A great resource for information on common typeclasses in Haskell is the typeclassopedia. We recommend reading that document, and following up here for additional pointers.

Section exercises:

- Write the
`foldMapM`

helper function - Implement the
`Validation`

`Applicative`

- Why isn't it a
`Monad`

?

- Why isn't it a

- Not going to cover these in depth
- Great resource on this: typeclassopedia

`Applicative`

wasn't a superclass of`Monad`

in the past`Semigroup`

wasn't a superclass of`Monoid`

in the past- Some unnecessary functions still lying around
- Sometimes functions aren't as general as we wish

- "Mappable"
- Provides
`fmap :: (a -> b) -> (f a -> f b)`

- Laws
`fmap id == id`

`fmap (g . h) == fmap g . fmap h`

- Cool fact: only one possible valid instance per type
- Can be derived automatically
- Covariant functor (contravariant also exists)
- See: Covariance and Contravariance

Provides:

```
pure :: a -> f a
(<*>) :: f (a -> b) -> f a -> f b
```

Compare:

```
fmap :: (a -> b) -> f a -> f b
(<*>) :: f (a -> b) -> f a -> f b
```

Also note that you can define `fmap`

using `Applicative`

```
fmap g x = pure g <*> x
```

Laws:

`pure id <*> x == x`

`pure f <*> pure x == pure (f x)`

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

`u <*> (v <*> w) = pure (.) <*> u <*> v <*> w`

Provides:

```
(>>=) :: m a -> (a -> m b) -> m b
```

Or flipped:

```
(=<<) :: (a -> m b) -> m a -> m b
```

Compare:

```
fmap :: (a -> b) -> f a -> f b
(<*>) :: f (a -> b) -> f a -> f b
(=<<) :: (a -> m b) -> m a -> m b
```

Laws:

`pure a >>= f == f a`

`m >>= pure == m`

`m >>= (\x -> f x >>= g) == (m >>= f) >>= g`

And we can define:

```
(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c)
```

And then restate these laws as:

```
f <=< pure == f
pure <=< f == f
(h <=< g) <=< f == h <=< (g <=< f)
```

Which are the same as the category laws:

```
f . id == f
id . f == f
(h . g) . f == h . (g . f)
```

Summary explanation: `Semigroup`

defines a binary, associative operator.

```
(<>) :: a -> a -> a
```

Law

```
(x <> y) <> z == x <> (y <> z)
```

`Monoid`

is a subclass of `Semigroup`

, and adds an identity to `Semigroup`

.

```
mempty :: a
```

The laws are the same again as `Monad`

and categories!

```
x <> mempty == x
mempty <> x == x
(x <> y) <> z == x <> (y <> z)
```

More worked explanation: `Semigroup`

is a typeclass that provides a single binary, associative operator.
For example, for integers, `+`

and `*`

are both valid `Semigroup`

implementations.
For lists, appending two lists forms a `Semigroup`

.

`Monoid`

builds on `Semigroup`

, but adds in an identity, where it follows the law that applying that binary operator as either the left or right value to the identity is a no-op

In other words: `0 + x = x`

, `x + 0 = x`

, and `(a + b) + c = a + (b + c)`

.
Therefore: `(<>) = (+)`

and `mempty = 0`

forms a valid `Semigroup`

/`Monoid`

pair of instances.
Some things are a `Semigroup`

, but not a Monoid.
A simple example: a non-empty list. While you can append together two non-empty lists, there's no identity value you can come up with where the identity laws hold.

That's all the technical definition. Intuition: `Semigroup`

and `Monoid`

let you define a way to slam data together!

**EXERCISE** Write a data type for calculating the average of a bunch of values. The data type will need to have two fields: one to keep the running sum, one the running total. Then write `Semigroup`

and `Monoid`

instances that Do The Right Thing, define an `average`

function that calculates the average from these two fields, and you're done. Try using `fold`

(part of the `Foldable`

typeclass we'll cover next) to summarize a list of values.

- "I can be turned into a list" but more efficient in some cases.
`foldMap :: Monoid m => (a -> m) -> f a -> m`

- Can be derived automatically
- No actual laws yet
- Could define a
`Vector`

that folds left-to-right or right-to-left `length`

of tuples and other things considered surprising/wrong by many

- "Map with effects"
- Generalizes
`mapM`

`traverse`

==`mapM`

, but works for`Applicative`

`for`

==`forM`

, but for`Applicative`

See start of section