[spire] mapping on spire types

[Marco Kuper]

Is it possible to map on the spire types, as long as the inner type that is mapped to is in the list of specialized types? E.g on a Field[A] the signature would look sth like

def map[B](f: A => B): Field[B]

I am asking because I would like to be able use operators combining types with different realized specialization, e.g. multiply a Field[Int] with a Field[Double]

Cheers 

[Erik Osheim]

Hi Marco,

Most of Spire's type classes can't correctly implement map. Let's look
at a simple example, Monoid[A], and see if we can do it.

trait Monoid[A] { self =>
def id: A // abstract
def op(x: A, y: A): A // abstract
def map[B](f: A => B): Monoid[B] =
new Monoid[B] {
def id: B = f(self.id)
def op(x: B, y: B): B =
f(self.op(x??, x??))
}
}

This would work except for the x?? and y?? parts. In order to use the
underlying Monoid to implement combine we also need a way to go from
(B => A), so we can convert the mapped combine's arguments into A
values so we can use them with the original Monoid.

We *could* give these types a method like imap:

trait Monoid[A] { self =>
def id: A // abstract
def op(x: A, y: A): A // abstract
def imap[B](f: A => B)(g: B => A): Monoid[B] =
new Monoid[B] {
def id: B = f(self.id)
def op(x: B, y: B): B =
f(self.op(g(x), g(y)))
}
}

Does this make sense?

To generalize this a bit (and move from Spire to Cats), we can
characterize any type that supports a map method as having a
cats.Functor (specificallly, a covariant functor). Types like
spire.algebra.Monoid do not have covariant functors, but they do have
a cats.functor.Invariant (an invariant functor) which has the more
restrictive imap method.

Spire itself does not currently put an imap method on these type
classes, but I would be open to adding it. Also, if you are using
Cats, you could define an instance of Invariant[Field] (for example)
which would give you access to .imap via enrichment.

-- Erik

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[Marco Kuper]

Thanks Erik, your explanation does make a lot of sense. I'll look into Cats to see if .imap is what I'm looking for.

Cheers,
Marco