# latex-pictures

A collection of TikZ drawings and other images 🖌️

## projective-system-universal-property.tikz (1268B)

 1 % This diagram represents the universal property of a projective limit
2 % \varphojlim X_i, i.e. the fact the a projective limit is the limit of the
3 % functor between I^op and the category of obejcts of the projective system
4 % given my the projective system itself (for each index i it yields X_i and for
5 % each inequality i <= j it yeilds \varphi_{i, j}
6 % Copyright Pablo (C) 2021
7 \begin{tikzpicture}[ampersand replacement=\&]
8   % The objects
9   \matrix(m)[matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em]
10   {     \&               X \&     \\
11         \& \varprojlim X_i \&     \\
12     X_i \&                 \& X_j \\};
13
14   % The morphism
15   \draw[->] (m-2-2) -- node[above right]{$\pi_j$} (m-3-3);
16   \draw[->] (m-2-2) -- node[above left]{$\pi_i$} (m-3-1);
17
18   % The projections
19   \draw[dotted, ->] (m-1-2) -- node[right]{$\theta$} (m-2-2);
20
21   % The arrow from the projective system
22   \draw[->] (m-3-3) -- node[below]{$\phi_{i j}$} (m-3-1);
23
24   % The arrows from the compatible family of morphisms
25   \draw[->] (m-1-2)
26             to[relative, out=-30, in=-150]
27             node[left]{$\theta_i$}
28             (m-3-1);
29   \draw[->] (m-1-2)
30             to[relative, out=30, in=150]
31             node[right]{$\theta_j$}
32             (m-3-3);
33 \end{tikzpicture}