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limitdef.tex
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%% 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}
\def\trilim{
\tikz[baseline=.1ex]{
\fill (0,1.75ex) coordinate (A);
\fill (2.5ex,1.75ex) coordinate (B);
\fill (1.25ex,-0.5ex) coordinate (C);
\draw[green] (1.25ex,1ex) node {$\commute$};
\draw[black] (A) -- (B);
\draw[green] (B) -- (C);
\draw[red] (C) -- (A);
}
}
\def\tricolim{
\tikz[baseline=.1ex]{
\fill (0,-0.5ex) coordinate (A);
\fill (2.5ex,-0.5ex) coordinate (B);
\fill (1.25ex,1.75ex) coordinate (C);
\draw[green] (1.25ex,0.25ex) node {$\commute$};
\draw[black] (A) -- (B);
\draw[green] (B) -- (C);
\draw[red] (C) -- (A);
}
}
\def\TriLimit{
\tikz[baseline=.1ex]{
\fill (0,-0.5ex) coordinate (A);
\fill (0,2ex) coordinate (B);
\fill (-2.25ex,0.75ex) coordinate (C);
\draw[orange] (-0.75ex,0.75ex) node {$\commute$};
\draw[orange] (A) -- (B);
\draw[red] (B) -- (C) ;
\draw[blue] (C) -- (A) ;
}
}
\begin{document}
% For every picture that defines or uses external nodes, you'll have to
% apply the 'remember picture' style. To avoid some typing, we'll apply
% the style to all pictures.
\tikzstyle{every picture}+=[remember picture]
% By default all math in TikZ nodes are set in inline mode. Change this to
% displaystyle so that we don't get small fractions.
\everymath{\displaystyle}
Let's begin from the dry abstract nonsense:
\begin{definition}[Diagram on $\cat$]
Functor $D: \cat[I] \rightarrow \cat$ from a small category $\cat[I]$ to $\cat[C]$.
\end{definition}
Basically it is a graph composed of objects and arrows in $\cat$. We call \emph{shape of $D$} the corresponding diagram.
% first column
\begin{minipage}[t]{0.5\textwidth}
\begin{definition}[Cone over diagram $D$]
\begin{displaymath}
(
\tikz[baseline]{
\node[fill=blue!20,anchor=base] (t1)
{$ V$};
}
,
\tikz[baseline]{
\node[fill=blue!20,anchor=base] (t2)
{$ \{\nu_i:V \rightarrow D_i\}_{i\in \cat[I]} $};
}
)
\end{displaymath}
\tikzstyle{na} = [baseline=-.5ex]
\begin{itemize}
\item[] \tikz[na] \node[coordinate] (n1) {};
Object $\in \cat$
\item[] Family of morphisms of $\cat$
\tikz[na]\node [coordinate] (n2) {};
\end{itemize}
% Now it's time to draw some edges between the global nodes. Note that we have to apply the 'overlay' style.
\begin{tikzpicture}[overlay]
\path[->] (n1) edge [bend left] (t1);
\path[->] (n2) edge [bend right] (t2);
%\path[->] (n3) edge [out=0, in=-90] (t3);
\end{tikzpicture}
\St
\parbox{0.4\textwidth}{
$$\forall \left( i \xrightarrow[]{f} j \right) \in \cat[I]$$
}
\begin{tikzcd}[sep=small]
& \textcolor{red}{V} \ar[dl,red,"\nu_i"'] \ar[d,phantom,blue,"\commute"] \ar[dr,red,"\nu_j"]& \\
D_i \ar[rr,"D(f)"']& \phantom. & D_j
\end{tikzcd}
\end{definition}
\begin{lemma}
Any cone is a n.t. $ \nu : \Delta_V \xrightarrow{\cdot} \cat[D]$
\begin{tikzcd}
i \ar[d,"f"] & & \textcolor{red}{V} \ar[d,equal,red] \ar[r,red,"\nu_i"] \ar[dr,phantom,blue,"\commute"]& D_i \ar[d,"D(f)"]\\
j & & \textcolor{red}{V} \ar[r,red,"\nu_j"'] & D_j
\end{tikzcd}
\end{lemma}
\end{minipage}
\vrule{}
\begin{minipage}[t]{0.5\textwidth}
