How to draw commutative diagrams in Tikz

Question

One of my friends can draw simple commutative diagrams in TikZ but he cannot draw this one:

I am not an expert in TikZ so I was wondering if someone could help with drawing this one in TikZ.

Answer 1

A solution with names like with a matrix but without matrix. It's more easy to scale the result. I use the scope library to draw the edges.

\documentclass[11pt]{scrartcl}
\usepackage[utf8]{inputenc} 
\usepackage{tikz}
\usetikzlibrary{arrows ,scopes}

\begin{document} 


\begin{tikzpicture}[xscale=1.5,yscale=3]

\path node   (m11) at (0,0) {$N_1$}         (-1,0) node[anchor=east] {$e_1$ \ :}       
      node   (m12) at (3,0) {$G_1$}         
      node   (m13) at (5,0) {$Q_1$}
      node   (m21) at (0,1) {$N_1\times A$} (-1,1) node[anchor=east] {$e_1\times A$ \ :}  
      node   (m22) at (3,1) {$G_1\times A$} 
      node   (m23) at (5,1) {$Q_1$} 
      node   (m31) at (0,2) {$\displaystyle\frac{ N_1\times A}{\alpha A}$} 
                      (-1,2)  node[anchor=east]{$nat'(e_1\times A)=
                                 \displaystyle\frac{ e_1\times A}{\alpha N_1}$ \ :}        
      node   (m32) at (3,2) {$\displaystyle\frac{ G_1\times A}{\beta\lambda N_1}$} 
      node   (m33) at (5,2) {$Q_1$} 
      node   (m41) at (0,3) {$N_1\times A$} (-1,3) node[anchor=east] {$e_1\times A$ \ :}           
      node   (m42) at (3,3) {$G_1\times A$} 
      node   (m43) at (5,3) {$Q_1$} 
      node   (m51) at (0,4) {$N_1\times N_2$} (-1,4) node[anchor=east] {$\tilde{e}$ \ :} 
      node   (m52) at (3,4) {$\tilde{G}$} 
      node   (m53) at (5,4) {$Q$} 
      node   (m61) at (0,5) {$N_2$} (-1,5) node[anchor=east] {$e_2$ \ :}           
      node   (m62) at (3,5) {$G_2$}         
      node   (m63) at (5,5) {$Q_2$} ;
     { [ thick]
      \draw[>->]   (m61) -- node[above]{$\kappa_2$}      (m62); 
      \draw[->>]   (m62) -- node[above]{$\pi_2$}    (m63);
      \draw[>->]   (m51) -- node[above]{$\lambda$}  (m52);  
      \draw[>->>]  (m52) -- node[above]{$\beta$}    (m53);    
      \draw[->>]   (m42) -- (m43);  
      \draw[>->]   (m31) -- (m32); 
      \draw[->>]   (m32) -- (m33);
      \draw[->>]   (m22) -- (m23);
      \draw[>->]   (m11) -- node[above]{$\kappa_1$}      (m12); 
      \draw[->>]   (m12) -- node[above]{$n_1$}      (m13);

      {[>->,shorten >=.5cm]
      \draw        (m41) -- (m42); 
      \draw        (m21) -- (m22); 
      }

      {[->>]       
      \draw   (m51) -- node[right]{$\sigma_2$} (m61); 
      \draw   (m52) -- node[right]{$\tau_2$}   (m62);
      \draw   (m21) -- node[right]{nat'}       (m31); 
      \draw   (m22) -- node[right]{nat}        (m32);  
      \draw   (m41) -- node[right]{nat'}       (m31); 
      \draw   (m42) -- node[right]{nat}        (m32);
      }
      \draw[->]    (m11) -- node[right]{$\mu'$}     (m21);  
      \draw[->]    (m12) -- node[right]{$\mu$}      (m22);  
      \draw[>->>]  (m53) -- node[right]{$\gamma_2$} (m63);  
      \draw[>->>]  (m53) -- node[right]{$\gamma_1$} (m43); 
      \draw[>->]   (m51) -- node[right]{$\alpha$}   (m41); 
      \draw[>->]   (m52) -- node[right]{$\beta$}    (m42);  
      {[dashed] 
       {[->]
       \draw   (m51) to [out=-140,in= 140]      (m31);
       \draw   (m52) to [out=-140,in= 140]      (m32);
       }
       {[>->]
       \draw  (m11) to [out= 140,in=-140]      (m31); 
       \draw  (m12) to [out= 140,in=-140]      (m32);} 
       }
       \draw[double]  (m43)--(m33)--(m23)--(m13);
       }
\end{tikzpicture}
\end{document}

