可換図式をLaTeXで書くときに使うxymatrixのテンプレート (3): 終対象や積などの特別な対象を書く - 疑念は探究の動機であり、探究の唯一の目的は信念の確定である。

今回の記事はLaTeXのxymatrixで可換図式を書くことについての一連のシリーズの最終回である。

前回の記事はこちら

第一回目の記事はこちら

今回は圏論に出てくる特別な概念---終対象と始対象、積と双対積、イクワライザーとコイクワライザー、そしてプルバックとプッシュアウト---の図式を書く。勝手に参照してくれて構わんよ~~~。

追記 2018/12/01

可換図式のまとめを見やすく書いた。こちらを参照。

終対象と始対象
積と双対積
イクワライザーとコイクワライザー
プルバックとプッシュアウト
終わりに
参考文献

終対象と始対象

最初は一番易しい概念である終対象と始対象のテンプレートを書く。

(40): 終対象(Terminal Objects) と始対象(Initial Objects)

INPUT

An object $T$ in a category $\textbf{C}$ is called a \textit{terminal object} if for any object $A$ in $\textbf{C}$ there is a unique arrow from $A$ to $T$. Dually an object $I$ in $\textbf{C}$ is called an \textit{initial object} if for any object $A$ in $\textbf{C}$ there is a unique arrow from $I$ to $A$.

¥begin{align}
&\xymatrix{
A\ar@{.>}[r]^-{t_A}&T
}
&
\xymatrix{
I\ar@{.>}[r]^-{i_A}&A
}
¥end{align}

OUTPUT

f:id:yoheiwatanabe0606:20180617081953p:plain

積と双対積

次に積と双対積(和)を書いてみよう。

(41): 積(Products)

INPUT

Let $A$ and $B$ be objects in a category $\textbf{C}$. An object $P$ in a category $\textbf{C}$ is called a \textit{product} of $A$ and $B$ if there are arrows $A\overset{p_A}{\leftarrow}P\overset{p_B}{\to}B$ such that for any pair of arrows $A\overset{f}{\leftarrow}X\overset{g}{\to}B$ there is a unique arrow $\langle f, g \rangle: X\dasharrow P$ making the following diagram commutative:
¥begin{equation}
\xymatrix{
&X\ar[dl]_-{f}\ar@{.>}[d]|{^{\exists !}\langle f, g \rangle}\ar[dr]^-{g}\ar@{}@<2.5ex>[ld]|{\circlearrowright}\ar@{}@<-2.5ex>[rd]|{\circlearrowleft}&\\
A&P\ar[l]^-{p_A}\ar[r]_-{p_B}&B}
¥end{equation}
That is, $p_A\circ \langle f, g \rangle = f$ and $p_B\circ \langle f, g \rangle = g$.

OUTPUT

f:id:yoheiwatanabe0606:20180617081143p:plain

(42) 双対積(Coproducts)

INPUT

Dually an object $Q$ in $\textbf{C}$ is called a \textit{coproduct} of $A$ and $B$ if there are arrows $A\overset{q_A}{\to}Q\overset{q_B}{\leftarrow}B$ such that for any pair of arrows $A\overset{f}{\to}X\overset{g}{\leftarrow}B$ there is a unique arrow $\langle f, g \rangle: Q\dasharrow X$ making the following diagram commutative:
¥begin{equation}
\xymatrix{
&X\ar@{}@<2.5ex>[ld]|{\circlearrowleft}\ar@{}@<-2.5ex>[rd]|{\circlearrowright}&\\
A\ar[ru]^-{f}\ar[r]_-{q_A}&Q\ar@{.>}[u]|{^{\exists !}\langle f, g \rangle}&B\ar[lu]_-{g}\ar[l]^-{q_B}
}
¥end{equation}
That is, $\langle f, g \rangle \circ q_A = f$ and $\langle f, g \rangle \circ q_B = g$.

OUTPUT

f:id:yoheiwatanabe0606:20180617081655p:plain

イクワライザーとコイクワライザー

次はイクワライザーとコイクワライザーを書く。

(43) イクワライザー(Equalizers)

INPUT

Let $A\overset{f}{\underset{g}{\rightrightarrows}}B$ be a pair of arrows in a category $\textbf{C}$. An \textit{equalizer} of $f$ and $g$ is a pair $(E, e)$ where $E$ is an object of $\textbf{C}$ and $e: E\to A$ is an arrow in $\textbf{C}$ with the following properties:
¥begin{enumerate}
\item
$f\circ e = g\circ e$
\item For any arrow $h: X\to A$ with $f\circ h = g\circ h$ in $\textbf{C}$, there is a unique arrow $\bar{h}: X\dasharrow E$ such that $e\circ \bar{h} = h$; i.e., the diagram below commutes:
¥begin{equation}
\xymatrix{
E\ar[r]^-{e}&A\ar@<0.6ex>[r]^-{f}\ar@<-0.6ex>[r]_-{g}&B\\
X\ar@{.>}[u]^-{\bar{h}}\ar[ur]_-{h}\ar@{}@<2.5ex>[ur]|{\circlearrowright}
}
¥end{equation}
¥end{enumerate}

