(\Gamma, g)$ is $\cH$. 
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 1126dee..885eb81 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -9,27 +9,27 @@
   \subsection{Definitions} 
   \begin{definition}
     Let $L$ be a signature and $M$ be an $L$-structure. The \emph{age} of $M$ is
-    the class $\cK$ of all finitely generated structures that embed into $M$.
-    The age of $M$ is also associated with class of all structures embeddable in
-    $M$ \emph{up to isomorphism}.
+    the class $\cC$ of all finitely generated structures that embed into $M$.
+    The age of $M$ is also associated with class of all finitely generated
+	structures embeddable in $M$ \emph{up to isomorphism}.
   \end{definition}
 
   \begin{definition}
-	We say that a class $\cK$ of finitely generated structures 
+	We say that a class $\cC$ of finitely generated structures 
 	is \emph{essentially countable} if it has countably many isomorphism types
 	of finitely generated structures.
   \end{definition}
 
   \begin{definition}
-    Let $\cK$ be a class of finitely generated structures. $\cK$ has the
-    \emph{hereditary property (HP)} if for any $A\in\cK$ and any finitely 
-	generated substructure $B$ of $A$ it holds that $B\in\cK$. 
+    Let $\cC$ be a class of finitely generated structures. $\cC$ has the
+    \emph{hereditary property (HP)} if for any $A\in\cC$ and any finitely 
+	generated substructure $B$ of $A$ it holds that $B\in\cC$. 
   \end{definition}
 
   \begin{definition}
-    Let $\cK$ be a class of finitely generated structures. We say that $\cK$ has
-	the \emph{joint embedding property (JEP)} if for any $A, B\in\cK$ there is 
-	a structure $C\in\cK$ such that both $A$ and $B$ embed in $C$.
+    Let $\cC$ be a class of finitely generated structures. We say that $\cC$ has
+	the \emph{joint embedding property (JEP)} if for any $A, B\in\cC$ there is 
+	a structure $C\in\cC$ such that both $A$ and $B$ embed in $C$.
 
     \begin{center}
       \begin{tikzcd}
@@ -39,18 +39,18 @@
     \end{center}
   \end{definition}
 
-  In terms of category theory we may say that $\cK$ is a category of finitely
+  In terms of category theory we may say that $\cC$ is a category of finitely
   generated structures where morphisms are embeddings of those structures.
-  Then the above diagram is a \emph{span} diagram in category $\cK$. 
+  Then the above diagram is a \emph{span} diagram in category $\cC$. 
 
   Fraïssé has shown fundamental theorems regarding age of a structure, one of 
   them being the following one:
   
   \begin{fact}
     \label{fact:age_is_hpjep}
-    Suppose $L$ is a signature and $\cK$ is a nonempty essentially countable set 
-    of finitely generated $L$-structures. Then $\cK$ has the HP and JEP if
-    and only if $\cK$ is the age of some finite or countable structure. 
+    Suppose $L$ is a signature and $\cC$ is a nonempty essentially countable set 
+    of finitely generated $L$-structures. Then $\cC$ has the HP and JEP if
+    and only if $\cC$ is the age of some finite or countable structure. 
   \end{fact}
 
   \begin{proof}
@@ -59,13 +59,13 @@
   \end{proof}
 
   Beside the HP and JEP Fraïssé has distinguished one more property of the 
-  class $\cK$, namely the amalgamation property.
+  class $\cC$, namely the amalgamation property.
 
   \begin{definition}
-    Let $\cK$ be a class of finitely generated $L$-structures. We say that $\cK$
-    has the \emph{amalgamation property (AP)} if for any $A, B, C\in\cK$ and
-    embeddings $e\colon C\to A, f\colon C\to B$ there exists $D\in\cK$ together
-    with embeddings $g\colon A\to D$ and $h\colon A\to D$ such that 
+    Let $\cC$ be a class of finitely generated $L$-structures. We say that $\cC$
+    has the \emph{amalgamation property (AP)} if for any $A, B, C\in\cC$ and
+    embeddings $e\colon C\to A, f\colon C\to B$ there exists $D\in\cC$ together
+    with embeddings $g\colon A\to D$ and $h\colon B\to D$ such that 
     $g\circ e = h\circ f$.
     \begin{center}
       \begin{tikzcd}
@@ -76,13 +76,13 @@
     \end{center}
   \end{definition}
 
