catdef.tex

%% A tentative, eye-catchy, one-page definition of category (I would like to replicate the page frome my handwritten notes.
\documentclass[preview]{standalone}

\usepackage{amsmath}
\usepackage{verbatim}
\usepackage[italian,english]{babel}
\usepackage[utf8]{inputenc}

\usepackage[basic,cat]{./Math-Symbols-List/toninus-math-symbols}
\usepackage{./Latex-Theorem/theoremtemplate}
\usepackage{./visualcat}


\begin{document} 
\begin{definition}[Category ]
Category is a tuple $($data,structure,axioms$)$.
\begin{itemize}
\item \emph{data}
	\begin{displaymath}
		\cat \leadsto
		\begin{cases}
			\Obj(\cat) : 
			    \text{\parbox{0.65\textwidth}{ collection of items called \emph{objects of $\cat$}}}
			\\
			\biggr\lbrace \hom_{\cat} (A,B) \biggr\rbrace_{A,B \in \Obj(\cat)} : 
				\text{\parbox{0.45\textwidth}{parametrized family of collections of items  called \emph{morphisms} or \emph{arrows}}}
		\end{cases}
	\end{displaymath}

\item Structure:
	(Composition of arrows)
	\begin{displaymath}
		\circ : \hom_{\cat}(A,B) \times \hom_{\cat}(B,C) \rightarrow \hom_{\cat} (A,C)
	\end{displaymath}
	Diagrammatically \footnote{This is more properly a diagrammatic definition of  $\commute$}: 
	$\forall \bulk \textcolor{red}{\exists \bulk} \textcolor{blue}{\St \commute}$\\
	\begin{tikzcd}
		A \arrow[d, "f"] \arrow[dd, bend left=60,red, "gf\textcolor{blue}{ = f \circ g}"] 	& \\
		B \arrow[d, "g"] \arrow[r, phantom,"\commute",, very near start,blue] 	& \phantom.\\
		C 																& 
	\end{tikzcd}
\item Axioms:
	\begin{itemize}
		\item \emph{Identity axiom}:
			$\forall \bulk \textcolor{red}{\exists \bulk} \textcolor{blue}{\St \commute}$\\
			\begin{tikzcd}
				A \arrow[rr, "\id_A",red] \arrow[rrdd, "f", bend right]& & A \arrow[dd, "f"] \arrow[rrdd, "f", bend left] & & \\
				 & |[blue]|\commute & & |[blue]|\commute & \\
				 & &  B \arrow[rr, "\id_B",red] & & B
			\end{tikzcd}
		\item \emph{associativity axiom}
			$\forall \bulk \textcolor{green}{h (gf) = (hg) f}$			\\
			\begin{tikzcd}
				A \arrow[r,"f"] \arrow[rr,"gf",bend left=40, blue] \arrow[rrr,"h(gf)",bend left=80, green] \arrow[rrr,"(hg)f",bend right=80, green] &
				B \arrow[r,"g"] \arrow[rr,"hg",bend right=40, blue] &
				C \arrow[r,"h"] &				
				D  
			\end{tikzcd}
	\end{itemize}
	
\end{itemize}
\end{definition}
\begin{notation}
	$\Arr(\cat) = \bigcup\limits_{A,B \in \Obj(\cat)}\hom_{\cat} (A,B) $
\end{notation}

\end{document}