The official tikz-cd is available here.

The following list shows all available arrows.

I tried all items in the list, and got error messages for the three items: dash, equal, and mapsfrom.

All the other items worked perfectly.

As pointed out by egreg in his answer to this question, one can use [-] instead of [dash].

I tried to use [=] instead of [equal], but it didn't work.

I got a not very good-looking mapsfrom using [<-|].

Of course I'd be delighted to have an explanation, but, above all, I'd like to find a way of obtaining the equal and mapsfrom arrows (mainly the equal one).

Here is a Minimal Working Example:

\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
$\begin{tikzcd} X \ar[equal]{r} & Y \end{tikzcd}$
\end{document}

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I suspect you have an outdated tikz-cd package, because with [equal] and [mapsfrom] I get the expected result. In the log file I have tikz-cd 2012/12/30 v0.3c –  egreg Mar 5 '13 at 13:14
Dear @egreg: I think you're right: in my log file I have tikz-cd 2012/03/18 v0.2b. I'll download the current version, but this may take a while because I have a slow connection. I'll let you know. Thank you very much for your edit and your comment (which, I'm pretty sure, is an answer). –  Pierre-Yves Gaillard Mar 5 '13 at 13:27
Dear @egreg: I haven't been able so far to update the package, but I don't have the slightest doubt about the correctness of what you say. If you upgrade your comment to an answer, I'll be delighted to upvote and accept it. Thanks again! –  Pierre-Yves Gaillard Mar 5 '13 at 15:45

This is what I get (tikz-cd version 0.3c, tikz version 2.10) from
\documentclass{article}
$\begin{tikzcd} X \ar[equal]{r} & Y \ar[mapsfrom]{r} & Z \end{tikzcd}$