-  In terms of category theory, $\cK$ has the amalgamation property if every
-  cospan diagram can be extended to a pushout diagram in category $\cK$.
+  In terms of category theory, $\cC$ has the amalgamation property if every
+  cospan diagram can be extended to a pushout diagram in category $\cC$.
   We will get into more details later, in the definition of canonical 
   amalgamation \ref{definition:canonical_amalgamation}.
 
   \begin{definition}
-	Class $\cK$ of finitely generated structure is a \emph{Fraïssé class}
+	Class $\cC$ of finitely generated structures is a \emph{Fraïssé class}
 	if it is essentially countable, has HP, JEP and AP.
   \end{definition}
 
@@ -96,11 +96,11 @@
 
   \begin{theorem}[Fraïssé theorem]
     \label{theorem:fraisse_thm}
-    Let L be a countable language and let $\cK$ be a nonempty countable set of
-    finitely generated $L$-structures which has HP, JEP and AP. Then $\cK$ is 
+    Let L be a countable language and let $\cC$ be a nonempty countable set of
+    finitely generated $L$-structures which has HP, JEP and AP. Then $\cC$ is 
     the age of a countable, ultrahomogeneous $L$-structure $M$. Moreover, $M$ is
-    unique up to isomorphism. We say that $M$ is a \emph{Fraïssé limit} of $\cK$
-    and denote this by $M = \Flim(\cK)$.
+    unique up to isomorphism. We say that $M$ is a \emph{Fraïssé limit} of $\cC$
+    and denote this by $M = \Flim(\cC)$.
   \end{theorem}
 
   \begin{proof}
@@ -135,8 +135,8 @@
   useful when recursively constructing the generic automorphism of a Fraïssé 
   limit.
 
-  % \begin{fact} If $\cK$ is a uniformly locally finite Fraïssé class, then
-  % $\Flim(\cK)$ is $\aleph_0$-categorical and has quantifier elimination.
+  % \begin{fact} If $\cC$ is a uniformly locally finite Fraïssé class, then
+  % $\Flim(\cC)$ is $\aleph_0$-categorical and has quantifier elimination.
   % \end{fact}
 
   \subsection{Random graph} 
@@ -149,8 +149,8 @@
   \emph{graph}, and its elements $\emph{vertices}$. If for some vertices 
   $u, v\in G$ we have $G\models uEv$ then we say that there is an $\emph{edge}$ 
   connecting $u$ and $v$. If $G\models \forall x\forall y (xEy\leftrightarrow yEx)$
-  then we say that $G$ is an \emph{undirected graph}. From now on we omit the word
-  \textit{undirected} and consider only undirected graphs.
+  then we say that $G$ is an \emph{undirected graph}. From now on we 
+  consider only undirected graphs and omit the word \textit{undirected}.
 
   \begin{proposition}
 	\label{proposition:finite-graphs-fraisse-class}
@@ -158,11 +158,11 @@
   \end{proposition}
   \begin{proof}
     $\cG$ is of course countable (up to isomorphism) and has the HP 
-    (graph substructure is also a graph). It has JEP: having two finite graphs 
+    (substructure of a graph is also a graph). It has JEP: having two finite graphs 
     $G_1,G_2$ take their disjoint union $G_1\sqcup G_2$ as the extension of them
     both. $\cG$ has the AP. Having graphs $A, B, C$, where $B$ and $C$ are
     supergraphs of $A$, we can assume without loss of generality that 
-    $(B\setminus A) \cap (C\setminus A) = \emptyset$. Then 
+    $B\cap C = A$. Then 
     $A\sqcup (B\setminus A)\sqcup (C\setminus A)$ is the graph we are looking
     for (with edges as in B and C and without any edges between $B\setminus A$
 	and $C\setminus A$).
@@ -183,10 +183,10 @@
   The random graph $\FrGr$ has one particular property that is unique to the
   random graph.
 
