# HG changeset patch # User Yasutaka Higa # Date 1423369772 -32400 # Node ID 6c0d32ef01cd519a3db694a05cb92daaf87ad47d # Parent 586f3ce1effe05f9f556ae157d39c69648cc80ea Add Functor diff -r 586f3ce1effe -r 6c0d32ef01cd category.tex --- 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。 + +% }}} + 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