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Add description natural transformation
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 08 Feb 2015 15:43:20 +0900
parents 6c0d32ef01cd
children e2afa8266ecc
files category.tex fig/delta_example.xbb fig/functor.xbb fig/morphism_composition_law.xbb fig/natural_transformation.graffle fig/natural_transformation.pdf fig/natural_transformation.xbb fig/non_delta_example.xbb fig/ryukyu.xbb fig/simple_category.xbb
diffstat 10 files changed, 729 insertions(+), 10 deletions(-) [+]
line wrap: on
line diff
--- a/category.tex	Sun Feb 08 13:29:32 2015 +0900
+++ b/category.tex	Sun Feb 08 15:43:20 2015 +0900
@@ -53,11 +53,11 @@
         改めて、morphism の合成の順序が結果に影響しないことを式にすると式\ref{exp:morphism_composition_law}のようになる。
 
         \begin{eqnarray}
+            \label{exp:morphism_composition_law}
             f : C \rightarrow D \nonumber \\
             g : B \rightarrow C \nonumber \\
             h : A \rightarrow B \nonumber \\
             (f \circ g) \circ h = f \circ (g \circ h) : A \rightarrow D
-            \label{exp:morphism_composition_law}
         \end{eqnarray}
 \end{itemize}
 
@@ -132,20 +132,20 @@
 category C の morphism は式\ref{exp:functor_categoryC}であるとする。
 
 \begin{eqnarray}
+    \label{exp:functor_categoryC}
     f : A \rightarrow C \\ \nonumber
     g : B \rightarrow B \\ \nonumber
     h : B \rightarrow C \\ \nonumber
     f = g \circ h
-    \label{exp:functor_categoryC}
 \end{eqnarray}
 
 次に、 2 つの object A', B' と morphism f', g' を持つ category D を考える。
 category D の morphism は式\ref{exp:functor_categoryD}であるとする。
 
 \begin{eqnarray}
+    \label{exp:functor_categoryD}
     f : B' \rightarrow A' \\ \nonumber
     g : A' \rightarrow B' \\ \nonumber
-    \label{exp:functor_categoryD}
 \end{eqnarray}
 
 C から D への functor F は式\ref{exp:functorF}のようなものがある。
@@ -153,6 +153,7 @@
 要素数が異なるため一見正しく mapping できないように思えるが、正しく Functor 則を満たしている。
 
 \begin{eqnarray}
+    \label{exp:functorF}
     F(A) = A' \\ \nonumber
     F(B) = B' \\ \nonumber
     F(C) = A' \\ \nonumber
@@ -163,7 +164,6 @@
     F(g) = g' \\ \nonumber
     F(h) = f' \\ \nonumber
     F(h \circ g) = F(h) \circ F(g) = f' \circ g' = id_{A'} = F(f)
-    \label{exp:functorF}
 \end{eqnarray}
 
 functor F を用いて category C を category D へと mapping した図が図\ref{fig:functor}である。
@@ -180,7 +180,44 @@
 
 % }}}
 
+% {{{ Natural Transformation
+
 \section{Natural Transformation}
+\label{section:natural_transformation}
+\ref{section:functor}節では category から category への mapping である functor について述べた。
+\ref{section:natural_transformation}節では functor と functor の関係である Natural Transformation(自然変換)について述べる。
+
+category C から D への functor F, G が存在する時、F から G への変換 t を考える。
+
+t は category C の A に対し、F(A) から G(A) への変換を提供する(式\ref{exp:trans})。
+
+\begin{equation}
+    t(A) : F(A) \rightarrow G(A)
+    \label{exp:trans}
+\end{equation}
+
+t が 式\ref{exp:natural_transformation}を満たす時、t を natural transformation を呼ぶ。
+
+\begin{eqnarray}
+    \label{exp:natural_transformation}
+    f : A \rightarrow B \\ \nonumber
+    G(f)t(A) = t(B)F(f)
+\end{eqnarray}
+
+t が natural transformation である時、 category A における A を functor F で移したのちに morphism を適用しても、 functor G で移した後に morphism を適用しても良いという可換性が得られる。
+式\ref{exp:natural_transformation}の可換図は図\ref{fig:natural_transformation}となる。
+
+\begin{figure}[htbp]
+    \begin{center}
+        \includegraphics[scale=1.0]{fig/natural_transformation.pdf}
+        \caption{自然変換の可換図(式\ref{exp:natural_transformation})}
+        \label{fig:natural_transformation}
+    \end{center}
+\end{figure}
+
+% }}}
+
 \section{Monad}
+\section{Kleisli Triple}
 \section{Monads in Functional Programming}
 
--- a/fig/delta_example.xbb	Sun Feb 08 13:29:32 2015 +0900
+++ b/fig/delta_example.xbb	Sun Feb 08 15:43:20 2015 +0900
@@ -4,5 +4,5 @@
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-%%CreationDate: Sun Feb  8 13:09:16 2015
+%%CreationDate: Sun Feb  8 15:38:39 2015
 
--- a/fig/functor.xbb	Sun Feb 08 13:29:32 2015 +0900
+++ b/fig/functor.xbb	Sun Feb 08 15:43:20 2015 +0900
@@ -4,5 +4,5 @@
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 %%PDFVersion: 1.3
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--- a/fig/morphism_composition_law.xbb	Sun Feb 08 13:29:32 2015 +0900
+++ b/fig/morphism_composition_law.xbb	Sun Feb 08 15:43:20 2015 +0900
@@ -4,5 +4,5 @@
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 %%PDFVersion: 1.3
 %%Pages: 1
-%%CreationDate: Sun Feb  8 13:09:16 2015
+%%CreationDate: Sun Feb  8 15:38:39 2015
 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/fig/natural_transformation.graffle	Sun Feb 08 15:43:20 2015 +0900
@@ -0,0 +1,674 @@
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