Typeclasses such as Bifunctor are often expressed in terms of whether they are covariant or contravariant. While these terms may appear intimidating to the unfamiliar, they are a precise language for discussing these concepts, and once explained are relatively easy to understand. Furthermore, the related topics of positive and negative position can greatly simplify how you think about complex data structures. This topic also naturally leads into subtyping.
This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. For more information, please see the Wikipedia page on covariance and contravariance.
This blog post is also part of the FP Complete Haskell Syllabus and part of our Haskell training.
The Functor typeclass: covariant functor
Let's consider the following functions (made monomorphic for clarity):
showInt :: Int > String showInt = show floorInt :: Double > Int floorInt = floor
Now suppose that we have a value:
maybeInt :: Maybe Int maybeInt = Just 5
We know Maybe
is an instance of
Functor
, providing us with the following function:
fmapMaybe :: (a > b) > Maybe a > Maybe b fmapMaybe = fmap
We can use fmapMaybe
and showInt
together to get a new, valid, welltyped value:
maybeString :: Maybe String maybeString = fmapMaybe showInt maybeInt
However, we can't do the same thing with floorInt
.
The reason for this is relatively straightforward: in order to use
fmapMaybe
on our Maybe Int
, we need to
provide a function that takes an Int
as an input,
whereas floorInt
returns an Int
as an
output. This is a longwinded way of saying that Maybe
is covariant on its type argument, or that the Functor
typeclass is a covariant functor.
Doesn't make sense yet? Don't worry, it shouldn't. In order to understand this better, let's contrast it with something different.
A noncovariant data type
Consider the following data structure representing how to create
a String
from something:
newtype MakeString a = MakeString { makeString :: a > String }
We can use this to convert an Int
into a
String
:
newtype MakeString a = MakeString { makeString :: a > String } showInt :: MakeString Int showInt = MakeString show main :: IO () main = putStrLn $ makeString showInt 5
The output for this program is, as expected, 5
. But
suppose we want to both add 3
to the Int
and turn it into a String
. We can do:
newtype MakeString a = MakeString { makeString :: a > String } plus3ShowInt :: MakeString Int plus3ShowInt = MakeString (show . (+ 3)) main :: IO () main = putStrLn $ makeString plus3ShowInt 5
But this approach is quite noncompositional. We'd ideally like to be able to just apply more functions to this data structure. Let's first write that up without any typeclasses:
newtype MakeString a = MakeString { makeString :: a > String } mapMakeString :: (b > a) > MakeString a > MakeString b mapMakeString f (MakeString g) = MakeString (g . f) showInt :: MakeString Int showInt = MakeString show plus3ShowInt :: MakeString Int plus3ShowInt = mapMakeString (+ 3) showInt main :: IO () main = putStrLn $ makeString plus3ShowInt 5
But this kind of mapping inside a data structure is exactly what
we use the Functor
type class for, right? So let's try
to write an instance!
instance Functor MakeString where fmap f (MakeString g) = MakeString (g . f)
Unfortunately, this doesn't work:
Main.hs:4:45:
Couldn't match type ‘b’ with ‘a’
‘b’ is a rigid type variable bound by
the type signature for
fmap :: (a > b) > MakeString a > MakeString b
at Main.hs:4:5
‘a’ is a rigid type variable bound by
the type signature for
fmap :: (a > b) > MakeString a > MakeString b
at Main.hs:4:5
Expected type: b > a
Actual type: a > b
Relevant bindings include
g :: a > String (bound at Main.hs:4:24)
f :: a > b (bound at Main.hs:4:10)
fmap :: (a > b) > MakeString a > MakeString b
(bound at Main.hs:4:5)
In the second argument of ‘(.)’, namely ‘f’
In the first argument of ‘MakeString’, namely ‘(g . f)’
To understand why, let's compare the type for fmap
(specialized to MakeString
) with our
mapMakeString
type:
mapMakeString :: (b > a) > MakeString a > MakeString b fmap :: (a > b) > MakeString a > MakeString b
Notice that fmap
has the usual a >
b
parameter, whereas mapMakeString
instead has
a b > a
, which goes in the opposite direction.
