Some functions I'm usin about extra

[ndmitchell]

While composeMany might be useful in certain situations, it doesn't seem a good fit for extra - there's nothing quite like it already, and nothing that even depends on alternative or category. In addition, I have almost no intuition of what it does.

For the second, I think you can generalise from Monad to Applicative, and generalise from a -> a to a -> b. Once you've done that you end up with something like liftA but the other way around, or even the dual of fmap in some sense. For that, I'd be curious to know what the operator should be called - I don't think $^ is necessarily the right intuition. FWIW, I have defined this function before, several times, but I tend to define a family of operators, which are the types:

Applicative m => m (a -> b) -> a -> m b
Applicative m => (a -> b) -> m a -> m b
Applicative m => m (a -> b) -> m a -> m b

The last two are <$> and <*>, but I define them with names like ^$ and ^$^ to have full symmetry. That said, because they don't fit in with the existing operators, I've never been a fan.

[marcosdumay]

On a second thought, I should have though composeMany for Monoid instead of Category anyway. It would be much easier to deal with.

I see your point about the operator. I'm left wondering if the ^$ set isn't better than the current <$> set, but yes, it's not something anybody would want to change now. On the <$> set, the $> operator seems to be unused, but because of the way <$ is defined, it may not be a good choice.

Well, I don't think I have anything valuable to add on this subject anymore. Thanks for the interaction.

[andreasabel]

Considering ($^), writing mf <*> pure (a r g) instead of mf $^ a r g does not seem that much extra to write. Of course with $^ you can save some parentheses.
Unfortunately, there is no good name available, as <* is taken for something else.

[ndmitchell]

I disagree about the extra syntactic-overhead of using pure - often the code is littered with $^ in many places, so you go from being mostly about your functions to just having a lot of pure. That said, without a good name, it isn't worth it.