Macro changing a string depending on letter case

Question

I would like to define a new mathmode command such that:

• the input is a string s;
• the output is the same string where the font of each caracter has been changed depending on whether the character is in lower case or upper case.

In pseudo-code, that would look like:

\newcommand{\cat}{
    forEach (character char in #1) {
        if (isLowerCase(char)) -> print \mathit{char}
        else -> print \mathcal{char}
    }
}

It seems I can use ifthenelse from the package ifthen for the if/else. I couldn't find how to:

• extract characters from string;
• loop for each of these characters;
• check whether a character is in lower case or not.

By the way, if anyone knows about a readable but nonetheless exhaustive reference on defining commands, I would appreciate (I could only find either too basic or "way over my head" references).

Thanks for your help!

PS: if anyone needs some context, I'm working in category theory, where the typical font for categories is \mathcal. Since lower case in mathcal is not defined, I need to find a way around it.

---

EDIT: I describe here my understanding of Ulrike Fischer's solution, as it might be useful to others.

\ExplSyntaxOn
\NewDocumentCommand\cat{m}
 {
   \tl_set:Nn\l_tmpa_tl {#1}
   \regex_replace_all:nnN {[a-z]}{\c{mathit}\cB\{\0\cE\}}\l_tmpa_tl
   \regex_replace_all:nnN {[A-Z]}{\c{mathcal}\cB\{\0\cE\}}\l_tmpa_tl
   \l_tmpa_tl
 }
\ExplSyntaxOff

• ExplSyntaxOn...ExplSyntaxOff: change code régime where spaces are ignored and ":" and "_" are treated as letters. Necessary to access functions and variables. See interface3.pdf, p.7.
• \NewDocumentCommand \cat {m} {...}: create a new document-level function. You shall first describe the name (\cat), the set of parameters ({m}, one parameter which is refered as #1 in the function code) and the function code itself ({...}). See xparse.pdf, p.7.
• \tl_set:Nn \l_tmpa_tl {#1}: \tl_set:Nn set the value of some variable (or "token list", hence tl). N and n define the usual type of parameters for this function, N standing for a single token and n for a list of tokens between brackets. \l_tmpa_tl is a standard name of local temporary assigned token list.
• the first argument of \regex_replace_all:nnN is a list of tokens to be replaced, e.g. {[a-z]}. The second is what to do with these tokens, e.g. {\c{mathit}\cB\{\0\cE\}}. \c{mathit} is the equivalent of \mathit (for some reason, you can't just write \mathit). \0 stands for everything that as been selected through the first argument. For some reason, it should be surrounded by \cB\{...\cE\}. Finally, the third argument is the token list on which the function is applied.
• the last \l_tpma_tl simply display the resulting token.

Answer 1

Assuming that you only have a-Z in the argument:

\documentclass{article}

\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand\mycal{m}
 {
   \tl_set:Nn\l_tmpa_tl {#1}
   \regex_replace_all:nnN {[a-z]}{\c{mathit}\cB\{\0\cE\}}\l_tmpa_tl
   \regex_replace_all:nnN {[A-Z]}{\c{mathcal}\cB\{\0\cE\}}\l_tmpa_tl
   \l_tmpa_tl
 }
\ExplSyntaxOff
\begin{document}
$\mycal{abcABC}$
\end{document}

Answer 2

Ulrike’s answer is fine, but we can improve it by removing unwanted kerns.

\documentclass{article}
\usepackage{amsmath}
\usepackage{xparse}

\AtBeginDocument{$\mathcal{\global\chardef\calfam=\fam}$}

\ExplSyntaxOn

\cs_new_protected:Nn \__trynopsis_cat_neg:n
 {
  \mathchoice
   { \__trynopsis_cat_neg_aux:Nn \textfont { #1 } }
   { \__trynopsis_cat_neg_aux:Nn \textfont { #1 } }
   { \__trynopsis_cat_neg_aux:Nn \scriptfont { #1 } }
   { \__trynopsis_cat_neg_aux:Nn \scriptscriptfont { #1 } }
 }
\cs_new_protected:Nn \__trynopsis_cat_neg_aux:Nn
 {
  \kern -\fontcharic#1\calfam`#2
 }

\NewDocumentCommand{\cat}{m}
 {
  \trynopsis_cat:n { #1 }
 }

\tl_new:N \l__trynopsis_cat_tl

\cs_new_protected:Nn \trynopsis_cat:n
 {
  \tl_set:Nn \l__trynopsis_cat_tl { #1 }
  \regex_replace_all:nnN
   { ([A-Z]) ([a-z]) } % search uppercase letter followed by a lowercase letter
   { \1 \c{__trynopsis_cat_neg:n} \1 \2 } % add a negative space in between
   \l__trynopsis_cat_tl
  \regex_replace_all:nnN
   { [A-Z]+ } % search any run of uppercase letters
   { \c{mathcal}\cB\{\0\cE\} } % enclose it in \mathcal
   \l__trynopsis_cat_tl
  \regex_replace_all:nnN
   { [a-z]+ } % search any run of lowercase letters
   { \c{mathit}\cB\{\0\cE\} } % enclose it in \mathit
   \l__trynopsis_cat_tl
  \regex_replace_all:nnN
   { \c{__trynopsis_cat_neg:n} \c{mathcal} }
   { \c{__trynopsis_cat_neg:n} }
   \l__trynopsis_cat_tl
  % deliver the new token list
  \tl_use:N \l__trynopsis_cat_tl
 }

\cs_new_protected:Nn \__trynopsis_cat_neg:
 {
  \mspace{-2mu}
 }

\ExplSyntaxOff

\begin{document}

$\cat{Set}\quad\cat{Grp}\quad\cat{Abc\_def}$

$\scriptstyle\cat{Set}\quad\cat{Grp}\quad\cat{Abc\_def}$

\end{document}

The fourth replacement transforms the unwanted \__trynopsis_cat_neg:n \mathcal {<letter>} into \__trynopsis_cat_neg:n {<letter>}.

The purpose of \__trynopsis_cat_neg:n is to remove the excess space when an uppercase (calligraphic) letter is followed by a lowercase (italic) one.