Commit 742146ee authored by Michael Kohlhase's avatar Michael Kohlhase

tweaks

parent 4c18aad5
\begin{mhmodnl}[creators=miko]{cartesian-closed-category}{en}
\begin{definition}
A \trefi[category]{category} $\cC$ is called \defii[name=CCC]{Cartesian}{closed} (a
\defi{CCC}) , iff it satisfies the following three properties:
\defi{CCC}), iff it satisfies the following three properties:
\begin{itemize}
\item $\cC$ has a \trefi[initerminal]{terminal} \trefi[category]{object}.
\item Any two \trefis[category]{object} $X$ and $Y$ of $\cC$ have a
......
......@@ -5,7 +5,7 @@
\mtrefi[category?arrow]{morphisms} $\catfun{\catprodprojectionFN1}X{X_1}$ and $\catfun{\catprodprojectionFN2}X{X_2}$
the \defi{product} of $X_1$ and $X_2$ and write it as $\catobjprod{X_1,X_2}$ if it
satisfies the following universal property:\\
\begin{minipage}{7cm}
\begin{minipage}{7.3cm}
For every object $Y$ and \trefi[pair]{pair} of \mtrefi[category?arrow]{morphisms}
$\catfun{f_1}Y{X_1}$ and $\catfun{f_2}Y{X_2}$ there exists a unique
\mtrefi[category?arrow]{morphism} $\catfun{f}Y{\catobjprod{X_1,X_2}}$ such that
......@@ -13,7 +13,7 @@
\end{minipage}
\begin{minipage}{4cm}
\mhtikzinput{tikz/product}
\end{minipage}
\end{minipage}\\
The unique morphism $f$ is called the \defiii{product}{of}{morphisms} $f_1$ and $f_2$
and is denoted $\catmorprod{f_1,f_2}$. The morphisms $\catprodprojectionFN1$ and
$\catprodprojectionFN2$ are called the
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......@@ -5,13 +5,13 @@
\item A \trefi[class]{class} $\catObj\cC$ of \defis{object}.
\item A \trefi[class]{class} $\catMorOp\cC$ of \defis{arrow} (also called
\defis[name=arrow]{morphism} or \defis[name=arrow]{map}).
\item For each \trefi{arrow} $f$, two \trefis{object} which are called \defi{domain} of
$f$; $\catdomain{f}$ and \defi{codomain}; $\catcodomain{f}$. We write
\item For each \trefi{arrow} $f$, two \trefis{object} which are called \defi{domain}
$\catdomain{f}$ and \defi{codomain} $\catcodomain{f}$ of $f$. We write
$\catfun{f}{\catdomain{f}}{\catcodomain{f}}$ and call two \trefis{arrow} $f$ and $g$
\defi{composable}, iff $\catdomain{f}=\catcodomain{g}$.
\item An \trefi[semigroup]{associative} \trefi[magma]{operation} $\catcompOp$ called
\defi{composition} assigning to each \trefi[pair]{pair} $\pair{f}g$ of
\trefi{composable} \trefis{arrow} another \trefi{arrow}; $\catcomp{g,f}$ such that
\trefi{composable} \trefis{arrow} another \trefi{arrow} $\catcomp{g,f}$ such that
$\catdomain{\catcomp{g,f}} = \catdomain{f}$ and
$\catcodomain{\catcomp{g,f}} = \catcodomain{g}$, i.e.
$\catfun{\catcomp{g,f}}{\catdomain{f}}{\catcodomain{g}}$.
......
......@@ -7,7 +7,7 @@
\begin{tikzpicture}[xscale=1.8,yscale=1.2]
\node (x1) at (-1,0) {$X_1$};
\node (x12) at (0,0) {$\catobjprod{X_1,X_2}$};
\node (x2) at (1,0) {$X_1$};
\node (x2) at (1,0) {$X_2$};
\node (y) at (0,1) {$Y$};
\draw[->] (x12) -- node[below]{$\pi_1$}(x1);
\draw[->] (x12) -- node[below]{$\pi_2$}(x2);
......
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