Categorical limits with smaller arrows

Question

I have always disliked the amsmath commands \varinjlim and \varprojlim for categorical limits. The arrows are too big and dominating, and the whole things take up too much vertical space, messing up the line spacing. I therefore usually define my own commands, see below, and would like to know if they can be improved and made more robust? In the current form, they use \raisebox which uses absolute dimensions and therefore causes issues when used e.g. in an index (see picture below). How can my construction be improved?

This question has some interesting solutions, but one uses PGF, which seems like extreme overkill to me, and the second one does not scale properly when used e.g. in indices. Oh yes, and there is a package called halloweenmath, but I’m not sure I trust a package with such a name. I think I prefer a solution by some TeX guru in here.

\documentclass{article}

\usepackage{amsmath,graphicx}

\makeatletter
\newcommand\dirlimformat[1]{
    \mathop{
        \smash{
            \operatorname*{#1}\limits_{
                {}
                \raisebox{.21em}[0pt][0pt]{\scalebox{.85}{$\m@th
                    \xrightarrow{%
                        \hphantom
                        {%
                            \!\!
                            \scalebox{1.17}{$\m@th
                                {\operatorname{#1}}
                            $}
                            \!\!
                        }
                    }
                $}}
            }
        }
        \vphantom{\textstyle\lim_n}
    }
}

\newcommand\invlimformat[1]{
    \mathop{
        \smash{
            \operatorname*{#1}\limits_{
                {}
                \raisebox{.21em}[0pt][0pt]{\scalebox{.85}{$\m@th
                    \xleftarrow{%
                        \hphantom
                        {%
                            \!\!
                            \scalebox{1.17}{$\m@th
                                {\operatorname{#1}}
                            $}
                            \!\!
                        }
                    }
                $}}
            }
        }
        \vphantom{\textstyle\lim_n}
    }
}
\makeatother

\newcommand\dirlim{\dirlimformat{lim}}
\newcommand\invlim{\invlimformat{lim}}

\begin{document}

\( \varinjlim X_n \) vs.\ \( \dirlim X_n \)

\( \varprojlim X_n \) vs.\ \( \invlim X_n \)

\( a^{\invlim X_n} \)

\end{document}

Answer 1

The definitions of \varinjim and \varptojlim in amsopn.sty are

\def\varinjlim{%
  \mathop{\mathpalette\varlim@{\rightarrowfill@\textstyle}}\nmlimits@
}
\def\varprojlim{%
  \mathop{\mathpalette\varlim@{\leftarrowfill@\textstyle}}\nmlimits@
}

The arrows are thus always in \textstyle. You can make your own definitions (or change the default ones---with the usual caveats) using \scriptstyle

\documentclass{article}

\usepackage{amsmath}

\makeatletter
\newcommand\dirlim{\mathop{\mathpalette\varlim@{\rightarrowfill@\scriptstyle}}\nmlimits@}
\newcommand\invlim{\mathop{\mathpalette\varlim@{\leftarrowfill@\scriptstyle}}\nmlimits@}
\makeatother

\begin{document}

\parskip=\medskipamount

\( \varinjlim X_n \) vs.\ \( \dirlim X_n \)

\( \varprojlim X_n \) vs.\ \( \invlim X_n \)

\( a^{\varprojlim X_n} \) vs.\ \( a^{\invlim X_n} \)

\end{document}

Of course this will be sub-optimal in \scriptscriptstyle; you could add a \mathpalette workaround which looks slightly less bad

\documentclass{article}

\usepackage{amsmath}

\makeatletter

% same as above
\newcommand\dirlimA{\mathop{\mathpalette\varlim@{\rightarrowfill@\scriptstyle}}\nmlimits@}
\newcommand\invlimA{\mathop{\mathpalette\varlim@{\leftarrowfill@\scriptstyle}}\nmlimits@}

% alternative version
\def\varlim@@#1#2{%
  \vtop{\m@th\ialign{##\cr
    \hfil$#1\operator@font lim$\hfil\cr
    \noalign{\nointerlineskip\kern1.5\ex@}
    \expandafter#2\ifx#1\scriptscriptstyle\scriptscriptstyle\else\scriptstyle\fi\cr
    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\newcommand\dirlimB{\mathop{\mathpalette\varlim@@\rightarrowfill@}\nmlimits@}
\newcommand\invlimB{\mathop{\mathpalette\varlim@@\leftarrowfill@}\nmlimits@}

\makeatother

\begin{document}

\parskip=\medskipamount

$\varinjlim_{\varinjlim_{\varinjlim}}$

$\dirlimA_{\dirlimA_{\dirlimA}}$

$\dirlimB_{\dirlimB_{\dirlimB}}$

\end{document}

Answer 2

You may want to use old-arrows. With the option old, the standard arrows are not modified, but we can patch \varinjlim and \varprojlim to use the smaller arrow tips.

\documentclass{article}
\usepackage{amsmath}
\usepackage[old]{old-arrows}
\usepackage{xpatch}

\makeatletter
\xpatchcmd{\varinjlim}{\rightarrowfill@}{\varrightarrowfill@}{}{}
\xpatchcmd{\varprojlim}{\leftarrowfill@}{\varleftarrowfill@}{}{}
\makeatother

\begin{document}

\[
\rightarrow\varrightarrow\varinjlim_{x\in X} \quad \varprojlim_{x\in X}
\]

\end{document}

With even smaller arrows:

\documentclass{article}
\usepackage{amsmath}
\usepackage[old]{old-arrows}
\usepackage{xpatch}

\makeatletter
\xpatchcmd{\varinjlim}{\rightarrowfill@}{\varrightarrowfill@}{}{}
\xpatchcmd{\varinjlim}{\textstyle}{\scriptstyle}{}{}
\xpatchcmd{\varprojlim}{\leftarrowfill@}{\varleftarrowfill@}{}{}
\xpatchcmd{\varprojlim}{\textstyle}{\scriptstyle}{}{}
\makeatother

\begin{document}

\[
\varinjlim_{x\in X} \quad \varprojlim_{x\in X}
\]

\end{document}