Bsrgvty

Background. I recently came across a Youtube video with a bunch of really nicely typeset 'diagram chases' that seem to have been made in TeX (perhaps TikZ?):

I'd like to create something similar for a presentation that I'm working on, but I feel a bit lost when it comes to figuring out a systematic/scalable way to this.

Own attempt. For small diagrams, it's possible to do this in a very ad hoc way, by just playing around with coordinates and the bend right and bend left attributes in TikZ. A quick example of what this might look like:

documentclassarticle

usepackagetikz
usetikzlibraryarrows.meta

begindocument

begintikzpicture[scale=1.5, bend left=15, bend right=15]

 node (B) at (0,0) $B$;
 node (B') at (0,-1) $B'$;
 node (C) at (1,0) $C$;
 node (C') at (1,-1) $C'$;

 draw[->,gray] (B) -- (B');
 draw[->,gray,shorten >= -1pt] (B') -- (C');
 draw[->,gray,shorten >= -1pt] (B) -- (C);
 draw[->,gray] (C) -- (C');

 node (b) at (-0.3,0.2) footnotesize $b$;
 node (bprime) at (-0.3,-1.2) footnotesize $b'$;
 node (cbar) at (0.7,0.2) footnotesize $barc$;
 node (cdiff) at (0.7,-1.3) footnotesize $c'-c''$;

 path (b) edge [->[scale=0.7],bend left] node [left] (cbar);
 path (b) edge [->[scale=0.7],bend right] node [left] (bprime);
 path (bprime) edge [->[scale=0.7],bend right] node [left] (cdiff);
 path (cbar) edge [->[scale=0.7],bend right, shorten >= -2pt, shorten <= -1pt] node [left] (cdiff);

endtikzpicture

enddocument

I think this gives a decent end result. But for larger diagrams, it would be an absolute nightmare to work with a code that is so messy and so heavily dependent on coordinates and tweaked parameters.

Thus, if anyone has suggestions (big or small) for a more clever way to do this (with or without TikZ) and/or a way to make my own approach a bit cleaner or more systematic somehow, that would be greatly appreciated!

We clearly can't avoid difficulties and complexity when dealing with such a complicated graph when using any kinds of tool, but by using matrices it has saved a lot of work.

documentclass[tikz]standalone
usetikzlibrarymatrix,positioning,arrows.meta
usepackagemathptmx
tikzsettoarrow/.style=->[scale=0.7],
 backarrow/.style=<[scale=0.7]-
begindocument
begintikzpicture
matrix[matrix of math nodes,row sep=2cm,column sep=2cm] (m) %
A & B & C & D & E\
A' & B' & C' & D' & E'\;
path (m-1-2) node[above left=1.5ex and 1.5ex] (b) $b$
 (m-1-3) node[above left=1.5ex and 1.5ex] (c) $c$
 (m-1-4) node[above left=1.5ex and 1.5ex] (d) $d$
 (m-1-5) node[above left=1.5ex and 1.5ex] (e) $e$ 
 (m-2-2) node[below left=1.5ex and 1.5ex] (b2) $b'$
 (m-2-3) node[below left=1.5ex and 1.5ex] (c2) $c'$
 (m-2-4) node[below left=1.5ex and 1.5ex] (d2) $d'$
 (m-2-5) node[below left=1.5ex and 1.5ex] (e2) $0$
 (m-2-3) node[above left=1.5ex and 1.5ex] (c3) $c''$
 (c) node[above left=1.5ex and -1.5ex] (cp) $overlinec+c$
 (c2) node[below=1.5ex] (cm) $c'-c''$
 (d2) node[below=1.5ex] (db) $0$
 (e.base east) node[above right=-.3333em and -1ex] $=0$
 (c.base west) node[above left=-.3333em and 1.5ex] (co) $overlinec$;
% Delete the following part to see what happens
foreach i [count=j from 2] in 1,2,3,4 
 draw[dotted,->] (m-1-i) -- (m-2-i);
 draw[dotted,->] (m-1-i) -- (m-1-j);
 draw[dotted,->] (m-2-i) -- (m-2-j);

draw[dotted,->] (m-1-5) -- (m-2-5);
draw[toarrow] (b) edge[bend right] (b2) edge[bend left] (co);
draw[toarrow] (b2) to[bend right] (cm);
draw[toarrow] (cm) to[bend right] (db);
draw[toarrow] (co) to[bend right] (cm);
draw[toarrow] (cp) to[bend right] (c2);
draw[toarrow] (c) edge[bend right] (c3) edge[bend left] (d);
draw[backarrow] (d2) edge[bend left] (c2) edge[bend right=20] (c3) edge[bend left] (d);
draw[backarrow] (e2) edge[bend left] (d2) edge[bend left] (e);
draw[toarrow] (d) edge[bend left] (e);
endtikzpicture
enddocument