In the last post about categories I presented the TS category (the TypeScript category) and the central problem with function composition
How can we compose two generic functions
f: (a: A) => B
andg: (c: C) => D
?
Why finding solutions to this problem is so important?
Because if categories can be used to model programmming languages, morphisms (i.e. functions in TS) can be used to model programs.
Therefore solving the problem also means to find how to compose programs in a general way. And this is pretty interesting for a developer, isn't it?
Functions as programs
We call pure program a function with the following signature
(a: A) => B
Such a signature models a program which accepts an input of type A
and yields a result of type B
, without any effect.
We call effectful program a function with the following signature
(a: A) => F<B>
Such a signature models a program which accepts an input of type A
and yields a result of type B
, along with an effect F
, where F
is some type constructor.
Recall that a type constructor is an n
ary type operator taking as argument zero or more types, and returning another type.
Example
Given the concrete type string
, the Array
type constructor returns the concrete type Array<string>
Here we are interested in n
ary type constructors with n >= 1
, for example
Type constructor  Effect (interpretation) 

Array<A> 
a non deterministic computation 
Option<A> 
a computation that may fail 
Task<A> 
an asynchronous computation 
Now back to our main problem
How can we compose two generic functions
f: (a: A) => B
andg: (c: C) => D
?
Since the general problem is intractable, we need to put some constraint on B
and C
.
We already know that if B = C
then the solution is the usual function composition
function compose<A, B, C>(g: (b: B) => C, f: (a: A) => B): (a: A) => C {
return a => g(f(a))
}
What about the other cases?
In which the constraint B = F<C>
leads to functors
Let's consider the following constraint: B = F<C>
for some type constructor F
, or in other words (and after some renaming)

f: (a: A) => F<B>
is an effectful program 
g: (b: B) => C
is a pure program
In order to compose f
with g
we could find a way to lift g
from a function (b: B) => C
to a function (fb: F<B>) => F<C>
so that we can use the usual function composition (the output type of f
would be the same as the input type of the lifted function)
So we turned the original problem into another one: can we find such a lift
function?
Let's see some examples
Example (F = Array
)
function lift<B, C>(g: (b: B) => C): (fb: Array<B>) => Array<C> {
return fb => fb.map(g)
}
Example (F = Option
)
import { Option, isNone, none, some } from 'fpts/Option'
function lift<B, C>(g: (b: B) => C): (fb: Option<B>) => Option<C> {
return fb => (isNone(fb) ? none : some(g(fb.value)))
}
Example (F = Task
)
import { Task } from 'fpts/Task'
function lift<B, C>(g: (b: B) => C): (fb: Task<B>) => Task<C> {
return fb => () => fb().then(g)
}
All those lift
functions almost look the same. It's not a coincidence, there's a functional pattern under the hood.
Indeed all those type constructors (and many others) admit a functor instance.
Functors
Functors are mappings between categories that preserve the categorical structure, i.e. that preserve identity morphisms and composition.
Since categories are constituted of two things (objects and morphisms) a functor is constituted of two things as well:
 a mapping between objects that associates to each object
X
in C an object in D  a mapping between morphisms that associates to each morphism in C a morphism in D
where C and D are two categories (aka two programming languages).
Even if a mapping between two different programming languages is an intriguing idea, we are more interested in a mapping where C and D coincide (with TS). In this case we talk about endofunctors ("endo" means "within", "inside").
From now on when I write "functor" I actually mean an endofunctor in TS.
Definition
A functor is a pair (F, lift)
where

