basicfunctorclassification.tex

%% A tentative, eye-catchy, one-page definition of category (I would like to replicate the page from 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}[Faithful Functor]
	\parbox{0.35\textwidth}{
		$F_{A,\; B} $ is injective. \ie : 
		$$\forall \bulk  \textcolor{blue}{s.t. \bulk} \textcolor{red}{\Rightarrow \bulk}$$
	}
	\begin{tikzcd}
		A \arrow[d, bend right=60, "f "'] \arrow[d, bend left=60, "g "] \arrow[d, phantom,"\commute",,red] & & 
		F(A) \arrow[d, bend right=60, "F(f) "'] \arrow[d, bend left=60, "F(g) "] \arrow[d, phantom,"\commute",,blue] \\
		B & & F(B)
	\end{tikzcd}
\end{definition}
%
\begin{definition}[Full Functor]
	\parbox{0.35\textwidth}{
		$F_{A,\; B} $ is surjective. \ie :
		$$\forall \bulk  \textcolor{blue}{\exists \bulk} \textcolor{red}{\St \bulk}$$
	}
	\begin{tikzcd}
		A \arrow[d, "f ",blue]  & &  
		F(A) \arrow[d, bend right=60, "F(f) "', blue] \arrow[d, bend left=60, "\eta "] \arrow[d, phantom,"\commute",,red] \\
		B & & F(B)
	\end{tikzcd}
\end{definition}
%
\begin{definition}[Costant Functor]
	$ \bigtriangleup_u : \cat \rightarrow \cat[D] $ s.t.
	\begin{tikzcd}
		A \arrow[d, "f "]  & &  
		\bigtriangleup_u(A) = u \arrow[d, "\bigtriangleup_u(f) = \id_u"]  \\
		B & & \bigtriangleup_u(A) = u
	\end{tikzcd}
\end{definition}
%


\end{document}