From 8c3772288630df347539eadde88dc22cc4ef2af0 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Tue, 28 Jun 2022 18:28:29 +0200 Subject: Nowe rzeczy, wstep do teorii kategorii --- .gitignore | 5 +++- lic_malinka.bib | 17 ++++++++++++ lic_malinka.pdf | Bin 398521 -> 404331 bytes lic_malinka.tex | 85 +++++++++++++++++++++++++++++++++++++++++++++++--------- licmalinka.bib | 10 ------- update-pdf.sh | 27 ------------------ 6 files changed, 93 insertions(+), 51 deletions(-) create mode 100644 lic_malinka.bib delete mode 100644 licmalinka.bib delete mode 100644 update-pdf.sh diff --git a/.gitignore b/.gitignore index a53d1a8..c8332b7 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,7 @@ lic_malinka.* +szkola-orlow* !lic_malinka.tex -!lic_malinka.pdf \ No newline at end of file +!lic_malinka.pdf +!lic_malinka.bib + diff --git a/lic_malinka.bib b/lic_malinka.bib new file mode 100644 index 0000000..34451a7 --- /dev/null +++ b/lic_malinka.bib @@ -0,0 +1,17 @@ +@book{hodges_1993, + place={Cambridge}, + series={Encyclopedia of Mathematics and its Applications}, + title={Model Theory}, + DOI={10.1017/CBO9780511551574}, + publisher={Cambridge University Press}, + author={Hodges, Wilfrid}, + year={1993}, + collection={Encyclopedia of Mathematics and its Applications} +} +@book{maclane_1978, + title={Categories for the Working Mathematics}, + DOI={10.1007/978-1-4757-4721-8}, + publisher={Springer New York, NY}, + author={Saunders Mac Lane}, + year={1978}, +} diff --git a/lic_malinka.pdf b/lic_malinka.pdf index 64f797a..a584720 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ 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} diff --git a/licmalinka.bib b/licmalinka.bib deleted file mode 100644 index 4c4c881..0000000 --- a/licmalinka.bib +++ /dev/null @@ -1,10 +0,0 @@ -@book{hodges_1993, - place={Cambridge}, - series={Encyclopedia of Mathematics and its Applications}, - title={Model Theory}, - DOI={10.1017/CBO9780511551574}, - publisher={Cambridge University Press}, - author={Hodges, Wilfrid}, - year={1993}, - collection={Encyclopedia of Mathematics and its Applications} -} \ No newline at end of file diff --git a/update-pdf.sh b/update-pdf.sh deleted file mode 100644 index f342079..0000000 --- a/update-pdf.sh +++ /dev/null @@ -1,27 +0,0 @@ -if [[ $# != 1 ]] -then - echo "Usage: $0 [.tex file]" - exit 1 -fi - -PRACA=$1 -if ! [[ -f $PRACA ]] -then - echo "No such file: $PRACA" - exit 1 -fi - -threshold=1 -while true -do - if (($(date +"%s")- $(stat --format="%Y" $PRACA) < $threshold)) - then - echo "updating" - pdflatex $PRACA - biber $PARACA - pdflatex $PRACA - pdflatex $PRACA - else - sleep 0.2 - fi -done -- cgit v1.2.3