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Diffstat (limited to 'sections/preliminaries.tex')
| -rw-r--r-- | sections/preliminaries.tex | 34 |
1 files changed, 18 insertions, 16 deletions
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex index b27cd69..b35b4d3 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} @@ -358,6 +359,7 @@ particularly interesting to us is the following fact. \begin{fact} + \label{fact:natural-automorphism} Let $\eta$ be a natural transformation of functors $F, G$ from category $\cC$ to $\cD$. Then $\eta$ is an isomorphism if and only if all of the component morphisms are isomorphisms. @@ -407,7 +409,7 @@ \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} |