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.