More on that next.
Exercise: Convince yourself that the
mapMakeString
function has the only valid type
signature we could apply to it, and that the implementation is the
only valid implementation of that signature. (It's true that you
can change the variable names around to cheat and make the first
parameter a > b
, but then you'd also have to
modify the rest of the type signature.)
Contravariance
What we just saw is that fmap
takes a function from
a > b
, and lifts it to f a > f b
.
Notice that the a
is always the "input" in both cases,
whereas the b
is the "output" in both cases. By
contrast, mapMakeString
has the normal f a >
f b
, but the initial function has its types reversed:
b > a
. This is the core of covariance vs
contravariance:
 In covariance, both the original and lifted functions point in
the same direction (from
a
tob
)  In contravariance, the original and lifted functions point in
opposite directions (one goes from
a
tob
, the other fromb
toa
)
This is what is meant when we refer to the normal
Functor
typeclass in Haskell as a covariant functor.
And as you can probably guess, we can just as easily define a
contravariant functor. In fact,
it exists in the contravariant package. Let's go ahead and use
that typeclass in our toy example:
import Data.Functor.Contravariant newtype MakeString a = MakeString { makeString :: a > String } instance Contravariant MakeString where contramap f (MakeString g) = MakeString (g . f) showInt :: MakeString Int showInt = MakeString show plus3ShowInt :: MakeString Int plus3ShowInt = contramap (+ 3) showInt main :: IO () main = putStrLn $ makeString plus3ShowInt 5
Our implementation of contramap
is identical to the
mapMakeString
used before, which hopefully isn't too
surprising.
Example:
filtering with Predicate
Let's say we want to print out all of the numbers from 1 to 10,
where the English word for that number is more than three
characters long. Using a simple helper function english ::
Int > String
and filter
, this is pretty
simple:
greaterThanThree :: Int > Bool
greaterThanThree = (> 3)
lengthGTThree :: [a] > Bool
lengthGTThree = greaterThanThree . length
englishGTThree :: Int > Bool
englishGTThree = lengthGTThree . english
english :: Int > String
english 1 = "one"
english 2 = "two"
english 3 = "three"
english 4 = "four"
english 5 = "five"
english 6 = "six"
english 7 = "seven"
english 8 = "eight"
english 9 = "nine"
english 10 = "ten"
main :: IO ()
main = print $ filter englishGTThree [1..10]
The contravariant package provides a newtype wrapper around such
a > Bool
functions, called Predicate
.
We can use this newtype to wrap up our helper functions and avoid
explicit function composition:
import Data.Functor.Contravariant greaterThanThree :: Predicate Int greaterThanThree = Predicate (> 3) lengthGTThree :: Predicate [a] lengthGTThree = contramap length greaterThanThree englishGTThree :: Predicate Int englishGTThree = contramap english lengthGTThree english :: Int > String english 1 = "one" english 2 = "two" english 3 = "three" english 4 = "four" english 5 = "five" english 6 = "six" english 7 = "seven" english 8 = "eight" english 9 = "nine" english 10 = "ten" main :: IO () main = print $ filter (getPredicate englishGTThree) [1..10]
NOTE: I'm not actually recommending this as a better practice than the original, simpler version. This is just to demonstrate the capability of the abstraction.
Bifunctor and Profunctor
We're now ready to look at something a bit more complicated.
Consider the following two typeclasses:
Profunctor and
Bifunctor. Both of these typeclasses apply to types of kind
* > * > *
, also known as "a type constructor
that takes two arguments." But let's look at their (simplified)
definitions:
class Bifunctor p where bimap :: (a > b) > (c > d) > p a c > p b d class Profunctor p where dimap :: (b > a) > (c > d) > p a c > p b d
They're identical, except that bimap
takes a first
parameter of type a > b
, whereas
dimap
takes a first parameter of type b >
a
. Based on this observation, and what we've learned
previously, we can now understand the documentation for these two
typeclasses:
Bifunctor
: Intuitively it is a bifunctor where both the first and second arguments are covariant.
Profunctor
: Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.
These are both bifunctors since they take two type parameters.