Answer 2

For this diagram, I would actually give up on using relative placements, and just spec out explicit node placements for everything. This, combined with a number of standard tikz styling tips, including:

• [>->], [>->>] and [->>] for arrow styling
• edge node {..} and edge node[swap] {..} to place labels on arrows
• edge [bend right] and edge [bend left] to produce the curvy arrows
• [dashed] and [double equal sign distance] for line styling
• [anchor=east] to make sure the left hand row labels can grow freely without running in over the diagram

makes the diagram straightforward:

\begin{tikzpicture}[scale=2,auto]
  \node [anchor=east] at (-.5,3) {$e_2 : $};
  \node (n2) at (0,3) {$N_2$};
  \node (g2) at (1,3) {$G_2$};
  \node (q2) at (2,3) {$Q_2$};
  \draw [>->] (n2) edge node {$\kappa_2$}  (g2);
  \draw [->>] (g2) edge node {$\pi_2$} (q2);

  \node [anchor=east] at (-.5,2) {$\tilde e : $};
  \node (n1xn2) at (0,2) {$N_1\times N_2$};
  \node (gn) at (1,2) {$\tilde G$};
  \node (q) at (2,2) {$Q$};
  \draw [>->] (n1xn2) edge node {$\lambda$}  (gn);
  \draw [->>] (gn) edge node {$\rho$} (q);
  \draw [->>] (n1xn2) edge node[swap] {$\sigma_2$} (n2);
  \draw [->>] (gn) edge node[swap] {$\tau_2$} (g2);
  \draw [>->>] (q) edge node[swap] {$\gamma_2$} (q2);

  \node [anchor=east] at (-.5,1) {$e_1\times A : $};
  \node (n1xa) at (0,1) {$N_1\times A$};
  \node (g1xa) at (1,1) {$G_1\times A$};
  \node (q11) at (2,1) {$Q_1$};
  \draw [>->] (n1xa) edge  (g1xa);
  \draw [->>] (g1xa) edge (q11);
  \draw [>->] (n1xn2) edge node {$\sigma$} (n1xa);
  \draw [>->] (gn) edge node {$\beta$} (g1xa);
  \draw [>->>] (q) edge node {$\gamma_1$} (q11);

  \node [anchor=east] at (-.5,0) {$nat'(e_1\times A)=\frac{e_1\times A}{\alpha N_1} : $};
  \node (n1xaf) at (0,0) {$\frac{N_1\times A}{\alpha N_1}$};
  \node (g1xaf) at (1,0) {$\frac{G_1\times A}{\beta\lambda N_1}$};
  \node (q10) at (2,0) {$Q_1$};
  \draw [>->] (n1xaf) -- (g1xaf);
  \draw [->>] (g1xaf) -- (q10);
  \draw [->>] (n1xa) edge node {$nat'$} (n1xaf);
  \draw [->>] (g1xa) edge node {$nat$} (g1xaf);
  \draw [double equal sign distance] (q11) -- (q10);

  \node [anchor=east] at (-.5,-1) {$e_1\times A : $};
  \node (n1xad) at (0,-1) {$N_1\times A$};
  \node (g1xad) at (1,-1) {$G_1\times A$};
  \node (q1d) at (2,-1) {$Q_1$};
  \draw [>->] (n1xad) -- (g1xad);
  \draw [->>] (g1xad) -- (q1d);
  \draw [->>] (n1xad) edge node[swap] {$nat'$} (n1xaf);
  \draw [->>] (g1xad) edge node[swap] {$nat$} (g1xaf);
  \draw [double equal sign distance] (q1d) -- (q10);

  \node [anchor=east] at (-.5,-2) {$e_1 : $};
  \node (n1) at (0,-2) {$N_1$};
  \node (g1) at (1,-2) {$G_1$};
  \node (q1dd) at (2,-2) {$Q_1$};
  \draw [>->] (n1) edge node {$\kappa_1$} (g1);
  \draw [->>] (g1) edge node {$\pi_1$} (q1dd);
  \draw [->] (n1) edge node[swap] {$\mu'$} (n1xad);
  \draw [->] (g1) edge node[swap] {$\mu$} (g1xad);
  \draw [double equal sign distance] (q1dd) -- (q1d);

  \draw[>->,dashed] (n1) edge[bend left] (n1xaf);
  \draw[>->,dashed] (g1) edge[bend left] (g1xaf);
  \draw[>->,dashed] (n1xn2) edge[bend right] (n1xaf);
  \draw[>->,dashed] (gn) edge[bend right] (g1xaf);

\end{tikzpicture}

A bit tedious, but entirely straightforward. Gives the resulting diagram:

Answer 3

I would use a \matrix for the node positioning. Matrices automatically name their cells, but I would give them their own names as I find it hard to remember which entry is in which cell but easier if they have names. I chose the names fairly arbitrarily since there are a few cells with repeated contents.

