"Co-lift" is not possible for functors generally. Comonads have a method extract :: Comonad w => w a -> a, which is somewhat related. Distributing and gathering are both really distributing. The first example distributes [] over f while the second example distributes f over []. Neither are possible for functors generally, but Applicative and Data.Traversable are strong enough to provide sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a). The problem is that the Applicative and Traversable laws alone are not enough to guarantee that this is well behaved. What you actually need is a stronger notion that f (g a) is an "f-structure" containing "g-structures" *all of the same shape*. If they aren't, the inner structures won't be distributed properly and some of the odd shaped bits might get cut off. For example, try to transpose a list of lists (iow, distribute [] over []) and you'll see that all of the inner lists must be the same length for your transposition to be well behaved. Conor McBride covers this in a very interesting way in this StackOverflow answer: http://stackoverflow.com/a/13100857/2225384 On Wed, Mar 2, 2016 at 2:50 PM Joel Neely <joel.neely@gmail.com> wrote:

IIUC, one would describe fmap as "lifting" a function g into a functor f, in the sense of

Functor f => (g a b) -> g a -> g b

Is there an inverse concept (? co-lift ?) that describes

Functor f => (g a -> g b) -> a -> b

Similarly is there standard terminology for "distributing" and "gathering" in the sense of

Functor f => f [a] -> [f a]

and

Functor f => [f a] -> f [a]

respectively?

References much appreciated! -jn-

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