From 488d16570b6cd0d1bdd265e22c87e19da6e179a0 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Wed, 29 Jun 2022 15:23:29 +0200 Subject: =?UTF-8?q?Kanoniczna=20amalgamacja=20ju=C5=BC=20prawie=20gotowa!?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- lic_malinka.pdf | Bin 415528 -> 419889 bytes lic_malinka.tex | 54 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 54 insertions(+) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index 0e26d72..4d7d77d 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ diff --git a/lic_malinka.tex b/lic_malinka.tex index 6672e83..4ce39e0 100644 --- a/lic_malinka.tex +++ b/lic_malinka.tex @@ -376,6 +376,10 @@ \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 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 $F\colon\cC\to\cD$ is a functor from category $\cC$ to category $\cD$ if it associates each object $A\in\cC$ @@ -388,6 +392,12 @@ functors. Here, we only consider \emph{covariant functors}, so we will simply say \emph{functor}. + \begin{fact} + \label{fact:functor_iso} + Functor $F\colon\cC\to\cD$ maps isomorphism $f\colon A\to B$ in $\cC$ + 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`` which is called \emph{natural transformation}. \begin{definition} @@ -726,6 +736,13 @@ & C \arrow[ur] \arrow[ul] \arrow[uu, dashed, "\gamma"] & \\ \end{tikzcd} \end{center} + % \begin{center} + % \begin{tikzcd} + % & A \ar[rrr, dashed, "\alpha"] \ar[drr, bend left=20, crossing over] & & & A' \ar[dr] & \\ + % C \ar[rr, dashed, "\gamma"] \ar[ur] \ar[dr] & & C \ar[rrd, bend right=20] \ar[rru, bend left=20] & A\otimes_C B \ar[rr, dashed, "\delta"] & & A' \otimes_C B' \\ + % & B \ar[rrr, dashed, "\beta"] \ar[urr, bend right=20, crossing over] & & & B' \ar[ur] & \\ + % \end{tikzcd} + % \end{center} \end{itemize} \end{definition} @@ -735,6 +752,43 @@ Then the class $\cH$ of $L$-structures with automorphism is a Fraïssé class. \end{theorem} + \begin{proof} + $\cH$ is obviously countable and has HP. It suffices to show that it + has AP (JEP follows by taking $C$ to be the empty structure). Take any + $(A,\alpha), (B,\beta), (C,\gamma)\in \cH$ such that $(C,\gamma)$ embeds + into $(A,\alpha)$ and $(B,\beta)$. Then $\alpha, \beta, \gamma$ yield + an automorphism $\eta$ (as a natural transformation) of a cospan: + \begin{center} + \begin{tikzcd} + A & & B \\ + % & C \ar[ur] \ar[ul] & \\ + A \ar[u, dashed, "\alpha"] & C \ar[ur] \ar[ul] & B \ar[u, dashed, "\beta"'] \\ + & C \ar[ur] \ar[ul] \ar[u, dashed, "\gamma"] & + % (A, \alpha) & & (B, \beta) \\ + % & (C, \gamma) \ar[ur] \ar[ul] & + \end{tikzcd} + \end{center} + + Then, by the fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism + of the pushout diagram: + + \begin{center} + \begin{tikzcd} + & A\otimes_C B \ar[loop above, "\delta"] & \\ + A \ar[ur] \ar[loop left, "\alpha"]& & B \ar[ul] \ar[loop right, "\beta"]\\ + & C \ar[ur] \ar[ul] \ar[loop below, "\gamma"] & + \end{tikzcd} + \end{center} + + TODO: ten diagram nie jest do końca taki jak trzeba, trzeba w zasadzie skopiować + ten z definicji kanonicznej amalgamcji. Czy to nie będzie wyglądać źle? + + This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$ + has to be automorphism. Thus, by the fact that the diagram commutes, + we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ + in $\cH$. + \end{proof} + \subsection{Graphs with automorphism} The language and theory of unordered graphs is fairly simple. We extend the language by one unary symbol $\sigma$ and interpret it as an arbitrary -- cgit v1.2.3