Kanoniczna amalgamacja już prawie gotowa!

1 files changed, 54 insertions, 0 deletions

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