They both treat their second parameter in the same way:
covariantly. However, the first parameter is treated differently by
the two: Bifunctor
is covariant, and
Profunctor
is contravariant.
Exercise Try to think of a few common datatypes in
Haskell that would be either a Bifunctor
or
Profunctor
, and write the instance.
Hint Some examples are Either
,
(,)
, and >
(a normal function from
a
to b
). Figure out which is a
Bifunctor
and which is a Profunctor
.
Solution
class Bifunctor p where bimap :: (a > b) > (c > d) > p a c > p b d class Profunctor p where dimap :: (b > a) > (c > d) > p a c > p b d instance Bifunctor Either where bimap f _ (Left x) = Left (f x) bimap _ f (Right x) = Right (f x) instance Bifunctor (,) where bimap f g (x, y) = (f x, g y) instance Profunctor (>) where  functions dimap f g h = g . h . f
Make sure you understand why these instances work the way they do before moving on.
Bivariant and invariant
There are two more special cases for variance: bivariant means "both covariant and contravariant," whereas invariant means "neither covariant nor contravariant." The only types which can be bivariant are phantoms, where the type doesn't actually exist. As an example:
import Data.Functor.Contravariant (Contravariant (..)) data Phantom a = Phantom instance Functor Phantom where fmap _ Phantom = Phantom instance Contravariant Phantom where contramap _ Phantom = Phantom
Invariance will occur if:

A type parameter is used multiple times in a data structure, both positively and negatively, e.g.:
data ToFrom a = ToFrom (a > Int) (Int > a)

A type parameter is used in type which is itself invariant in the parameter, e.g.:
newtype ToFromWrapper a = ToFromWrapper (ToFrom a)
Note that even though the parameter only appears once here, it appears twice in
ToFrom
itself. 
In special types like references, e.g.:
data IORef a  a is invariant newtype RefWrapper a = RefWrapper (IORef a)  also invariant
Exercise Convince yourself that you can not make an
instance of either Functor
nor
Contravariant
for ToFrom
or
IORef
.
Exercise Explain why there's also no way to make an
instance of Bifunctor
or Profunctor
for
these datatypes.
As you can see, the a
parameter is used as both the
input to a function and output from a function in
ToFrom
. This leads directly to our next set of
terms.
NOTE There's a good Reddit discussion which led to clarification of these section.
Positive and negative position
Let's look at some basic covariant and contravariant data types:
data WithInt a = WithInt (Int > a) data MakeInt a = MakeInt (a > Int)
By now, you should hopefully be able to identify that
WithInt
is covariant on its type parameter
a
, whereas MakeInt
is contravariant.
Please make sure you're confident of that fact, and that you know
what the relevant Functor
and
Contravariant
instance will be.
Can we give a simple explanation of why each of these is
covariant and contravariant? Fortunately, yes: it has to do with
the position the type variable appears in the function. In fact, we
can even get GHC to tell us this by using Functor
deriving:
{# LANGUAGE DeriveFunctor #} data MakeInt a = MakeInt (a > Int) deriving Functor
This results in the (actually quite readable) error message:
Can't make a derived instance of ‘Functor MakeInt’:
Constructor ‘MakeInt’ must not use the type variable in a function argument
In the data declaration for ‘MakeInt’
Another way to say this is "a
appears as an input
to the function." An even better way to say this is that
"a
appears in negative position." And now we get to
define two new terms:
 Positive position: the type variable is the result/output/range/codomain of the function
 Negative position: the type variable is the argument/input/domain of the function
When a type variable appears in positive position, the data type is covariant with that variable. When the variable appears in negative position, the data type is contravariant with that variable. To convince yourself that this is true, go review the various data types we've used above, and see if this logic applies.
But why use the terms positive and negative? This is where things get quite powerful, and drastically simplify your life. Consider the following newtype wrapper intended for running callbacks:
type Callback a = a > IO ()  newtype CallbackRunner a = CallbackRunner (Callback a > IO ())  Expands to: newtype CallbackRunner a = CallbackRunner ((a > IO ()) > IO ())
Is it covariant or contravariant on a
? Your first
instinct may be to say "well, a
is a function
parameter, and therefore it's contravariant. However, let's break
things down a bit further.
