If “List” is a monoid, what is its “set”?

Question

Just reading through category theory book , and decided to apply it to haskell.

The author defines Monoid as:

Monoid is a set L equipped with a binary operation *:LxL->L and a distinguished unit element u in L such that etc...

Taking a "List" structure as a monoid, it is clear that binary operation is concat and unit is [] .

But what is the set M here? I tried L = {set of all lists} but I think that leads me into trouble with "is L in L?" question, which seems to be the same problem as sets have.

Or am I thinking of something incorrectly?

EDIT: As pointed out by @applicative, Haskell's lists are monoids called the Free monoids !

Answer 1

Instead of saying "List is a Monoid", it would be more accurate to say "For all types a, the type [a] is a Monoid". So for any particular type a , your L will be L = {set of all lists of as} . And with that definition, L can of course not contain itself.

Answer 2

For any type t you can have that

L = all elements of the type [t]

then L is a monoid in the trivial way using ++ . In fact, we formalize this in Haskell

class Monoid m where
   mempty  :: m
   mappend :: m -> m -> m

this is a "class" of types that have the requisite operations to form a monoid, so

instance Monoid [a] where
   mempty = []
   mappend a b = a ++ b

in fact, this is known as the "free monoid on a"