ていへんかけるたかさわるに

モナドの概念をまとめておく.
定義等は Emily Riehl の教科書 "Category Theory in Context" に従う.

定義.圏 $\mathrm{C}$ におけるモナド (monad) は 3 つ組 $(T,\eta,\mu)$:
・ 関手 $T : \mathrm{C} \rightarrow \mathrm{C}$;
・ 単位 (unit)と呼ばれる自然変換 $\eta : 1_{\mathrm{C}} \Rightarrow T$;
・ 積 (multiplication)と呼ばれる自然変換 $\mu : T^2 \Rightarrow T$
で, 以下の図式を可換にするものである.
\begin{equation*}
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\begin{xy}
\xymatrix@=24pt {
T^3 \ar@{=>}[d]_{\mu T} \ar@{=>}[r]^{T \mu} & T^2 \ar@{=>}[d]^{\mu} \\
T^2 \ar@{=>}[r]_{\mu} & T
}
\qquad
\xymatrix@=24pt {
T \ar@{=>}[r]^{\eta T} \ar@{=>}[dr]_{\Un{T}}
& T^2 \ar@{=>}[d]^{\mu}
& T \ar@{=>}[l]_{T \eta} \ar@{=>}[dl]^{\Un{T}} \\
& T &
}
\end{xy}
\end{equation*}
補題. 任意の随伴
\begin{equation*}
\begin{xy}
\xymatrix@=24pt {
\rC \ar@<1ex>[r]^F \ar@{}[r]|{\bot} & \rD \ar@<1ex>[l]^U
}
\end{xy}
\qquad
\eta : \Un{\rC} \Rightarrow UF,
\quad
\epsilon : FU \Rightarrow \Un{\rD}
\end{equation*} は以下の構成により左随伴の定義域の圏 $\rC$ 上のモナドを引き起こす:
・ 関手 $T : \rC \rightarrow \rC$ は $UF$ により与えられる.
・ 単位 $\eta : \Un{\rC} \Rightarrow UF$ がモナドの単位 $\eta : \Un{\rC} \Rightarrow T$ となる.
・ $U \epsilon F : UFUF \Rightarrow UF$ がモナドの積 $\mu : T^2 \Rightarrow T$ となる.

証明 (概略).
まず, 図式
\begin{equation}
\label{dgm:1}
\begin{xy}
\xymatrix@=24pt {
UF \ar@{=>}[r]^{\eta UF} \ar@{=>}[dr]_{\Un{UF}} & UFUF \ar@{=>}[d]^{U \epsilon F} & UF \ar@{=>}[l]_{UF\eta} \ar@{=>}[dl]^{\Un{UF}} \\
& UF &
}
\end{xy} \tag{1}
\end{equation}
を考える.
随伴の単位 (unit) $\eta : \Un{\rC} \Rightarrow UF$, 余単位 (counit) $\epsilon : FU \Rightarrow \Un{\rD}$ は三角恒等式 (triangle identities)
\begin{equation*}
\begin{xy}
\xymatrix@=24pt {
F \ar@{=>}[r]^{F\eta} \ar@{=>}[dr]_{\Un{F}} & FUF \ar@{=>}[d]^{\epsilon F} \\
& F
}
\end{xy}
\qquad
\begin{xy}
\xymatrix@=24pt {
U \ar@{=>}[r]^{\eta U} \ar@{=>}[dr]_{\Un{U}} & UFU \ar@{=>}[d]^{U\epsilon} \\
& U
}
\end{xy}
\end{equation*} を可換にする.
左の三角形に $U$ を適用すると (\ref{dgm:1}) の右側の三角形が得られる.
右の三角形を $F$ に適用すると (\ref{dgm:1}) の左側の三角形が得られる.
よって, 図式 (\ref{dgm:1}) は可換である.

次に図式
\begin{equation}
\label{dgm:2}
\begin{xy}
\xymatrix@=24pt {
UFUFUF \ar@{=>}[d]_{U \epsilon FUF} \ar@{=>}[rr]^{UFU \epsilon F} && UFUF \ar@{=>}[d]^{U \epsilon F} \\
UFUF \ar@{=>}[rr]_{U \epsilon F} && UF
}
\end{xy} \tag{2}
\end{equation} を考える.
任意の対象 $c\in\rC$ にこの図式を適用すると,
\begin{equation*}
\begin{xy}
\xymatrix@=24pt {
UFU(FUFc) \ar[d]_{U\epsilon_{FUFc}} \ar[rr]^{UFU(\epsilon_{Fc})} && UFU(Fc) \ar[d]^{U\epsilon_{Fc}} \\
U(FUFc) \ar[rr]_{U(\epsilon_{Fc})} && U(Fc)
}
\end{xy}
\end{equation*} となり, これは自然変換 $U\epsilon : UFU \Rightarrow U$ の $c$ における可換性を示す図式に他ならない.
よって, 図式 (\ref{dgm:2}) も可換となり, $(UF,\eta,U\epsilon F)$ はモナドとなる.

これにより, 関手の随伴が与えられれば, それに伴うモナドが構成されることになる.