Which monad would be best suited to a structure where lifting any value with pure, return will always yield the identity element of the trivial group?

I need to meet the following requirements:

1)Trivial group: Only a single element exists, which serves as its own identity (e.g., for addition, 0; for multiplication, 1).

2)Lifting (return/pure): Any input value, when lifted, always becomes this identity element, regardless of the original value.

I tried the Identity monad, but it returns the value I specify, not the fixed identity.

I can't describe correctly the monad that best fits the algebraic structure where pure/return always returns only the identity element in a given group. I need to get a monad with a single possible value that always returns the identity element of the group regardless of the input value.

Answer

The Const family of monads does this for any monoid (including groups). For the trivial group specifically, you can use Const (); there is also the specialization to the Proxy monad.