From cc29818ac950b734bba958c35ba1057cc6a73476 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Tue, 28 Jun 2022 21:36:53 +0200
Subject: Teoria kategorii, kanoniczna amalgamacja

---
 lic_malinka.tex | 129 ++++++++++++++++++++++++++++++++++++++++++++++++++++----
 1 file changed, 121 insertions(+), 8 deletions(-)

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
index bc4d53b..6672e83 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -3,7 +3,7 @@
 
 \usepackage[utf8]{inputenc}
 \usepackage[backend=biber]{biblatex}
-\addbibresource{lic_malinka.bib}
+\addbibresource{licmalinka.bib}
 
 \usepackage[T1]{fontenc} 
 \usepackage{mathtools}
@@ -67,6 +67,8 @@
 \DeclareMathOperator{\lin}{{Lin}}
 \DeclareMathOperator{\Th}{{Th}}
 \DeclareMathOperator{\Obj}{{Obj}}
+\DeclareMathOperator{\Cospan}{{Cospan}}
+\DeclareMathOperator{\Pushout}{{Pushout}}
 \DeclareMathOperator{\Mor}{{Mor}}
 
 \newtheorem{theorem}{Theorem} 
@@ -374,13 +376,18 @@
   \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
+  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 $A\in\cC$
   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 $\cC$
   $F(f\circ g) = F(f) \circ F(g)$.
 
+  In category theory we distinguish \emph{covariant} and \emph{contravariant}
+  functors. Here, we only consider \emph{covariant functors}, so we will simply
+  say \emph{functor}.
+
   Notion that will be very important for us is a ``morphism of functors``
   which is called \emph{natural transformation}.
   \begin{definition}
@@ -397,7 +404,53 @@
       \end{tikzcd}
     \end{center}
   \end{definition}
-  
+
+  \begin{definition}
+	In category theory, a \emph{diagram} of type $\mathcal{J}$ in category $\cC$
+	is a functor $D\colon \mathcal{J}\to\cC$. $\mathcal{J}$ is called the
+	\emph{index category} of $D$. In other words, $D$ is of \emph{shape} $\mathcal{J}$.
+
+	For example, $\mathcal{J} = \{-1\leftarrow 0 \rightarrow 1\}$, then a diagram
+	$D\colon\mathcal{J}\to \cC$ is called a \emph{cospan}. For example,
+	if $A, B, C$ are objects of $\cC$ and $f\in\Mor(C, A), g\in\Mor(C, B)$, then
+	the following diagram is a cospan:
+
+    \begin{center}
+      \begin{tikzcd}
+		A &  & B \\
+		& C \arrow[ur, "g"'] \arrow[ul, "f"] &
+      \end{tikzcd}
+    \end{center}
+  \end{definition}
+
+  From now we omit explicit definition of the index category, as it is easily
+  referable from a picture.
+
+  \begin{definition}
+	Let $A, B, C, D$ be objects in the category $\cC$ with morphisms
+	$e\colon C\to A, f\colon C\to B, g\colon A\to D, h\colon B\to D$ such
+	that $g\circ e = h\circ f$.
+	Then the following diagram:
+    \begin{center}
+      \begin{tikzcd}
+		& D & \\
+		A \arrow[ur, "g"] &  & B \arrow[ul, "h"'] \\
+		& C \arrow[ur, "e"'] \arrow[ul, "f"] &
+      \end{tikzcd}
+    \end{center}
+	
+	is called a \emph{pushout} diagram 
+  \end{definition}
+
+  \begin{definition}
+	The \emph{cospan category} of category $\cC$, refered to as $\Cospan(\cC)$,
+	is the category of cospan diagrams of $\cC$, where morphisms between
+	two cospans are normal transformations of the underlying functors.
+
+	We define \emph{pushout category} analogously and call it $\Pushout(\cC)$.
+  \end{definition}
+
+  TODO: dodać tu przykład?
 
   \section{Fraïssé classes}
 
@@ -429,6 +482,15 @@
     Let $\bK$ be a class of finitely generated structures. We say that $\bK$ has
     \emph{joint embedding property (JEP)} if for any $A, B\in\bK$ there is a
     structure $C\in\bK$ such that both $A$ and $B$ embed in $C$.
+
+    \begin{center}
+      \begin{tikzcd}
+                                  & C & \\
+        A \arrow[ur, dashed, "f"] &   & B \arrow[ul, dashed, "g"'] 
+	  \end{tikzcd}
+    \end{center}
+
+	In terms of category theory, this is a \emph{span} in category $\bK$.
   \end{definition}
 