F
is an
ary type constructor (n >= 1
) which maps each typeX
to the typeF<X>
(mapping between objects) 
lift
is a function with the following signature
lift: <A, B>(f: (a: A) => B) => ((fa: F<A>) => F<B>)
which maps each function f: (a: A) => B
to a function lift(f): (fa: F<A>) => F<B>
(mapping between morphisms).
The following properties must hold

lift(identity
_{X})
=identity
_{F(X)} (identities map to identities) 
lift(g ∘ f) = lift(g) ∘ lift(f)
(mapping a composition is the composition of the mappings)
The lift
function is also known through a variant called map
, which is basically lift
with the arguments rearranged
lift: <A, B>(f: (a: A) => B) => ((fa: F<A>) => F<B>)
map: <A, B>(fa: F<A>, f: (a: A) => B) => F<B>
Note that map
can be derived from lift
(and viceversa).
Functors in fpts
How can we define a functor instance in fpts
? Let's see a practical example.
The following declaration defines a model for the response of an API call
interface Response<A> {
url: string
status: number
headers: Record<string, string>
body: A
}
Note that the body
field is parametrized, this makes Response
a good candidate for a functor instance since Response
is a n
ary type constructors with n >= 1
(a necessary precondition).
In order to define a functor instance for Response
we must define a map
function (along with some technicalities required by fpts
)
// `Response.ts` module
import { Functor1 } from 'fpts/Functor'
export const URI = 'Response'
export type URI = typeof URI
declare module 'fpts/HKT' {
interface URItoKind<A> {
Response: Response<A>
}
}
export interface Response<A> {
url: string
status: number
headers: Record<string, string>
body: A
}
function map<A, B>(fa: Response<A>, f: (a: A) => B): Response<B> {
return { ...fa, body: f(fa.body) }
}
// functor instance for `Response`
export const functorResponse: Functor1<URI> = {
URI,
map
}
Is the general problem solved?
Not at all. Functors allow us to compose an effectful program f
with a pure program g
, but g
must be unary, that is it must accept only one argument as input. What if g
accepts two arguments? Or three?
Program f  Program g  Composition 

pure  pure  g ∘ f 
effectful  pure (unary)  lift(g) ∘ f 
effectful  pure (n ary, n > 1 ) 
? 
In order to handle such circumstances we need something more: in the next post I'll talk about another remarkable abstraction of functional programming: applicative functors.
TLDR: functional programming is all about composition
Top comments (9)
What's functorResponse good for? You defined it while dropping a ton of boilerplate from HKT without explaining it and without providing any use case. That's the point in your series when I felt like handwaving started and wasn't able to say 'Yes. I understand fully what you mean.'
However otherwise it's really great write up about category theory concepts. It got me way closer to understanding who it works and what might be the value of it than any other thing I read earlier. I especially like the parallels you drew between increasingly complex contexts (like how lift relates to map, and ap relates to unpack) and examples with Arrays, Options, Promises (I'm not sure why but Arrays seem to be hardest for me to understand, especially when talking about applicative functors, others are just specific boxes over single value, arrays are weird because they have multiple values). I also very much liked the drawing in last parts that can give intuition why a thing is called 'lift'. You might want to add some of those in this part where you first introduce 'lift'.
Great job! Thank you for that!
I spent four years learning functional programming, but, I
I really understand it after reading your blog. Thank you!
Small typo:
Great article. For typescripters, talking about generic types can be simpler than "
n
ary type constructors withn
>= 1", don't you think?"How can we compose two generic functions f: (a: A) => B and g: (c: C) => D"
Don't you mean "g: (b: B) => C"?
And thank you for writing these articles, they are very helpful :)
Composing
f: (a: A) => B
andg: (b: B) => C
is trivial by normal function composition.What is challenging is composing
f
andg
when output type off
isn't the same as input type ofg
. And that is impossible to do if these outputinput types are completely arbitrary.So what we do is we try to introduce some dependency between them ("what if output of
f
isF<input of g>
?") and see if we can somehow compose them NOW, with that constraint in place.Turns out we CAN  if F is a functor.
Taking the concrete problem of two generic functions f: (a: A) => B and g: (c: C) => D
Can you show how a Response Functor solves the composition when B != C
First time I see a map :: (m a, a > b) > m b. How do you relate it to lift precisely?
I always saw map :: (a > b) > m a > m b. Basically, the canonical map is "your" map curried.
What is the advantage of using map with your types? How does it help understand function composition?
So we can interpret Map < A , B > as a computation that returns a mapping function on A and return B?