From 78636273b73556313468caf37a938a8a408e4b59 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sat, 9 Jul 2022 14:50:30 +0200
Subject: Corrects to preliminaries

---
 sections/preliminaries.tex | 72 +++++++++++++++++++++++++++++++---------------
 1 file changed, 49 insertions(+), 23 deletions(-)

(limited to 'sections/preliminaries.tex')

diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 5f53c0a..47e889e 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -52,6 +52,13 @@
     }x\}$ is comeagre in $X$.  
   \end{definition}
 
+  \begin{definition}
+	\label{definition:generic_automorphism}
+	Let $G = \Aut(M)$ be the automorphism group of structure $M$. We say
+	that $f\in G$ is a \emph{generic automorphism}, if the conjugacy
+	class of $f$ is comeagre in $G$.
+  \end{definition}
+
   \begin{definition} 
 	Let $X$ be a nonempty topological space and let
     $A\subseteq X$. The \emph{Banach-Mazur game of $A$}, denoted as
@@ -71,8 +78,7 @@
     $T$ is \emph{the tree of all legal positions} in the Banach-Mazur game
     $G^{\star\star}(A)$ when $T$ consists of all finite sequences $(W_0,
     W_1,\ldots, W_n)$, where $W_i$ are nonempty open sets such that
-    $W_0\supseteq W_1\supseteq\ldots\supseteq W_n$. In other words, $T$ is
-    a pruned tree on $\{W\subseteq X\mid W \textrm{is open nonempty}\}$.
+    $W_0\supseteq W_1\supseteq\ldots\supseteq W_n$. 
   \end{definition} 
 
   \begin{definition}
@@ -109,9 +115,14 @@
   playing any $U_1\subseteq V_0$ and $\textit{II}$ plays unique $V_1$
   such that $(U_0, V_0, U_1, V_1)\in\sigma$, etc.
 
+  We will often denote a sequence 
+  $U_0 \supseteq V_0 \supseteq U_1 \supseteq V_1 \supseteq\ldots$ of open sets
+  as \emph{an instance} of a Banach-Mazur game, or just simply by a \emph{game}.
+
   \begin{definition} 
     A strategy $\sigma$ is a \emph{winning strategy for $\textit{II}$} if for
-    any game $(U_0, V_0\ldots)\in [\sigma]$ player $\textit{II}$ wins, i.e.
+    any instance $(U_0, V_0\ldots)\in [\sigma]$ of the Banach-Mazur game
+	player $\textit{II}$ wins, i.e.
     $\bigcap_{n}V_n \subseteq A$.
   \end{definition}
 
@@ -133,7 +144,7 @@
     \emph{comprehensive for p} if the family $\cV_p = \{V_{n+1}\mid (U_0,
     V_0,\ldots, V_n, U_{n+1}, V_{n+1})\in S\}$ (it may be that $n=-1$, which
     means $p=\emptyset$) is pairwise disjoint and $\bigcup\cV_p$ is dense in
-    $V_n$ (where we think that $V_{-1} = X$).
+    $V_n$ (where we put $V_{-1} = X$).
     We say that $S$ is \emph{comprehensive} if it is comprehensive for
     each $p=(U_0, V_0,\ldots, V_n)\in S$.  
   \end{definition}
@@ -149,8 +160,9 @@
       \item $\emptyset\in S$, 
       \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 $I$'s
-        move $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the unique set
+      \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 
+		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
         $U_{n+1}\subseteq V_n$ such that the set $\{U^\star_{n+1}\mid
@@ -176,10 +188,12 @@
     \begin{enumerate}[label=(\roman*)]
       \item For any open $V_n\subseteq X$ there is at most one $p=(U_0, V_0,
         \ldots, U_n, V_n)\in S$.  
-      \item Let $S_n = \{V_n\mid (U_0, V_0, \ldots, V_n)\in S\}$ for $n\in\bN$
-        (i.e. $S_n$ is a family of all possible choices player $\textit{II}$
-        can make in its $n$-th move according to $S$). Then $\bigcup S_n$ is
-        open and dense in $X$.  
+	  \end{enumerate}
+	  Let $S_n = \{V_n\mid (U_0, V_0, \ldots, V_n)\in S\}$ for $n\in\bN$
+	  (i.e. $S_n$ is a family of all possible choices player $\textit{II}$
+	  can make in its $n$-th move according to $S$).
+	  \begin{enumerate}[resume, label=(\roman*)]
+      \item  $\bigcup S_n$ is open and dense in $X$.  
       \item $S_n$ is a family of pairwise disjoint sets.  
     \end{enumerate} 
   \end{lemma}
@@ -218,7 +232,7 @@
     V'_{n+1}$. Moreover, there is no such set in
     $S_{n+1}\setminus\cV_{p_{V_n}}$, because those sets are disjoint from
     $V_{n}$. Hence there is no $V'_{n+1}\in S_{n+1}$ other than $V_n$
-    such that $x\in V'_{n+1}$. We've chosen $x$ and $V_{n+1}$ arbitrarily,
+    such that $x\in V'_{n+1}$. We have chosen $x$ and $V_{n+1}$ arbitrarily,
     so $S_{n+1}$ is pairwise disjoint.  
   \end{proof}
 
