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Diffstat (limited to 'sections')
| -rw-r--r-- | sections/introduction.tex | 4 | ||||
| -rw-r--r-- | sections/preliminaries.tex | 72 |
2 files changed, 51 insertions, 25 deletions
diff --git a/sections/introduction.tex b/sections/introduction.tex index 433f3e9..0605356 100644 --- a/sections/introduction.tex +++ b/sections/introduction.tex @@ -22,8 +22,8 @@ Rado graph), the Fraïssé limit of the class of finite undirected graphs. It serves as a useful example, gives an intuition of the Fraïssé limits, weak Hrushovski property and free amalgamation. Perhaps most importantly, - the random graph has a so-called generic automorphism (DODAĆ DEFINICJĘ - GENERYCZNEGO AUTOMORFIZMU I ZLINKOWAĆ TUTAJ), which was first proved + the random graph has a so-called generic automorphism + \ref{definition:generic_automorphism}, which was first proved by Truss in \cite{truss_gen_aut}, where he also introduced the term. The key theorem \ref{theorem:generic_aut_general} 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} |