\begin{definition}[CoCone over diagram $D$]
\begin{displaymath}
(
\tikz[baseline]{
\node[fill=blue!20,anchor=base] (t1)
{$ W$};
}
,
\tikz[baseline]{
\node[fill=blue!20,anchor=base] (t2)
{$ \{\omega_i:D_i \rightarrow W\}_{i\in \cat[I]} $};
}
)
\end{displaymath}
\tikzstyle{na} = [baseline=-.5ex]
\begin{itemize}
\item[] \tikz[na] \node[coordinate] (n1) {};
Object $\in \cat$
\item[] Family of morphisms of $\cat$
\tikz[na]\node [coordinate] (n2) {};
\end{itemize}
\begin{tikzpicture}[overlay]
\path[->] (n1) edge [bend left] (t1);
\path[->] (n2) edge [bend right] (t2);
%\path[->] (n3) edge [out=0, in=-90] (t3);
\end{tikzpicture}
\St
\parbox{0.4\textwidth}{
$$\forall \left( i \xrightarrow[]{f} j \right) \in \cat[I]$$
}
\begin{tikzcd}[sep=small]
D_i \ar[rr,"D(f)"] \ar[dr,red,"\omega_i"']& \phantom. & D_j \ar[dl,red,"\omega_j"]\\
& \textcolor{red}{W} \ar[u,phantom,blue,"\commute"] &
\end{tikzcd}
\end{definition}
\begin{lemma}
Any co-cone is a n.t. $ \omega : \cat[D] \xrightarrow{\cdot} \Delta_W$
\begin{tikzcd}
i \ar[d,"f"] & & D_i \ar[d,"D(f)"'] \ar[r,red,"\omega_i"] \ar[dr,phantom,blue,"\commute"]& \textcolor{red}{W}\ar[d,equal,red] \\
j & & D_j \ar[r,red,"\omega_j"'] & \textcolor{red}{W}
\end{tikzcd}
\end{lemma}
\end{minipage}
\begin{center}
\begin{tikzcd}[row sep= 5pt]%small]
e.g.:& & \textcolor{green}{V} \ar[ddl, green, bend right]\ar[dd, green] \ar[dr, green, bend left] \ar[dddr, green,bend left] & &
\textcolor{green}{\textrm{\small cone}} \\
& & & D_3 & \\
& D_1 \ar[r] & D_2 \ar[ru] \ar[rd]& \phantom. & \textrm{\small diagram}\\
& & & D_4 \\
& & \textcolor{red}{W} \ar[uul, red, bend left,leftarrow]\ar[uu, red,leftarrow] \ar[ur, red, bend right,leftarrow] \ar[uuur, red, bend right,leftarrow] & &
\textcolor{red}{\textrm{\small co-cone}}
\end{tikzcd}
\end{center}
% first column
\begin{minipage}[t]{0.5\textwidth}
\begin{definition}[Limit of diagram $D$]
\emph{Universal cone over $D$}
$$\textcolor{red}{cone \bulk} s.t. \textcolor{blue}{\forall cone \bulk} \quad \textcolor{orange}{\exists! \bulk} \St \TriLimit$$
\begin{tikzcd}[row sep= small]
i \ar[dd,"f"] & & D_i \ar[dd,"D(f)"']& \\
& & \phantom. \ar[r,phantom,red,"\commute"] \ar[rdd,phantom,blue,"\commute"]& \textcolor{red}{V} \ar[lu,red] \ar[ld,red] \\
j & & D_j & \phantom. \ar[l,phantom,near start, orange,"\commute"]\\
& & & \textcolor{blue}{V'} \ar[uuul,blue] \ar[ul,blue] \ar[uu,orange,"\exists!"']
\end{tikzcd}
\end{definition}
\end{minipage}
\vrule{}
\begin{minipage}[t]{0.5\textwidth}
\begin{definition}[Co-Limit of diagram $D$]
\emph{Universal co-cone over $D$}
$$\textcolor{red}{cocone \bulk} s.t. \textcolor{blue}{\forall cone \bulk} \quad \textcolor{orange}{\exists! \bulk} \St \TriLimit$$
\begin{tikzcd}[row sep= small]
i \ar[dd,"f"] & & D_i \ar[rd,red] \ar[rddd,blue] \ar[dd,"D(f)"']& \\
& & \phantom. \ar[r,phantom,red,"\commute"] \ar[rdd,phantom,blue,"\commute"] & \textcolor{red}{W} \ar[dd,orange,"\exists!"] \\
j & & D_j \ar[ru,red] \ar[rd,blue]& \phantom. \ar[l,phantom,near start, orange,"\commute"]\\
& & & \textcolor{blue}{W'}
\end{tikzcd}
\end{definition}
\end{minipage}
\end{document}