-  \begin{fact}[random graph property]
+  \begin{fact}[Random graph property]
 	For each finite disjoint $X, Y\subseteq \FrGr$ there exists $v\in\FrGr\setminus (X\cup Y)$
     such that $\forall u\in X$ we have that $\FrGr\models vEu$ and 
-	$\forall u\in Y$ We have that $\FrGr\models \neg vEu$.
+	$\forall u\in Y$ we have that $\FrGr\models \neg vEu$.
   \end{fact}
   \begin{proof}
     Take any finite disjoint $X, Y\subseteq\FrGr$. Let $G_{XY}$ be the
@@ -216,8 +216,8 @@
 	Suppose we have $f_n$. We seek $b\in G$ such that $f_n\cup \{\langle
     a_{n+1}, b\rangle\}$ is a partial isomorphism. 
     If $a_{n+1}\in\dom{f_n}$, then simply $b = f_n(a_{n+1})$. Otherwise,
-	let $X = \{a\in\FrGr\mid
-    aE_{\FrGr} a_{n+1}\}\cap \dom{f_n}, Y = X^{c}\cap \dom{f_n}$, i.e. $X$ are
+	let $X = \{a\in\dom{f_n}\mid
+    aE_{\FrGr} a_{n+1}\}, Y = \dom{f_n}\setminus X$, i.e. $X$ are
     vertices of $\dom{f_n}$ that are connected with $a_{n+1}$ in $\FrGr$ and
     $Y$ are those vertices that are not connected with $a_{n+1}$. Let $b$ be
     a vertex of $G$ that is connected to all vertices of $f_n[X]$ and to none
@@ -234,11 +234,11 @@
   Using this fact one can show that the graph constructed in the probabilistic
   manner is in fact isomorphic to the random graph $\FrGr$.
 
-  \begin{definition} We say that a Fraïssé class $\cK$ has the \emph{weak
-    Hrushovski property} (\emph{WHP}) if for every $A\in \cK$ and an isomorphism
+  \begin{definition} We say that a Fraïssé class $\cC$ has the \emph{weak
+    Hrushovski property} (\emph{WHP}) if for every $A\in \cC$ and an isomorphism
 	of its finitely generated substructures 
 	$p\colon A\to A$ (also called a partial automorphism of $A$),
-	there is some $B\in \cK$ such 
+	there is some $B\in \cC$ such 
     that $p$ can be extended to an automorphism of $B$, i.e.\ there is an 
     embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following
     diagram commutes: 
@@ -262,7 +262,7 @@
   to a supergraph so that every partial automorphism of the graph extend
   to an automorphism of the supergraph.
 
-  Moreover, there is a theorem saying that every Fraïssé class $\cK$, in a
+  Moreover, there is a theorem saying that every Fraïssé class $\cC$, in a
   relational language $L$, with \emph{free amalgamation} (see the definition
   \ref{definition:free_amalgamation} below) has WHP. The statement and
   proof of this theorem can be found in 
@@ -272,9 +272,9 @@
 
   \begin{definition}
 	\label{definition:free_amalgamation} 
-	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:
+	Let $L$ be a relational language and $\cC$ a class of $L$-structures.
+	$\cC$ has \emph{free amalgamation} if for every
+	$A, B, C\in\cC$ such that $C = A\cap B$ the following diagram commutes:
 	\begin{center}
 	  \begin{tikzcd}
 		& A\sqcup_C B & \\
@@ -282,7 +282,6 @@
 		& C \ar[ur, hook] \ar[ul, hook'] & 
 	  \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 $\bar{a}\subseteq A\cup B$,
@@ -294,6 +293,8 @@
   Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that
   the class of finite graphs is indeed a Fraïssé class.
 
