spacing in tikz

Question

I'm trying to typeset the naturality property of the cap product, I'm hoping to achieve something like this:

but the best I can do is something like this:

\begin{equation*}
\begin{tikzcd}
H_j(X,A\cup B; W) \arrow[d] &\otimes_\mathbb{Z} &H^i(X,A;V) \arrow[r]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) \arrow[d] \\
H_j(X,A\cup B; W) &\otimes_\mathbb{Z} &H^i(X,A;V) \arrow[u] \arrow[r]  
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) 
\end{tikzcd}
\end{equation*}

I would like to group the two entries on either side of the tensor product at the beginning in a single entry, but then I wouldn't be able to get arrows to and from each one of them individually. I also don't want to change the general column spacing, just have those behave as a single entry but with arrows departing from each one. Thank you very much for any advice!

Answer 1

You can reduce the column separation with the column sep option. In this case, you can also pretend that the \otimes characters are like arrows between two cells, so that you can remove the second column, like so:

\begin{tikzcd}[column sep=small]
H_j(X,A\cup B; W) \arrow[d] \arrow[r,phantom,"\otimes_\mathbb{Z}" description] &H^i(X,A;V) \arrow[r]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) \arrow[d] \\
H_j(X,A\cup B; W) \arrow[r,phantom,"\otimes_\mathbb{Z}" description] &H^i(X,A;V) \arrow[u] \arrow[r]  
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) 
\end{tikzcd}

As you can see, I replaced FIRST CELL & op & SECOND CELL by

FIRST CELL \arrow[r,phantom,"op" description] & SECOND CELL

The phantom option avoid the arrow to be typeset, and the description ensures that the operator is placed at the correct position.

Answer 2

This requires some manual adjusting, but arrows are standard ones, so they can easily receive labels:

\documentclass{article}
\usepackage{amsmath,amssymb,tikz-cd}

\begin{document}
\begin{equation*}
\begin{tikzcd}
H_j(X,A\cup B; W) \otimes_\mathbb{Z} H^i(X,A;V) \arrow[r] \arrow[d,shift right=3em]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) \arrow[d] \\
H_j(X,A\cup B; W) \otimes_\mathbb{Z} H^i(X,A;V)
 \arrow[u,shift right=5em] \arrow[r]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W)
\end{tikzcd}
\end{equation*}
\end{document}