I’m not sure we should worry about generalizing with high-powered machinery. Pick a collection, and you can represent it with a monad. But pick a monad, and it’s probably not a collection (I think?).
E.g. consider a “whoops I lost some of them monad”—for each set you choose some subset, plus an extra element (could call it {*}, as in the maybe monad). So if my original set is (1,2,3,4,5,6,7), there will be some whoops monad that maps this to (1,2,*). Functions work as normal except when they would involve * or lost elements, in which case they get mapped to *. Seems like a perfectly good monad, but it’s the diametric opposite of a collection.
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?
I’m not sure we should worry about generalizing with high-powered machinery. Pick a collection, and you can represent it with a monad. But pick a monad, and it’s probably not a collection (I think?).
E.g. consider a “whoops I lost some of them monad”—for each set you choose some subset, plus an extra element (could call it {*}, as in the maybe monad). So if my original set is (1,2,3,4,5,6,7), there will be some whoops monad that maps this to (1,2,*). Functions work as normal except when they would involve * or lost elements, in which case they get mapped to *. Seems like a perfectly good monad, but it’s the diametric opposite of a collection.
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.