@@ -255,11 +269,11 @@
     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 --
-    take $V_n$ a basic open subset of $U_n\cap A_n$).  
+    take for $V_n$ a basic open subset of $U_n\cap A_n$).  
   \end{proof}
 
   This corollary will be important in using the theorem in practice --
-  it's much easier to work with basic open sets rather than any open
+  it's much easier to work with basic open sets rather than arbitrary open
   sets.
 
   \subsection{Category theory}
@@ -268,23 +282,23 @@
   category theory that will be necessary to generalize the key result of the
   paper.
 
-  We will use a standard notation. If the reader is interested in detailed
+  We will use a standard notation. If the reader is interested in a more detailed
   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}
   $\Mor(A, B)$ between each pair of objects $A, B\in \cC$. We require that
-  for each morphisms $f\colon B\to C$, $g\colon A\to B$ there is a morphism
-  $f\circ g\colon A\to C$. For every $A\in\cC$ there is an 
+  for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there is a 
+  morphism $f\circ g\colon A\to C$. For every $A\in\cC$ there is an 
   \emph{identity morphism} $\id_A$ such that for any morphism $f\in \Mor(A, B)$
-  it follows that $f\circ id_A = \id_B \circ f$.
+  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
   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 $A = B$.
 
-  A \emph{functor} is a ``homeomorphism`` of categories. We say that 
+  A \emph{functor} is a ``(homo)morphism`` of categories. We say that 
   $F\colon\cC\to\cD$ is a functor
   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
@@ -293,7 +307,7 @@
   $F(f\circ g) = F(f) \circ F(g)$.
 
   In category theory we distinguish \emph{covariant} and \emph{contravariant}
-  functors. Here, we only consider \emph{covariant functors}, so we will simply
+  functors. Here, we only consider covariant functors, so we will simply
   say \emph{functor}.
 
   \begin{fact}
@@ -302,7 +316,7 @@
 	to the isomorphism $F(f)\colon F(A)\to F(B)$ in $\cD$.
   \end{fact}
 
-  Notion that will be very important for us is a ``morphism of functors``
+  A notion that will be very important for us is a ``morphism of functors``
   which is called \emph{natural transformation}.
   \begin{definition}
 	Let $F, G$ be functors between the categories $\cC, \cD$. A \emph{natural 
@@ -353,16 +367,28 @@
       \end{tikzcd}
     \end{center}
 	
-	is called a \emph{pushout} diagram 
+	is called a \emph{pushout diagram}.
   \end{definition}
 
+  In both definitions of cospan and pushout diagrams we say that the object $C$
+  is the \emph{base} of the diagram.
+
   \begin{definition}
 	The \emph{cospan category} of category $\cC$, referred to as $\Cospan(\cC)$,
 	is the category of cospan diagrams of $\cC$, where morphisms between
-	two cospans are normal transformations of the underlying functors.
+	two cospans are natural transformations of the underlying functors.
 
 	We define \emph{pushout category} analogously and call it $\Pushout(\cC)$.
   \end{definition}
 
-  TODO: dodać tu przykład?
+  From now on we work in subcategories of cospan diagrams and pushout diagrams
+  where we fix the base structure. Formally, for a fixed
+  $C\in\cC$, category $\Cospan_C(\cC)$ is the category of all cospans in 
+  $\Cospan(\cC)$ such that the base of the diagram is $C$. 
+  Natural transformation $\eta$ of two diagrams in $\Cospan_C(\cC)$ are 
+  such that 
+  the morphism $\eta_C\colon C\to C$ is an automorphism of $C$. 
+  $\Pushout_C(\cC)$ is defined analogously. In most contexts we consider 
+  only one base structure, 
+  hence we will often write $\Pushout(\cC)$ instead of $\Pushout_C(\cC)$.
 \end{document}
-- 
cgit v1.2.3