+  \vspace{3cm}
+
   \subsection{Canonical amalgamation}
 
   Recall, $\Cospan(\cC)$, $\Pushout(\cC)$ are the categories of cospan
@@ -303,10 +304,10 @@
 
   \begin{definition}
 	\label{definition:canonical_amalgamation}
-	Let $\cK$ be a class finitely generated $L$-structures. We say that
-	$\cK$ has \emph{canonical amalgamation} if for every $C\in\cK$ there
-	is a functor $\otimes_C\colon\Cospan_C(\cK)\to\Pushout_C(\cK)$ such that
-	it has the following properties:
+	Let $\cC$ be a class of finitely generated $L$-structures. We say that
+	$\cC$ has \emph{canonical amalgamation} if for every $C\in\cC$ there
+	is a functor $\otimes_C\colon\Cospan_C(\cC)\to\Pushout_C(\cC)$ with
+	following properties:
 	\begin{itemize}
 	  \item Let $A\leftarrow C\rightarrow B$ be a cospan. Then $\otimes_C$ sends
 		it to a pushout that preserves ``the bottom'' structures and embeddings, i.e.:
@@ -359,16 +360,17 @@
 
   \begin{theorem}
 	\label{theorem:canonical_amalgamation_thm}
-	Let $\cK$ be a Fraïssé class of $L$-structures with canonical amalgamation. 
-	Then the class $\cH$ of $L$-structures with automorphism is a Fraïssé class.
+	Let $\cC$ be a Fraïssé class of $L$-structures with canonical amalgamation. 
+	Then the class $\cD$ of $L$-structures with automorphism is a Fraïssé class.
   \end{theorem}
 
   \begin{proof}
-	$\cH$ is obviously countable and has HP. It suffices to show that it
+	$\cD$ is obviously countable and has HP. It suffices to show that it
 	has AP (JEP follows by taking $C$ to be the empty structure). Take any
-	$(A,\alpha), (B,\beta), (C,\gamma)\in \cH$ such that $(C,\gamma)$ embeds
+	$(A,\alpha), (B,\beta), (C,\gamma)\in \cD$ such that $(C,\gamma)$ embeds
 	into $(A,\alpha)$ and $(B,\beta)$. Then $\alpha, \beta, \gamma$ yield
-	an automorphism $\eta$ (as a natural transformation) of a cospan:
+	an automorphism $\eta$ (as a natural transformation, see \ref{fact:natural-automorphism}) 
+	of a cospan:
 	\begin{center}
 	  \begin{tikzcd}
 		A & & B \\
@@ -383,11 +385,10 @@
 	Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
 	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 $\delta\colon A\otimes_C B\to A\otimes_C B$
 	has to be automorphism. Thus, by the fact that the diagram commutes, 
 	we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ 
-	in $\cH$.
+	in $\cD$.
   \end{proof}
 
   The following theorem is one of the most important in construction of
@@ -416,7 +417,7 @@
 	$(\Pi, \sigma)$, then obviously $B\in\cC$ by the definition of $\cD$.
 	Hence, $\cC$ is indeed the age of $\Pi$.
 
-	Now, to show that $\Pi$ is weakly homogeneous, take any structures $A, B\in\cC$ 
+	Now, to show that $\Pi$ is weakly ultrahomogeneous, take any structures $A, B\in\cC$ 
 	such that $A\subseteq B$ with a fixed embedding of $A$ into $\Pi$. 
 	Without the
 	loss of generality assume that $A = B\cap \Pi$ (i.e. $A$ embeds into $\Pi$
@@ -462,8 +463,8 @@
 	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures
 	with WHP and canonical amalgamation. Let
 	$\cD$ be the class consisting of structures from $\cC$ with an additional
-	automorphism. Let $\Gamma = \Flim(\cC)$ and $\Pi = \Flim(\cD)$.
-	Then $\Gamma \cong \Pi\mid_L$.
+	automorphism. Let $\Gamma = \Flim(\cC)$ and $(\Pi,\sigma) = \Flim(\cD)$.
+	Then $\Gamma \cong \Pi$.
   \end{corollary}
 
   \begin{proof}
diff --git a/sections/introduction-pl.tex b/sections/introduction-pl.tex
index 1b06733..7e8243e 100644
--- a/sections/introduction-pl.tex
+++ b/sections/introduction-pl.tex
@@ -19,7 +19,7 @@
 
