redefine caret ^ while using mathtools package

Question

consider the following MWE:

\documentclass{article}
%\usepackage{mathtools}

\newcommand\newCaret{X}

\catcode`^=\active
\let^=\newCaret

\begin{document}
    $x^2$
\end{document}

this works fine, the output is xX2 as expected.

When I now enable the mathtools package, I obtain the following error:

Package: graphics 2016/07/10 v1.0t Standard LaTeX Graphics (DPC,SPQR)

! Missing number, treated as zero.
<to be read again> 
                   X
l.33 \catcode`\^^
                 A=\catcode`\%
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

(At the beginning of the graphics package, which is used by mathtools, there is some catcode-voodoo which seems to conflict with my \catcode...)

possible solution 1

When I now move the two lines \catcode... and \let... after the \begin{document}, everything is fine again. I need them, however, to stay in the preamble, so this does not solve the problem for me.

possible solution 2

I tried wrapping the \catcode... and \let... inside an \AtBeginDocument, but then I get:

\Gin@req@height=\dimen123
\Gin@req@width=\dimen124
)
! Missing control sequence inserted.
<inserted text> 
                \inaccessible 
l.11 \begin{document}

Please don't say `\def cs{...}', say `\def\cs{...}'.
I've inserted an inaccessible control sequence so that your
definition will be completed without mixing me up too badly.
You can recover graciously from this error, if you're
careful; see exercise 27.2 in The TeXbook.

! Undefined control sequence.
l.12    $x^
         2$

background

I had the, maybe stupid, idea to simplify the work with upright and italic sub- and superscripts. Until now, I had

\newcommand\sub[1]{_\text{#1}}
\let\supremum\sup
\renewcommand\sup[1]{^\text{#1}}

for stuff like k\sub{b} and f_i. In combination with my question here I managed get equivalent output with k__b and f_i: when I double the underscore, I get upright subscripts like with my \sub command, and it's super fast to type/change. It also works with superscripts (^ and ^^), but only if I change the catcode/let the caret after the preamble, so I cannot bundle my code neatly inside a package.

Answer 1

What's the problem in changing the category code of ^ in the preamble? That some packages load files at begin document and these files might contain constructions such as

\catcode`\^^A=\catcode`\%

that are used to make byte 1 into a comment character or some `\^^M trickery.

You can set the catcode in the preamble, but delaying it with \AtBeginDocument (be sure to do it after loading all packages):

\newcommand\NewCaret{X}
\AtBeginDocument{%
  \begingroup\lccode`\~=`\^
  \lowercase{\endgroup\let~}\NewCaret
  \catcode`\^=\active
}

The \lowercase trick is explained elsewhere on the site, see The \lowercase trick. On the other hand, if the definition of ^ as an active character is only needed in math mode, it's better to make it only math active, see Underscore in textmode vs mathmode

\newcommand\NewCaret{X}
\begingroup\lccode`\~=`\^
\lowercase{\endgroup\let~}\NewCaret
\AtBeginDocument{\mathcode`\^="8000 }