Nowe rzeczy, wstep do teorii kategorii

1 files changed, 72 insertions, 13 deletions

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}