Updates updates

1 files changed, 28 insertions, 1 deletions

diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 82e64b4..05aa3ed 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -306,7 +306,11 @@
   $\Obj(\cC)$, 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 pair of 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 
+  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 $f\in \Mor(A, B)$
   we have that $f\circ id_A = \id_B \circ f$.
 
@@ -349,6 +353,29 @@
     \end{center}
   \end{definition}
 
+  Natural transformation has, \textit{nomen omen}, natural properties. One
+  particularly interesting to us is the following fact.
+
+  \begin{fact}
+	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.
+  \end{fact}
+
+  \begin{proof}
+	Suppose that $\eta_(A)$ is an isomorphism for every $A\in\cC$, where 
+	$\eta_{A}\colon F(A)\to G(A)$ is the morphism of the natural transformation
+	coresponding to $A$. Then $\eta^{-1}$ is simply given by the morphisms
+	$\eta^{-1}_A$.
+
+	Now assume that $\eta$ is an isomorphism, i.e. $\eta^{-1}\circ\eta = \id_F$.
+	\textit{Ad contrario} assume that there is $A\in\cC$ such that the component
+	morphism $\eta_A\colon F(A)\to G(A)$ is not an isomorphism. It means
+	that $\eta_A^{-1}\circ\eta_A \neq id_A$, hence 
+	$F(A) = \dom(\eta^{-1}\circ\eta)(A) \neq \rng(\eta^{-1}\circ\eta)(A) = F(A)$,
+	which is obviously a contradiction.
+  \end{proof}
+
   \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