Further questions about mathrm and operatorname: spacing after \colon

Question

EDIT: As clarified in the answers below, this appears to be a bug with amsmath and colon and thus doesn't really have anything to do with mathrm versus operatorname.

This question contains an enlightening discussion between using \mathrm and \operatorname. The tl;dr version is: whenever you have an operator, use \operatorname.

However it seems bad to use \operatorname if what you are defining is a set, since in general \operatorname adds a little space before it. This means that an expression like:

f \colon \operatorname{End}(V) \to \mathbb{R}

renders badly, as there is too much white space between the colon and the operatorname{End}. Using \mathrm (the RHS) is more visually appealing:

Thus if one views \operatorname{End}(V) as a set, then it makes more sense to write \mathrm{End}(V).

Now one can view \operatorname{End} also a functor on the category of vector spaces, for instance, then you would want the \operatorname:

My question is this: What is the best practice when dealing with a quantity that is used both as a set and an operator? Should one really switch between \mathrm and \operatorname as appropriate? Or is there a better option.

Answer 1

That's a very interesting observation.

The definition of \colon in amsmath is

\renewcommand{\colon}{
  \nobreak
  \mskip 2mu
  \mathpunct{}
  \nonscript
  \mkern-\thinmuskip
  {:}
  \mskip 6mu plus 1mu\relax
}

(edited to show the various parts). The space you see is a consequence of the fact that \colon ends with an ordinary atom, namely {:}.

It would have been better to use \mathopen, as the following test shows. I first get an alias of \colon and patch it to change {:} into \mathopen:.

\documentclass{article}
\usepackage{amsmath}
\usepackage{etoolbox}

\let\pcolon\colon
\patchcmd{\pcolon}{{:}}{\mathopen:}{}{}

\begin{document}

$f \colon \operatorname{End}(V) \to R$

$f \pcolon \operatorname{End}(V) \to R$

\bigskip

\begin{tabular}{@{}*{8}{l}@{}}
&0&1&2&3&4&5&6 \\
\verb|\colon| &
  $\colon A$ &
  $\colon \max$ &
  $\colon +$ &
  $\colon \sim$ &
  $\colon ($ &
  $\colon )$ &
  $\colon ,$
\\
\verb|\pcolon| &
  $\pcolon A$ &
  $\pcolon \max$ &
  $\pcolon +$ &
  $\pcolon \sim$ &
  $\pcolon ($ &
  $\pcolon )$ &
  $\pcolon ,$
\end{tabular}

\end{document}

In the tests, \colon is followed by a symbol in each class.

The only noticeable differences are in the cases when \colon is followed by atoms of class 1 (operators) and 3 (relations), performing perhaps better in both.

If you want to follow this suggestion, your document can load etoolbox and do

\patchcmd{\colon}{{:}}{\mathopen:}{}{}

I don't think that it's possible to fix amsmath as many existing documents depend on it and such a change would affect the output and line breaks. It might be possible to add an option for using a fixed definition.