CatCheatSheet.tex

%% A tentative, eye-catchy, one-page definition of category (I would like to replicate the page frome my handwritten notes.

\documentclass[a4paper,12pt,fleqn]{scrartcl}  %Per La Stampa
\usepackage[a4paper, margin=0.75cm]{geometry}

\usepackage{amsmath}

\usepackage[subpreambles=true]{standalone}
\usepackage{commath}

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


\usepackage{graphicx}
\usepackage{hyperref}

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


\usepackage[toc,page]{appendix}

\usepackage{epigraph} %funny quotes


%%__________________________________
%% To transform in a package.
%%__________________________________
%%
	\newcommand{\catdef}[3]{
		\begin{displaymath}
		#1 \leadsto
		\begin{cases}
			Obj = #2\\
			Arr = #3
		\end{cases}
		\end{displaymath}
	}
	\newcommand{\smalltikzcd}[2]{
		\begin{tikzpicture}[baseline= (a).base]
			\node[scale=#1] (a) at (0,0){
				\begin{tikzcd} #2 \end{tikzcd}
			};
		\end{tikzpicture}
	}


	\newcommand{\id}{\textrm{id}}

%Diagrammatic
\usepackage{tikz-cd}
\usepackage{xcolor}
\usepackage{fdsymbol}
	% bulk of arrows and objects
	\newcommand{\bulk}{\blacklozenge}
	% such that
	\newcommand{\St}{\textrm{s.t.}}
	% id est
	\newcommand{\ie}{\textrm{\emph{i.e.}}}
	% commuting
	\newcommand{\commute}{\circledequal}%\circlearrowleft



\begin{document} 
\begin{abstract}
	The preliminary material for my talk on the category of diffeological spaces.
\end{abstract}


\begin{appendices}
\epigraph{Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams and arrows.}{Saunders Mac Lane}
\epigraph{Poetry says more with less words.\\
Mathematics too.}{Gabriele Lolli}

%------------------------------------------------------------------------------------BASIC ---------------------------------
\section{Basic definitions}
	\input{catdef}
We can not delve here on the foundational issues related to this definitions. Let us simple look at it a triple composed of \emph{data}, \emph{structure}, \emph{axiom}.
\\
One of these problem is the problem of "size". We use the term "collections" when we want to remain vague about the axiomatic framework of sets considered.
We use "mapping" when we want to consider correspondences between collections.
\\
Words like "set" and "function" are reserved to objects and morhpisms in the set category.


\paragraph{How to compare cat?}
	\input{functordef}

	\subparagraph{Basic Classification:}
	\input{basicfunctorclassification}

\subparagraph{How to compare Functors?}
	\input{naturaltransfdef}

%------------------------------------------------------------------------------------Limits ---------------------------------
\newpage
\section{Limits}
	\input{limitdef}
	The best way to grasp the meaning (...) to understand what is a limit it is through examples. The model category is $\cat[Set]$ which turn out to be complete and co-complete.
	\input{limitbestiary}

\newpage
\section{Classification of Morphisms}
	In any $\cat$ there is a basic classification of morphisms $f: C \rightarrow D$.\\
	

	\input{basicmorphismclassification}

\newpage
\section{Subobjects}

\newpage
\section{Cartesian Closedness}



\section{Eternal TODO}
	\input{slicecatdef}
	
	



\end{appendices}
			\nocite{*}
			\bibliographystyle{plain}
			\bibliography{biblio}

\end{document}