functorclassification_1.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}[Iso-reflecting]
	\parbox{0.35\textwidth}{
		$$\forall \bulk  \textcolor{blue}{\Rightarrow \commute}$$
	}

	\begin{tikzcd}
		A \arrow[d, "f ", bend right=60]  & & F(A) \arrow[d, "F(f) "', bend right=60] \arrow[drr, phantom,"\commute",blue] & & \\
		B \arrow[u, "f^{-1}", bend right=60] & & F(B) \arrow[u, "F(f^{-1})"', bend right=60] \arrow[u, "F(f)^{-1}"', bend right=90, blue, looseness=5] & & \phantom{A}
	\end{tikzcd}
	\\ or \\
	\begin{tikzcd}
		A \arrow[d, "f "']   \arrow[dd, "id_A"', bend right=60]& & F(A) \arrow[d, "F(f) "'] \arrow[dd, "id_{F(A)} ", bend left=60]  & \\
		B \arrow[d, "f^{-1}"'] & & F(B)  \arrow[d, "F(f^{-1}) "'] \arrow[r, phantom,"\commute",blue, near start]&\phantom{A}\\
		A & & F(A) &
	\end{tikzcd}
	
\end{definition}

\begin{proposition}
	Given F a functor
	$$ \textrm{Fully Faithfull} \Leftrightarrow \textrm{Iso-reflecting}$$
\end{proposition}


\end{document}