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changeset 15:6c0d32ef01cd

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Add Functor
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 08 Feb 2015 13:29:32 +0900
parents 586f3ce1effe
children cff9f6f0ff25
files category.tex fig/delta_example.xbb fig/functor.graffle fig/functor.pdf fig/functor.xbb fig/morphism_composition_law.xbb fig/non_delta_example.xbb fig/ryukyu.xbb fig/simple_category.xbb
diffstat 9 files changed, 1456 insertions(+), 8 deletions(-) [+]
line wrap: on
line diff
--- a/category.tex	Sun Feb 08 12:22:56 2015 +0900
+++ b/category.tex	Sun Feb 08 13:29:32 2015 +0900
@@ -8,9 +8,10 @@
 % {{{ Category
 
 \section{Category}
+\label{section:category}
 まずは Monad の定義に必要な Category (圏)について述べる。
 
-Category は2つの要素を持つ。
+category は2つの要素を持つ。
 
 \begin{itemize}
     \item object (要素)
@@ -30,9 +31,8 @@
 
 \end{itemize}
 
-
+そして category は object と moprhism に対して2つの法則を満たす。
 
-そして2つの法則を満たす。
 \begin{itemize}
     \item 全ての object について identity mapping が存在する
 
@@ -91,8 +91,95 @@
 
 % }}}
 
+% {{{ Functor
 
 \section{Functor}
+\label{section:functor}
+\ref{section:category} 節では category を定義した。
+\ref{section:functor} 節では category から category への mapping である Functor (関手)について解説する。
+
+functor F とはある category C から D への mapping である。
+functor F を用いて category C の object もしくは morphism である x を category D の object もしくは morphism に対応させることを式\ref{exp:apply_functor}のように記述する。
+
+\begin{equation}
+    F(x)
+    \label{exp:apply_functor}
+\end{equation}
+
+functor は以下の functor 則を満たす。
+
+\begin{itemize}
+    \item id を保存する
+
+        category C における object A への id $ 1_A $ は F A においても id として振る舞う。
+
+        \begin{equation}
+            F(1_A) = 1_{F(A)}
+        \end{equation}
+
+    \item morphism の合成を保存する
+
+        category C における morphism f g の合成を考える。
+        合成された $ f \circ g $ を F により mapping することと、f と g を個別に F により mapping した後に合成した結果は同じである。
+
+        \begin{equation}
+            F(f \circ g) = F(f) \circ F(g)
+        \end{equation}
+\end{itemize}
+
+functor の例を挙げる。
+まず、 3つの object A,B,C と morphism f,g,h を持つ category C を考える。
+category C の morphism は式\ref{exp:functor_categoryC}であるとする。
+
+\begin{eqnarray}
+    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}
+    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。
+
+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}
+    F(A) = A' \\ \nonumber
+    F(B) = B' \\ \nonumber
+    F(C) = A' \\ \nonumber
+    F(id_A) = F(id_{A'}) \\ \nonumber
+    F(id_B) = F(id_{B'}) \\ \nonumber
+    F(id_C) = F(id_{A'}) \\ \nonumber
+    F(f) = id_{A'} \\ \nonumber
+    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。
+
+\begin{figure}[htbp]
+    \begin{center}
+        \includegraphics[scale=0.8]{fig/functor.pdf}
+        \caption{Functor の例}
+        \label{fig:functor}
+    \end{center}
+\end{figure}
+
+functor により category から category への mapping を考えることが可能となった。
+
+% }}}
+
 \section{Natural Transformation}
 \section{Monad}
 \section{Monads in Functional Programming}
--- a/fig/delta_example.xbb	Sun Feb 08 12:22:56 2015 +0900
+++ b/fig/delta_example.xbb	Sun Feb 08 13:29:32 2015 +0900
@@ -4,5 +4,5 @@
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+%%CreationDate: Sun Feb  8 13:09:16 2015
 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/fig/functor.graffle	Sun Feb 08 13:29:32 2015 +0900
@@ -0,0 +1,1353 @@
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