   Fraïssé has shown fundamental theories regarding age of a structure, one of 
@@ -447,18 +509,21 @@
   \begin{definition}
     Let $\bK$ be a class of finitely generated $L$-structures. We say that $\bK$
     has the \emph{amalgamation property (AP)} if for any $A, B, C\in\bK$ and
-    embeddings $f\colon A\to B, g\colon A\to C$ there exists $D\in\bK$ together
-    with embeddings $h\colon B\to D$ and $j\colon C\to D$ such that 
-    $h\circ f = j\circ g$.
+    embeddings $e\colon C\to A, f\colon C\to B$ there exists $D\in\bK$ together
+    with embeddings $g\colon A\to D$ and $h\colon A\to D$ such that 
+    $g\circ e = h\circ f$.
     \begin{center}
       \begin{tikzcd}
                                   & D & \\
-        B \arrow[ur, dashed, "h"] &   & C \arrow[ul, dashed, "j"'] \\
-                                  & A \arrow[ur, "g"'] \arrow[ul, "f"]
+        A \arrow[ur, dashed, "g"] &   & B \arrow[ul, dashed, "h"'] \\
+                                  & C \arrow[ur, "f"'] \arrow[ul, "e"]
       \end{tikzcd}
     \end{center}
   \end{definition}
 
+  In terms of category theory, amalgamation over some structure $C$ is a
+  pushout diagram.
+
   \begin{definition}
     Let $M$ be an $L$-structure. $M$ is \emph{ultrahomogeneous} if every
     isomorphism between finitely generated substructures of $M$ extends to an
@@ -623,6 +688,53 @@
     It may be there some day, but it may not!
   \end{proof}
 
+  \subsection{Canonical amalgamation}
+
+  \begin{definition}
+	Let $\bK$ be a class finitely generated $L$-structures. We say that
+	$\bK$ has \emph{canonical amalgamation} if for every $C\in\bK$ there
+	is a functor $\otimes_C\colon\Cospan(\bK)\to\Pushout(\bK)$ such that
+	it has the following properties:
+	\begin{itemize}
+	  \item Let $A\leftarrow C\rightarrow B$ be a cospan. Then $\otimes_C$ sends
+		it to a pushout that preserves ``the bottom`` structures and embeddings, i.e.:
+		\begin{center}
+		  \begin{tikzcd}
+			& & 	 							& 		 & A\otimes_C B & \\
+			A & & B  \arrow[r, dashed, "A\otimes_C B"] & A \arrow[ur, dashed] & & B \arrow[ul, dashed] \\
+			& C \arrow[ul] \arrow[ur]  & & & C \arrow[ul] \arrow[ur] &
+		  \end{tikzcd}
+		\end{center}
+
+	  We have deliberately omited names for embeddings of $C$. Of course, 
+	  the functor has to take them into account, but this abuse of notation
+	  is convenient and should not lead into confusion.
+	  \item Let $A\leftarrow C\rightarrow B$, $A'\leftarrow C\rightarrow B'$ be cospans
+		with a natural transformation given by $\alpha\colon A\to A', \beta\colon B\to B',
+		\gamma\colon C\to C$. Then $\otimes_C$ preserves the morphisms of
+		when sending the natural transformation of those cospans to natural
+		transformation of pushouts by adding the 
+		$\delta\colon A\otimes_C B\to A'\otimes_C B'$ morphism:
+
+		\begin{center}
+		  \begin{tikzcd}
+			& A'\otimes_C B' & \\
+			A' \arrow[ur] & & B' \arrow[ul] \\
+			& A\otimes_C B  \ar[uu, dashed, "\delta"] & \\
+			& C \arrow[uul, bend left] \arrow[uur, bend right] & \\
+			A \arrow[uuu, dashed, "\alpha"] \arrow[uur, bend left, crossing over] & & B \arrow[uuu, dashed, "\beta"'] \arrow[uul, bend right, crossing over] \\
+			& C \arrow[ur] \arrow[ul] \arrow[uu, dashed, "\gamma"] & \\ 
+		  \end{tikzcd}
+		\end{center}
+	\end{itemize}
+  \end{definition}
+
+  \begin{theorem}
+	\label{theorem:canonical_amalgamation_thm}
+	Let $\bK$ be a Fraïssé class of $L$-structures with canonical amalgamation. 
+	Then the class $\cH$ of $L$-structures with automorphism is a Fraïssé class.
+  \end{theorem}
+
   \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
@@ -694,6 +806,7 @@
     $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$.
   \end{proof}
 
+
   \begin{theorem}
 	\label{theorem:isomorphic_fr_lims}
 	Let $\cC$ be a Fraïssé class of finite structures in a relational language
-- 
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