Suppose we're just trying to deal with a > IO
()
. As we've established many times above: this function is
contravariant on a
, and equivalently a
is
in negative position. This means that this function expects on
input of type a
.
But now, we wrap up this entire function as the input to a new
function, via: (a > IO ()) > IO ()
. As a
whole, does this function consume an a
, or does
it produce an a
? To get an intuition, let's
look at an implementation of CallbackRunner Int
for
random numbers:
supplyRandom :: CallbackRunner Int supplyRandom = CallbackRunner $ \callback > do int < randomRIO (1, 10) callback int
It's clear from this implementation that
supplyRandom
is, in fact, producing an
Int
. This is similar to Maybe
, meaning we
have a solid argument for this also being covariant. So let's go
back to our positive/negative terminology and see if it explains
why.
In a > IO ()
, a
is in negative
position. In (a > IO ()) > IO ()
, a
> IO ()
is in negative position. Now we just follow
multiplication rules: when you multiply two negatives, you get a
positive. As a result, in (a > IO ()) > IO ()
,
a
is in positive position, meaning that
CallbackRunner
is covariant on a
, and we
can define a Functor
instance. And in fact, GHC agrees
with us:
{# LANGUAGE DeriveFunctor #} import System.Random newtype CallbackRunner a = CallbackRunner { runCallback :: (a > IO ()) > IO () } deriving Functor supplyRandom :: CallbackRunner Int supplyRandom = CallbackRunner $ \callback > do int < randomRIO (1, 10) callback int main :: IO () main = runCallback supplyRandom print
Let's unwrap the magic, though, and define our
Functor
instance explicitly:
newtype CallbackRunner a = CallbackRunner { runCallback :: (a > IO ()) > IO () } instance Functor CallbackRunner where fmap f (CallbackRunner aCallbackRunner) = CallbackRunner $ \bCallback > aCallbackRunner (bCallback . f)
Exercise 1: Analyze the above Functor
instance and understand what is occurring.
Exercise 2: Convince yourself that the above
implementation is the only one that makes sense, and similarly that
there is no valid Contravariant
instance.
Exercise 3: For each of the following newtype wrappers, determine if they are covariant or contravariant in their arguments:
newtype E1 a = E1 (a > ()) newtype E2 a = E2 (a > () > ()) newtype E3 a = E3 ((a > ()) > ()) newtype E4 a = E4 ((a > () > ()) > ()) newtype E5 a = E5 ((() > () > a) > ())  trickier: newtype E6 a = E6 ((() > a > a) > ()) newtype E7 a = E7 ((() > () > a) > a) newtype E8 a = E8 ((() > a > ()) > a) newtype E9 a = E8 ((() > () > ()) > ())
Lifting
IO
to MonadIO
Let's look at something seemingly unrelated to get a feel for
the power of our new analysis tools. Consider the base function
openFile
:
openFile :: FilePath > IOMode > IO Handle
We may want to use this from a monad transformer stack based on
top of the IO
monad. The standard approach to that is
to use the MonadIO
typeclass as a constraint, and its
liftIO
function. This is all rather
straightforward:
import System.IO import Control.Monad.IO.Class openFileLifted :: MonadIO m => FilePath > IOMode > m Handle openFileLifted fp mode = liftIO (openFile fp mode)
But of course, we all prefer using the withFile
function instead of openFile
to ensure resources are
cleaned up in the presence of exceptions. As a reminder, that
function has a type signature:
withFile :: FilePath > IOMode > (Handle > IO a) > IO a
So can we somehow write our lifted version with type signature:
withFileLifted :: MonadIO m => FilePath > IOMode > (Handle > m a) > m a
Try as we might, this can't be done, at least not directly (if
you're really curious, see liftedbase and
its implementation of bracket
). And now, we have the
vocabulary to explain this succinctly: the IO
type
appears in both positive and negative position in
withFile
's type signature. By contrast, with
openFile
, IO
appears exclusively in
positive position, meaning our transformation function
(liftIO
) can be applied to it.
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