How do I make a character active only in text mode?

Question

I want to make the character > active only in text mode (to create markdown-style block quotes) but not in math mode. I know the \mathcode`>="8000 way to make a character active only in math mode, but what about the reverse? I tried \ifmmode but that doesn't seem to work.

Here's what I have now, having to escape it as \> in math mode.

\usepackage[many]{tcolorbox}

\makeatletter

\newtcolorbox{QuoteBlock}{
    blanker,
    before skip=\topsep,
    after skip=\topsep,
    borderline west={2pt}{0pt}{black!50!white},
    breakable,
    left=\parindent,
}

\begingroup
    \catcode`\>=\active
    \catcode`\^^M=\active
    \gdef>#1^^M{%
        \begin{QuoteBlock}%
            \noindent\@ifnextchar\par\@gobble\relax#1
        \end{QuoteBlock}
    }
\endgroup
\def\>{\string>}

\AtBeginDocument{\catcode`\>=\active \catcode`\^^M=\active}

\makeatother

\begin{document}
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
    > Lorem ipsum dolor sit amet, consectetuer adipiscing elit
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
    \[x\>0\]
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
\end{document}

Answer 1

Changing catcode values changes the tokenization of characters yet to be read from teh file but has no effect on any tokens already created.

You want it to be active always but do what it did before in math mode, so:

\documentclass{article}

\usepackage[many]{tcolorbox}

\makeatletter

\newtcolorbox{QuoteBlock}{
    blanker,
    before skip=\topsep,
    after skip=\topsep,
    borderline west={2pt}{0pt}{black!50!white},
    breakable,
    left=\parindent,
}

\begingroup
    \catcode`\>=\active
    \catcode`\^^M=\active
    \gdef>{\relax\ifmmode\string>\else\expandafter\quotegt\fi}%
    \gdef\quotegt#1^^M{%
        \begin{QuoteBlock}%
            \noindent\@ifnextchar\par\@gobble\relax#1
        \end{QuoteBlock}
    }
\endgroup

\AtBeginDocument{\catcode`\>=\active \catcode`\^^M=\active}

\makeatother

\begin{document}
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
    > Lorem ipsum dolor sit amet, consectetuer adipiscing elit
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
    \[x>0\]
    Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi.
\end{document}