liftA4, which seem perfectly natural when viewed in the math lab, but hold the clarity of an indecisive Darth Vader at the drive thru in any other context.
Containerwith a partially applied function inside. More specifically, we have a
Container(add(2))and we'd like to apply its
Container(3)to complete the call. In other words, we'd like to apply one functor to another.
mapthe partially applied
Container(3)merely to satisfy the monad's sequential demands.
apis a function that can apply the function contents of one functor to the value contents of another. Say that five times fast.
Container(3)as it's been set free from the jail of the nested monadic function. It's worth mentioning again that
add, in this case, gets partially applied during the first
mapso this only works when
this.$valuewill be a function and we'll be accepting another functor so we need only
mapit. And with that we have our interface definition:
An applicative functor is a pointed functor with an
apcharacter will prove useful. Before we get into it, let's explore a nice property.
fis equivalent to
aping a functor of
f. Or in properer English, we can place
xinto our container and
map(f)OR we can lift both
xinto our container and
apthem. This allows us to write in a left-to-right fashion:
of, each value gets transported to the magical land of containers, this parallel universe where each application can be async or null or what have you and
apwill apply functions within this fantastical place. It's like building a ship in a bottle.
Taskin our example. This is a prime situation where applicative functors pull their weight. Let's look at a more in-depth example.
Httpcalls will happen instantly and
renderPagewill be called when both are resolved. Contrast this with the monadic version where one
Taskmust finish before the next fires off. Since we don't need the destinations to retrieve events, we are free from sequential evaluation.
renderPageis curried or it will not wait for both
Tasksto finish. Incidentally, if you've ever had to do such a thing manually, you'll appreciate the astonishing simplicity of this interface. This is the kind of beautiful code that takes us one step closer to the singularity.
signInis a curried function of 3 arguments so we have to
apaccordingly. With each
signInreceives one more argument until it is complete and runs. We can continue this pattern with as many arguments as necessary. Another thing to note is that two arguments end up naturally in
IOwhereas the last one needs a little help from
ofto lift it into
apexpects the function and all its arguments to be in the same type.
mapis equal to
of/ap, we can write generic functions that will
apas many times as we specify:
liftA2is a strange name. It sounds like one of the finicky freight elevators in a rundown factory or a vanity plate for a cheap limo company. Once enlightened, however, it's self explanatory: lift these pieces into the applicative functor world.
liftA(N)itself, so it cannot vary in argument length.
createUsertakes two arguments, we use the corresponding
liftA2. The two statements are equivalent, but the
liftA2version has no mention of
Either. This makes it more generic and flexible since we are no longer married to a specific type.
ap. This allows for a more natural function application style and can help remove some parenthesis.
of/apis equivalent to
map. We can use this knowledge to define
chain, we get functor and applicative for free:
ap's appeal is the ability to run things concurrently so defining it via
chainis missing out on that optimization. Despite that, it's good to have an immediate working interface while one works out the best possible implementation.
apwill never change container types on us (yet another reason to favor over monads). That's not to say we cannot have multiple different effects - we can stack our types knowing that they will remain the same during the entirety of our application.
idall from within a functor shouldn't alter the value in
v. For example:
Identity.of(id)makes me chuckle at its futility. Anyway, what's interesting is that, as we've already established,
of/apis the same as
mapso this law follows directly from functor identity:
map(id) == id.
chain, and now
apfunctions. In the next chapter, we'll learn how to work better with multiple functors and disassemble them in a principled way.
safeAddfrom exercise_b to use