Drawing a 3D commutative diagram

Question

I was drawing a commutative-diagram when I had the (questionable?) idea of drawing a tetrahedral version of it. I used Asymptote in the hope of getting good-looking thumbnail and a nice embedded 3D model at the same time. This is what I got so far:

After activating the diagram and rotating around freely, I realized that my 3D model doesn't really look pleasant:

I would like to make the following improvements:

1. Arrow start and end positions are automatically calculated like in tikz-cd such that I don't need to use Fill on the labels or use the convexcomb routine.
2. Allow arrows in the front to cross over the ones in the back, analogous to the tikz-cd option crossing over.
3. Arrows look good without activating the model.
4. Arrows look good in 3D with camera rotation.

I don't really know how to do 1 & 2. Moreover, 3 & 4 seem to be contradictory goals. What should my realistic goal be and what improvements should I make? Source:

settings.render = 0;
import solids;
import three;
size(240,120);
defaultpen(basealign);

real side = 1;
triple origin = (0,0,0);
triple voa = (0,0,sqrt(6)/3*side);
triple vob = (sqrt(3)/4*side,-.5*side,0);
triple voc = (sqrt(3)/4*side,.5*side,0);
triple vod = (-sqrt(3)/3*side,0,0);

triple convexcomb(triple a, triple b, real x)
{
  return (1-x) * a + x * b;
}

// A: U
// B: V1x...xVn
// C: W
// D: V

draw(convexcomb(vob,voa,.15)--convexcomb(vob,voa,.85),L=Label("$\varphi$",align=NW)         ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(vob,voc,.15)--convexcomb(vob,voc,.85),L=Label("$\psi$"   ,align=unit(4*S+W)),Arrow3(TeXHead2(normal=X)));
draw(convexcomb(vob,vod,.15)--convexcomb(vob,vod,.85),L=Label("$\phi$"   ,align=unit(4*N+E)),Arrow3(TeXHead2(normal=O)));
draw(convexcomb(voa,voc,.15)--convexcomb(voa,voc,.85),L=Label("$T_{0}$"  ,align=E)          ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(voa,vod,.15)--convexcomb(voa,vod,.85),L=Label("$\Pi$"    ,align=unit(E+NE)) ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(vod,voc,.15)--convexcomb(vod,voc,.85),L=Label("$T$"      ,align=unit(E+SE)) ,Arrow3(TeXHead2(normal=O)));

label(Label("$U$",Fill(white)),voa);
label(Label("$V_{1}\times\cdots\times V_{n}$",Fill(white)),vob);
label(Label("$W$",Fill(white)),voc);
label(Label("$V$",Fill(white)),vod);

Answer 1

Here is a proposal for a pseudo-3D tikz-cd diagram.

\documentclass[border=2pt]{standalone}
\usepackage{amssymb,amsmath}
\usepackage{tikz-cd}
\usetikzlibrary{arrows}

\usepackage{tikz}

\begin{document}
\tikzset{zshift/.style={xshift={-0.3*#1},yshift={-0.9*#1}}}

\begin{tikzcd}[row sep=2cm,column sep=2cm,inner sep=1ex]
& U \arrow{d}{T_0} \arrow{dr}{\Pi} & \\
V_1\times\cdots V_n \arrow{ur}{\varphi} 
\arrow{r}{\phi} \arrow{rr}{\psi}&
|[zshift=-1.5cm]| V \arrow{r}{T}& |[zshift=1cm]|W
\end{tikzcd}

\end{document}

Answer 2

Here are similarly-shaped diagrams taken from another context; maybe they are helpful:

\begin{tikzcd}[row sep={7.2em,between origins}, column sep={9.0em,between origins}, background color=backgroundColor, ampersand replacement=\&,execute at end picture={
  \foreach \Number in  {A,B,...,D}
      {\coordinate (\Number) at (\Number.center);}
  \begin{pgfonlayer}{background}
      \fill[left color=grey!98!black, right color=grey!80!white, shading angle=60] (A) -- (C) -- (B) -- cycle;
      \fill[left color=grey!85!black, right color=grey!85!white, shading angle=-60] (D) -- (C) -- (B) -- cycle;
      \fill[left color=grey!80!black, right color=grey!98!white, shading angle=180] (A) -- (B) -- (D) -- cycle;
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (A);
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (C);
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (D);
  \end{pgfonlayer}
  \filldraw[black!30!white] (A) circle (6pt) node[black,align=center] {$A$};
  \filldraw[black!30!white] (B) circle (6pt) node[black,align=center] {$B$};
  \filldraw[black!30!white] (C) circle (6pt) node[black,align=center] {$C$};
  \filldraw[black!30!white] (D) circle (6pt) node[black,align=center] {$D$};
}]
    \&
    |[alias=C]|C
    \arrow[rdd, "h"]
    \&\\[4.5em]\&
    |[alias=B]|B
    \arrow[u, dashed, "\textcolor{white}{\phantom{g}}"{description,background color=grey,opacity=0.75}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[u, phantom, "\textcolor{white}{g}"]
    \arrow[dr, dashed, "\textcolor{white}{\phantom{h\circ g}}"{description,background color=grey,opacity=0.5}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[dr, dashed, "\textcolor{white}{h\circ g}", phantom]
    \&\\[-1.85em]
    |[alias=A]|A
    \arrow[ruu, "g\circ f"]
    \arrow[rr, "h\circ g\circ f"']
    \arrow[ru, dashed, "\textcolor{white}{\phantom{f}}"{description,background color=grey,opacity=0.75}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[ru, dashed, "\textcolor{white}{f}", phantom]
    \&\&
    |[alias=D]|D\mathrlap{.}
\end{tikzcd}

2. (Ignore the 2-morphisms):

   \begin{tikzcd}[row sep={7.2em,between origins}, column sep={9.0em,between origins}, background color=backgroundColor, ampersand replacement=\&]
       \&
       A_{2}
       \arrow[rdd, "f_{23}", bend left=20]
       \&\\\&
       A_{1}
       \arrow[u, "f_{12}"{description}, dash pattern=on 4.0pt off 4.0pt]
       \arrow[rd, "f_{13}"'{description,name=f13}, dash pattern=on 4.0pt off 4.0pt]
       \&\\
       A_{0}
       \arrow[ruu, "f_{02}"{name=f02}, bend left=20]
       \arrow[rr, "f_{03}"'{name=f03}, bend right=20]
       \arrow[ru, "f_{01}"{description}, dash pattern=on 4.0pt off 4.0pt]
       \&\&
       A_{3}
       % 2-Arrows
       \arrow[from=2-2, to=f02, Rightarrow, shorten=1.0em, "\theta_{012}"description, pos=0.475]
       \arrow[from=1-2, to=f13, Rightarrow, yshift=-0.25em, xshift=0.75em, shorten=3.5em, "\theta_{123}"description, pos=0.475]
       \arrow[from=2-2, to=f03, Rightarrow, shorten=2.5em, "\theta_{013}"description, pos=0.475]
       \arrow[from=1-2, to=f03, bend left=20, yshift=+0.0em, xshift=0.0em, crossing over, Rightarrow, shorten=2.0em, "\hspace{+0.625em}\theta_{023}"description, pos=0.425, crossing over clearance=1.5ex]
   \end{tikzcd}