OUTPUT f:id:yoheiwatanabe0606:20180617083228p:plain

(44) コイクワライザー(Coequalizers)

INPUT

Dually A \textit{coequalizer} of $f$ and $g$ is a pair $(C, c)$ where $C$ is an object in $\textbf{C}$ and $c: B\to C$ is an arrow in $\textbf{C}$ with the following properties:
¥begin{enumerate}
\item $c\circ f = c\circ g$
\item For any arrow $h: B\to X$ with $h\circ f = h\circ g$ in $\textbf{C}$, there is a unique arrow $\bar{h}: C\dasharrow X$ in $\textbf{C}$ such that $\bar{h}\circ c = h$; i.e., the diagram below commutes:
¥begin{equation}
\xymatrix{
A\ar@<0.6ex>[r]^-{f}\ar@<-0.6ex>[r]_-{g}&B\ar[r]^-c\ar[rd]_-{h}\ar@{}@<2.5ex>[rd]|{\circlearrowright}&C\ar@{.>}[d]^-{\bar{h}}\\
&&X
}
¥end{equation}
¥end{enumerate}

OUTPUT

f:id:yoheiwatanabe0606:20180617083353p:plain

プルバックとプッシュアウト

最後にプルバックとプッシュアウトを書く。

(45) プルバック(Pullbacks)

INPUT

Let $A\overset{f}{\to}C\overset{g}{\leftarrow}B$ be arrows in a category $\textbf{C}$. A \textit{pullback} of $f$ and $g$ is an object $P$ in $\textbf{C}$ together with arrows $A\overset{p_A}{\leftarrow}P\overset{p_B}{\to}B$ in $\textbf{C}$ satisfying the following properties:
¥begin{enumerate}
\item $f\circ p_A = g\circ p_B$
\item For any pair of arrows $h: X\to A$ and $k: X\to B$ with $f\circ h = g\circ k$, there is a unique arrow $l: X\dasharrow P$ such that $p_A\circ l = h$ and $p_B\circ l = k$; namely,
¥begin{equation}
\xymatrix{
X\ar@/^10pt/[rrd]^-{k}\ar@/_10pt/[rdd]_-{h}\ar@{.>}[rd]_-{^{\exists !}l}\ar@{}@<-0.5ex>[rrd]|{\circlearrowright}\ar@{}@<-0.5ex>[rdd]|{\circlearrowright}&&\\
&P\ar[r]^-{p_B}\ar[d]_-{p_A}\ar@{}[rd]|{\circlearrowright}&B\ar[d]^-{g}\\
&A\ar[r]_-{f}&C
}
¥end{equation}
¥end{enumerate}
We denote a pullback $P$ of $f$ and $g$ by
¥begin{equation}
\xymatrix{
P\ar[r]^-{p_B}\ar@{}[rd]|{\text{PB}}\ar[d]_-{p_A}&B\ar[d]^-{g}\\
A\ar[r]_-{f}&C
}
¥end{equation}

OUTPUT

f:id:yoheiwatanabe0606:20180617083639p:plain

(46) プッシュアウト(Pushouts)

INPUT

Dually a \textit{pushout} of $f$ and $g$, where $A\overset{f}{\leftarrow}C\overset{g}{\to}B$ are arrows in $\textbf{C}$, is an object $P$ in $\textbf{C}$ together with arrows $A\overset{p_A}{\to}P\overset{p_B}{\leftarrow}B$ in $\textbf{C}$ satisfying the following properties:
¥begin{enumerate}
\item $p_A\circ f = p_B\circ g$
\item For any pair of arrows $h: A\to X$ and $k: B\to X$ with $h\circ f = k\circ g$, there is a unique arrow $l: P\dasharrow X$ such that $l\circ p_A = h$ and $l\circ p_B = k$. That is, the following diagram commutes:
¥begin{equation}
\xymatrix{
C\ar[r]^-{g}\ar[d]_-{f}&B\ar[d]^-{p_B}\ar@{}[ld]|{\circlearrowright}\ar@/^10pt/[rdd]^-{k}\ar@{}@<0.5ex>[rdd]|{\circlearrowright}&\\
A\ar[r]_-{p_A}\ar@/_10pt/[rrd]_{h}\ar@{}@<-0.5ex>[rrd]|{\circlearrowright}&P\ar@{.>}[rd]^-{^{\exists !}l}&\\
&&X
}
¥end{equation}
¥end{enumerate}
We denote a pushout $P$ of $f$ and $g$ by
¥begin{equation}
\xymatrix{
C\ar[r]^-{g}\ar[d]_-{f}\ar@{}[rd]|{\text{PO}}&B\ar[d]^-{p_B}\\
A\ar[r]_-{p_A}&P
}
¥end{equation}