   Graf losowy \ref{definition:random_graph}, 
   zwany również grafem Rado, jest prototypową strukturą tej
-  prac. Graf losowy można skonstruować jako granicę Fraïsségo klasy skończonych
+  pracy. Graf losowy można skonstruować jako granicę Fraïsségo klasy skończonych
   grafów nieskierowanych. Służy on jako użyteczny przykład, daje intuicję
   stojącą za konstrukcją granicy Fraïsségo, słabej własności Hrushovskiego
   oraz wolnej amalgamacji. Ponadto, co najważniejsze dla niniejszej pracy,
diff --git a/sections/introduction.tex b/sections/introduction.tex
index 6cb432e..8886847 100644
--- a/sections/introduction.tex
+++ b/sections/introduction.tex
@@ -30,7 +30,8 @@
   says that a Fraïssé class with canonical amalgamation and weak Hrushovski
   property has a generic automorphism. The fact that such an automorphism
   exists in this case follows from the classical results of Ivanov \cite{ivanov_1999}
-  and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}, we show a new way to construct
+  and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}.
+  In this work we show a new way to construct
   a generic automorphism by expanding the structures of the class by a (total)
   automorphism and considering limit of such extended Fraïssé class. We achieve
   this by using the Banach-Mazur games, a well known method in the descriptive
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index b27cd69..b35b4d3 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -53,7 +53,7 @@
   \end{definition}
 
   Let $M$ be a structure. We define a topology on the automorphism group 
-  $\Aut(M)$ of $M$ by the basis of open sets: for a finite function
+  $\Aut(M)$ by the basis of open sets: for a finite function
   $f\colon M\to M$ we have a basic open set 
   $[f]_{\Aut(M)} = \{g\in\Aut(M)\mid f\subseteq g\}$. This is a standard
   definition.
@@ -178,7 +178,7 @@
       \item if $(U_0, V_0, \ldots, U_n)\in S$, then $(U_0, V_0, \ldots, U_n,
         V_n)\in S$ for the unique $V_n$ given by the strategy $\sigma$, 
       \item let $p = (U_0, V_0, \ldots, V_n)\in S$. For a possible player
-        move of player I $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the 
+		move of player \textit{I} $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the 
 		unique set
         player $\mathit{II}$ would respond with by $\sigma$. Now, by Zorn's
         Lemma, let $\cU_p$ be a maximal collection of nonempty open subsets
@@ -257,7 +257,7 @@
 
   \begin{proof}[Proof of Theorem \ref{theorem:banach_mazur_thm}]
     $\Rightarrow$: Let $(A_n)$ be a sequence of dense open sets with
-    $\bigcap_n A_n\subseteq A$.  The simply $\textit{II}$ plays $V_n
+    $\bigcap_n A_n\subseteq A$.  Then $\textit{II}$ simply plays $V_n
     = U_n\cap A_n$, which is nonempty by the denseness of $A_n$.
 
     $\Leftarrow$: Suppose $\textit{II}$ has a winning strategy $\sigma$.
@@ -284,8 +284,8 @@
 
   \begin{proof} 
     If one adds the word \textit{basic} before each occurrence
-    of word \textit{open} in previous proofs and theorems then they all
-    will still be valid (except for $\Rightarrow$, but its an easy fix --
+    of word \textit{open} in previous proofs and theorems then they 
+	still will be valid (except for $\Rightarrow$, but its an easy fix --
     take for $V_n$ a basic open subset of $U_n\cap A_n$).  
   \end{proof}
 
@@ -303,19 +303,20 @@
   introduction to the category theory, then it's recommended to take a look
   at \cite{maclane_1978}. Here we will shortly describe the standard notation.
 