I would also gather together the arrows into "types" and draw them with a single command. To get lots of arrows with a single command it is necessary to use the edge path construction (not necessary if only drawing a single arrow per command).

Other than that, this is fairly similar to Mikael's version.

\documentclass{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix}

\DeclareMathOperator{\nat}{nat}

\tikzset{
  label lines/.style={
    every node/.style={
      auto, % puts the label to one side of the line
% sometimes we need to "swap" the side with the 'swap' key
      maths labels
    }
  },
% This means we don't have to put the arrow labels in \(...\)
  maths labels/.style={
    execute at begin node=\(,
    execute at end node=\)
  }
}

\begin{document}
\begin{tikzpicture}
\matrix[
  matrix of math nodes, % to avoid having to put everything in \(..\)
  row sep={between origins, 2cm}, % space things out a bit vertically
  column sep={between origins,3cm}, % and horizontally
  nodes={anchor=center}, % makes the arrows straighter
  column 1/.style={every node/.append style={left}} % line up first column on colons
] (cd) {
|[name=e2]|  e_2 \colon &[-2cm] % squeeze the first column in a bit
|[name=n2]| N_2 &|[name=g2]| G_2 &|[name=q2]| Q_2 \\
%
|[name=e]|  \tilde{e} \colon &|[name=n1n2]| N_1 \times N_2 &|[name=g]| \tilde{G} &|[name=q]| Q \\
%
|[name=e1a]|  e_1 \times A \colon &|[name=n1a]| N_1 \times A &|[name=g1a]| G_1 \times A &|[name=q1]| Q_1 \\
%
|[name=ne1a]|  \displaystyle\nat'(e_1 \times A) = \frac{e_1 \times A}{\alpha N_1} \colon &|[name=nn1a]| \displaystyle\frac{N_1 \times A}{\alpha N_1} &|[name=ng1a]| \displaystyle\frac{G_1 \times A}{\beta \lambda N_1} &|[name=nq1]| Q_1 \\
%
|[name=e1a2]|  e_1 \times A \colon &|[name=n1a2]| N_1 \times A &|[name=g1a2]| G_1 \times A &|[name=q1a2]| Q_1 \\
%
|[name=e12]|  e_1 \colon &|[name=n12]| N_1 &|[name=g12]| G_1 &|[name=q12]| Q_1\\
};
\draw[>->,label lines]
(n2) edge node {\kappa_2} (g2)
(n1n2) edge node {\lambda} (g)
(n1a) edge (g1a)
(nn1a) edge (ng1a)
(n1a2) edge (g1a2)
(n12) edge node {\kappa_1} (g12)
(n1n2) edge node {\alpha} (n1a)
(g) edge node {\beta} (g1a)
;
\draw[->>,label lines]
(g2) edge node{\pi_2} (q2)
(g) edge node{\rho} (q)
(g1a) edge (q1)
(ng1a) edge (nq1)
(g1a2) edge (q1a2)
(g12) edge node {\pi_1} (q12)
(n1n2) edge node[swap] {\sigma_2} (n2) % the 'swap' puts the label on the other side of the line
(g) edge node[swap] {\tau_2} (g2)
(n1a2) edge node[swap] {\nat'} (nn1a)
(n1a) edge node {\nat'} (nn1a)
(g1a2) edge node[swap] {\nat} (ng1a)
(g1a) edge node {\nat} (ng1a)
;
\draw[>->>,label lines]
(q) edge node {\gamma_1} (q1)
(q) edge node[swap] {\gamma_2} (q2)
;
\draw[->, label lines,swap]
(n12) edge node {\mu'} (n1a2)
(g12) edge node {\mu} (g1a2)
;
\draw[double,double distance=.5ex] (q1) -- (nq1) -- (q1a2) -- (q12);
\draw[->,dashed,bend right]
(n1n2) edge (nn1a)
(g) edge (ng1a)
;
\draw[>->,dashed,bend left]
(n12) edge (nn1a)
(g12) edge (ng1a)
;
\end{tikzpicture}
\end{document}

Result: