MWE:
\documentclass{article}
\usepackage{mathtools}
\DeclarePairedDelimiter{\abs}{\left|}{\right|}
\DeclarePairedDelimiter{\absl}{\lvert}{\rvert}
\begin{document}
$\abs{\cdot}$
$\absl{\cdot}$
\end{document}
The output is as expected:
However, I have the following error with regard to the usage of \abs:
Missing { inserted. $\abs{\cdot}
Missing { inserted. $\abs{\cdot}
Missing } inserted. $\abs{\cdot}
Missing } inserted. $\abs{\cdot}$
Why? How to prevent this?

\leftand\rightwithinDeclarePairedDelimiter, was it? Just use the second definition – user202729 May 15 '22 at 03:29\left|? – Atom May 15 '22 at 03:35