sorry i’m not getting this whoops monad. can you spell out the details, or pick a more standard example to illustrate your point?
i think “every monad formalises a different notion of collection” is a bit strong. for example, the free vector space monad V (see section 3.2) — is 2⋅milk+1⋅eggs−3⋅sugar a collection of the elements, for some notion of collection?
is every element of a free algebraic structure a “collection” of the generators? would you hear someone say that a quantum state is a collection of eigenstates? at a stretch maybe.
The identity monad probably works about as well as an illustration, but has less of the flavor of “not only did you not make this more like a collection, you made it worse” :P But advantage is you didn’t need the axiom of choice to specify it.
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?
sorry i’m not getting this whoops monad. can you spell out the details, or pick a more standard example to illustrate your point?
i think “every monad formalises a different notion of collection” is a bit strong. for example, the free vector space monad V (see section 3.2) — is 2⋅milk+1⋅eggs−3⋅sugar a collection of the elements, for some notion of collection?
is every element of a free algebraic structure a “collection” of the generators? would you hear someone say that a quantum state is a collection of eigenstates? at a stretch maybe.
The identity monad probably works about as well as an illustration, but has less of the flavor of “not only did you not make this more like a collection, you made it worse” :P But advantage is you didn’t need the axiom of choice to specify it.
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.