  We propose another interpretation of well-known derivatives computations from
regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch,
in order to abstract the underlying data structures (e.g. sets or linear
combinations) using the notion of monad. As an example of this generalization
advantage, we introduce a new derivation technique based on the graded module
monad.
  We also extend operators defining expressions to any n-ary functions over
value sets, such as classical operations (like negation or intersection for
Boolean weights) or more exotic ones (like algebraic mean for rational
weights).
  Moreover, we present how to compute a (non-necessarily finite) automaton from
such an extended expression, using the Colcombet and Petrisan categorical
definition of automata. These category theory concepts allow us to perform this
construction in a unified way, whatever the underlying monad.
  Finally, to illustrate our work, we present a Haskell implementation of these
notions using advanced techniques of functional programming, and we provide a
web interface to manipulate concrete examples.
