note that there are only two exceptions to the claim “the unit of a monad is componentwise injective”. this means (except these two weird exceptions), that the singleton collections ηX(x1) and ηX(x2) are always distinct for x1≠x2. hence, M(X), the set of collections over X, always “contains” the underlying set X. by “contains” i mean there is a canonical injection ηX:X→M(X), i.e. in the same way the real numbers contains the rational .
in particular, i think this should settle the worry that “there should be more collections than singleton elements”. is that your worry?
https://math.stackexchange.com/questions/1840104/regarding-the-injectivity-of-units-of-monads-on-mathbfset
note that there are only two exceptions to the claim “the unit of a monad is componentwise injective”. this means (except these two weird exceptions), that the singleton collections ηX(x1) and ηX(x2) are always distinct for x1≠x2. hence, M(X), the set of collections over X, always “contains” the underlying set X. by “contains” i mean there is a canonical injection ηX:X→M(X), i.e. in the same way the real numbers contains the rational .
in particular, i think this should settle the worry that “there should be more collections than singleton elements”. is that your worry?
I wouldn’t say it’s my worry exactly, but it does deal with the most forceful reasons for worrying, yeah.