From 3379b5851e1dd3096e832215b5685ec9f058ebb0 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 17 Jul 2022 22:04:55 +0200
Subject: Poprawki male

---
 sections/preliminaries.tex | 33 +++++++++++++++++----------------
 1 file changed, 17 insertions(+), 16 deletions(-)

(limited to 'sections/preliminaries.tex')

diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index b27cd69..d6d5376 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}
@@ -407,7 +408,7 @@
       \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}
 	
-- 
cgit v1.2.3