-  A \emph{category} $\cC$ consists of the collection of objects (denoted as
-  $\Obj(\cC)$, but most often simply as $\cC$) and collection of \emph{morphisms}
+  A \emph{category} $\cC$ consists of a collection of objects (denoted as
+  $\Obj(\cC)$, but most often simply as $\cC$) and a collection of \emph{morphisms}
   $\Mor(A, B)$ between each pair of objects $A, B\in \cC$. We require that
-  for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there is a 
+  for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there was a 
   morphism $f\circ g\colon A\to C$. If $f\colon A\to B$ then we say
   that $A$ is the domain of $f$ ($\dom{f}$) and that $B$ is the range of
   $f$ ($\rng{f}$).
 
   For every $A\in\cC$ there is an 
-  \emph{identity morphism} $\id_A$ such that for any morphism $f\in \Mor(A, B)$
+  \emph{identity morphism} $\id_A\colon A\to A$ 
+  such that for any morphism $f\in \Mor(A, B)$
   we have that $f\circ id_A = \id_B \circ f$.
 
-  We say that $f\colon A\to B$ is \emph{isomorphism} if there is (necessarily
+  We say that $f\colon A\to B$ is an \emph{isomorphism} if there is (necessarily
   unique) morphism $g\colon B\to A$ such that $g\circ f = id_A$ and $f\circ g = id_B$.
   Automorphism is an isomorphism where $A = B$.
 
@@ -324,8 +325,8 @@
   from category $\cC$ to category $\cD$ if it associates each object $A\in\cC$
   with an object $F(A)\in\cD$, associates each morphism $f\colon A\to B$ in
   $\cC$ with a morphism $F(f)\colon F(A)\to F(B)$. We also require that 
-  $F(\id_A) = \id_{F(A)}$ and that for any (compatible) morphisms $f, g$ in $\cC$
-  $F(f\circ g) = F(f) \circ F(g)$.
+  $F(\id_A) = \id_{F(A)}$ and that for any (compatible) morphisms $f, g$ in $\cC$,
+  $F(f\circ g) = F(f) \circ F(g)$ should hold.
 
   In category theory we distinguish \emph{covariant} and \emph{contravariant}
   functors. Here, we only consider covariant functors, so we will simply
@@ -342,14 +343,14 @@
   \begin{definition}
 	Let $F, G$ be functors between the categories $\cC, \cD$. A \emph{natural 
 	transformation}
-	$\tau$ is function that assigns to each object $A$ of $\cC$ a morphism $\tau_A$
+	$\eta$ is function that assigns to each object $A$ of $\cC$ a morphism $\eta_A$
 	in $\Mor(F(A), G(A))$ such that for every morphism $f\colon A\to B$ in $\cC$
 	the following diagram commutes:
 
     \begin{center}
       \begin{tikzcd}
-		A \arrow[d, "f"] & F(A) \arrow[r, "\tau_A"] \arrow[d, "F(f)"] & G(A) \arrow[d, "G(f)"] \\
-		B & F(B) \arrow[r, "\tau_B"] & G(B) \\ 
+		A \arrow[d, "f"] & F(A) \arrow[r, "\eta_A"] \arrow[d, "F(f)"] & G(A) \arrow[d, "G(f)"] \\
+		B & F(B) \arrow[r, "\eta_B"] & G(B) \\ 
       \end{tikzcd}
     \end{center}
   \end{definition}
@@ -358,6 +359,7 @@
   particularly interesting to us is the following fact.
 
   \begin{fact}
+	\label{fact:natural-automorphism}
 	Let $\eta$ be a natural transformation of functors $F, G$ from category
 	$\cC$ to $\cD$. Then $\eta$ is an isomorphism if and only if
 	all of the component morphisms are isomorphisms.
@@ -407,7 +409,7 @@
       \begin{tikzcd}
 		& D & \\
 		A \arrow[ur, "g"] &  & B \arrow[ul, "h"'] \\
-		& C \arrow[ur, "e"'] \arrow[ul, "f"] &
+		& C \arrow[ul, "e"'] \arrow[ur, "f"] &
       \end{tikzcd}
     \end{center}