The Universe of Discourse

'The' reader monad does not exist

Reading over my recent article complaining about the environment functor I realized there's yet another terminology problem that makes the discussion unnecessarily confusing. “The” environment functor isn't unique. There is a family of environment functors, one for each possible environment type e. If g is the environment functor at type e, a value of type g t is a function e → t. But e could be anything and if g and h are environment functors at two different types e and e’ they are of course different functors.

This is even obvious from the definition:

    data Environ e t = Env (e -> t)
    instance Functor (Environ e) where
      fmap f (Env x) = Env $ \e -> f (x e)

The functor isn't Environ, it's Environ e, and the functor instance declaration, as it says on line 2. (It seems to me that the notation is missing a universal quantifier somewhere, but I'm not going to open that issue.)

We should speak of Environ e as an environment functor, not the environment functor. So for example instead of:

When operating in the environment functor, fmap has the type (a -> b) -> g a -> g b

I should have said:

When operating in an environment functor, fmap has the type (a -> b) -> g a -> g b

And instead of:

A function p -> q is a q parcel in the environment functor

I should have said:

A function p -> q is a q parcel in an environment functor

or

A function p -> q is a q parcel in the environment functor at p

although I'm not sure I like the way the prepositions are proliferating there.

The same issue affects ⸢the⸣ reader monad, ⸢the⸣ state monad, and many others.

I'm beginning to find remarkable how much basic terminology Haskell is missing or gets wrong. Mathematicians have a very keen appreciation of the importance of specific and precise terminology, and you'd think this would have filtered into the Haskell world. People are forever complaining that Haskell uses unfamiliar terms like “functor”, and the community's response is (properly, I think) that these terms are pre-existing and there is no point to inventing a new term that will be just as unfamiliar, or, worse, lure people into thinking that the know what it means when they don't. You don't want to call a functor a “container”, says the argument, because many functors (environment functors for example) are nothing at all like containers. I think this is wise.

But having planted their flag on that hill, the Haskell folks don't then use their own terminology correctly. I complained years ago that the term “monad” was used interchangeably for four subtly different concepts, and here we actually have a fifth. I pointed out that in the case of Environment e t, common usage refers to both Environment e and Environment e t as monads, and only the first is correct. But when people say “the environment monad” they mean that Environment itself is a monad, which it is not.

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