---
title:
- Applicatives
author:
- Sanchayan Maity
theme:
- default
classoption:
- aspectratio=169
---

# Agenda

- Recap of Functors
- Applicative

# Functor[^1][^2]

```haskell
class Functor f where
    fmap :: (a -> b) -> f a -> f b
    (<$) :: a -> f b -> f a
```

Functors Laws

- Must preserve identity

```haskell
fmap id = id
```

- Must preserve composition of morphism

```haskell
fmap (f . g)  ==  fmap f . fmap g
```

[^1]: [Category Design Pattern](https://www.haskellforall.com/2012/08/the-category-design-pattern.html)
[^2]: [Functor Design Pattern](https://www.haskellforall.com/2012/09/the-functor-design-pattern.html)

# Higher order kinds[^3]

- For something to be a functor, it has to be a first order kind.

[^3]: [Haskell's Kind System](https://diogocastro.com/blog/2018/10/17/haskells-kind-system-a-primer/)

# Applicative

```haskell
class Functor f => Applicative (f :: TYPE -> TYPE) where
  pure :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b
```

```haskell
(<$>) :: Functor f =>       (a -> b) -> f a -> f b
(<*>) :: Applicative f => f (a -> b) -> f a -> f b
```

```haskell
fmap f x = pure f <*> x
```

# Examples

```haskell
pure (+1) <*> [1..3]
[2, 3, 4]

[(*2), (*3)] <*> [4, 5]
[8,10,12,15]

("Woo", (+1)) <*> (" Hoo!", 0)
("Woo Hoo!", 1)

(Sum 2, (+1)) <*> (Sum 0, 0)
(Sum {getSum = 2}, 1)

(Product 3, (+9)) <*> (Product 2, 8)
(Product {getProduct = 6}, 17)

(,) <$> [1, 2] <*> [3, 4]
[(1,3),(1,4),(2,3),(2,4)]
```

# Lifting

- Seeing Functor as unary lifting and Applicative as n-ary lifting

```haskell
liftA0 :: Applicative f => (a)                -> (f a)
liftA1 :: Functor     f => (a -> b)           -> (f a -> f b)
liftA2 :: Applicative f => (a -> b -> c)      -> (f a -> f b -> f c)
liftA3 :: Applicative f => (a -> b -> c -> d) -> (f a -> f b -> f c -> f d)
liftA4 :: Applicative f => ..
```

Where `liftA0 = pure` and `liftA1 = fmap`.

# Monoidal functors

- Remember Monoid?

```haskell
class Monoid m where
  mempty :: m
  mappend :: m -> m -> m
```

```haskell
($)   ::   (a -> b) ->   a ->   b
(<$>) ::   (a -> b) -> f a -> f b
(<*>) :: f (a -> b) -> f a -> f b

mappend ::     f          f      f
($) ::      (a -> b) ->   a ->   b
<*> ::    f (a -> b) -> f a -> f b

instance Monoid a => Applicative ((,) a) where
  pure x = (mempty, x)
  (u, f) <*> (v, x) = (u `mappend` v, f x)
```

# Function apply

- Applying a function to an `effectful` argument

```haskell
(<$>) :: Functor m     =>   (a -> b)   -> m a -> m b
(<*>) :: Applicative m => m (a -> b)   -> m a -> m b
(=<<) :: Monad m       =>   (a -> m b) -> m a -> m b
```

# Contrasts with monad

- No data dependency between `f a` and `f b`
- Result of `f a` can't possibly influence the behaviour of `f b`
- That needs something like `a -> f b`


# Applicative laws

```haskell
-- Identity
pure id <*> v = v

-- Composition
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)

-- Homomorphism
pure f <*> pure x = pure (f x)

-- Interchange
u <*> pure y = pure ($ y) <*> u
```

# Applicative vs monads

- Applicative
  * Effects
  * Batching and aggregation
  * Concurrency/Independent
    - Parsing context free grammar
    - Exploring all branches of computation (see `Alternative`)

- Monads
  * Effects
  * Composition
  * Sequence/Dependent
    - Parsing context sensitive grammar
    - Branching on previous results

# Weaker but better

- Weaker than monads but thus also more common
- Lends itself to optimisation (See Facebook's [Haxl](https://hackage.haskell.org/package/haxl) project)
- Always opt for the least powerful mechanism to get things done
- No dependency issues or branching? just use applicative

# Resources

- [Applicative Programming with Effects](https://www.staff.city.ac.uk/~ross/papers/Applicative.html)
- [optparse-applicative](https://hackage.haskell.org/package/optparse-applicative)
- [Control Applicative](https://hackage.haskell.org/package/base-4.19.1.0/docs/Control-Applicative.html)

# Questions

- Reach out on
  * Email: sanchayan@sanchayanmaity.net
  * Mastodon: [sanchayanmaity.com](https://sanchayanmaity.com/@sanchayan)
  * Telegram: [t.me/SanchayanMaity](https://t.me/SanchayanMaity)
  * Blog: [sanchayanmaity.net](https://sanchayanmaity.net/)