From 8c3772288630df347539eadde88dc22cc4ef2af0 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Tue, 28 Jun 2022 18:28:29 +0200
Subject: Nowe rzeczy, wstep do teorii kategorii

---
 lic_malinka.tex | 85 ++++++++++++++++++++++++++++++++++++++++++++++++---------
 1 file changed, 72 insertions(+), 13 deletions(-)

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 9cce430..bc4d53b 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -1,9 +1,9 @@
-\documentclass[11pt, a4paper, final]{amsart} 
+\documentclass[12pt, a4paper, final]{amsart} 
 \setlength{\emergencystretch}{2em}
 
 \usepackage[utf8]{inputenc}
 \usepackage[backend=biber]{biblatex}
-\addbibresource{licmalinka.bib}
+\addbibresource{lic_malinka.bib}
 
 \usepackage[T1]{fontenc} 
 \usepackage{mathtools}
@@ -11,6 +11,7 @@
 \microtypecontext{spacing=nonfrench}
 % \usepackage[utf8]{inputenc}
 
+
 \usepackage{amsfonts} 
 \usepackage{amsmath} 
 \usepackage{amssymb}
@@ -65,6 +66,8 @@
 \DeclareMathOperator{\id}{{id}} 
 \DeclareMathOperator{\lin}{{Lin}}
 \DeclareMathOperator{\Th}{{Th}}
+\DeclareMathOperator{\Obj}{{Obj}}
+\DeclareMathOperator{\Mor}{{Mor}}
 
 \newtheorem{theorem}{Theorem} 
 \numberwithin{theorem}{section}
@@ -353,6 +356,49 @@
   it's much easier to work with basic open sets rather than any open
   sets.
 
+  \subsection{Category theory}
+
+  In this section we will give a short introduction to the notions of
+  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
+  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 
+  \emph{identity morphism} $\id_A$ such that for any morphism $f\in \Mor(A, B)$
+  it follows that $f\circ id_A = \id_B \circ f$.
+
+  A \emph{functor} is a ``homeomorphism`` of categories. $F\colon\cC\to\cD$ 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
+  $\cC$ 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 $\cC$
+  $F(f\circ g) = F(f) \circ F(g)$.
+
+  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 
+	transformation}
+	$\tau$ is function that assigns to each object $A$ of $\cC$ a morphism $\tau_A$
+	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) \\ 
+      \end{tikzcd}
+    \end{center}
+  \end{definition}
+  
+
   \section{Fraïssé classes}
 
   In this section we will take a closer look at classes of finitely
@@ -701,12 +747,6 @@
 
   \section{Conjugacy classes in automorphism groups}
 
-  TODO:
-  \begin{itemize}
-	\item w głównym dowodzie mogę użyć wprost AP, nie muszę tego uzasadniać
-	  jeszcze raz.
-  \end{itemize}
-
   Let $M$ be a countable $L$-structure. We define a topology on the $G=\Aut(M)$:
   for any finite function $f\colon M\to M$ we have a basic open set 
   $[f]_{G} = \{g\in G\mid f\subseteq g\}$.  
@@ -925,15 +965,34 @@
 
   \subsection{Properties of the generic automorphism}
 
+  Let $\cC$ be a Fraïssé class in a finite relational language $L$ with
+  weak Hrushovski property. Let $\cH$ be the Fraïssé class of the $L$-structures
+  with additional automorphism symbol. Let $\Gamma = \Flim(\cC)$.
+
+ %  \begin{proposition}
+	% Let $\sigma$ be the generic automorphism of the random graph $\FrGr$. Then
+	% the graph induced by the set of the fixed points of $\sigma$ is isomorphic
+	% to $\FrGr$.
+ %  \end{proposition}
+	%
+ %  \begin{proof}
+	% Let $F = \{v\in\FrGr\mid \sigma(v) = v\}$. It suffices to show that $F$ is
+	% infinite and has the random graph property.
+ %  \end{proof}
   \begin{proposition}
-	Let $\sigma$ be the generic automorphism of the random graph $\FrGr$. Then
-	the graph induced by the set of the fixed points of $\sigma$ is isomorphic
-	to $\FrGr$.
+	Let $\sigma$ be the generic automorphism of $\Gamma$. Then the set
+	of fixed points of $\sigma$ is isomorphic to $\Gamma$.
   \end{proposition}
 
   \begin{proof}
-	Let $F = \{v\in\FrGr\mid \sigma(v) = v\}$. It suffices to show that $F$ is
-	infinite and has the random graph property.
+	Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$.
+	First we need to show that it is an infinite. By the theorem \ref{theorem:generic_aut_general}
+	we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cH$, thus we
+	can embedd finite $L$-structures of any size with identity as an 
+	automorphism of the	structure into $(\Gamma, \sigma)$. Thus $S$ has to be
+	infinite. Also, the same argument shows that the age of the structure is
+	exactly $\cC$. It is weakly ultrahomogeneous, also by the fact that
+	$(\Gamma, \sigma)$ is in $\cH$.
   \end{proof}
   \printbibliography
 \end{document}
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