## Concept:

# Applicative functor

## Headline

A functor with function application within the functor

### Description

Applicative functors are described here briefly in Haskell's sense.

The corresponding type class (modulo some simplifications) looks as follows.

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

The expectation is that *pure* promotes a value to a functorial value whereas "<*>" can be seen as a variation of fmap such that a function within the functor (as opposed to just a plain function) is applied to a functorial value.

The following laws are assumed.

```
pure f <*> x = fmap f x
pure id <*> v = v
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
pure f <*> pure x = pure (f x)
u <*> pure y = pure ($ y) <*> u
```

## Illustration

We make *Maybe* and lists applicative functors:

```
instance Applicative Maybe where
pure = Just
Nothing <*> _ = Nothing
(Just f) <*> x = fmap f x
instance Applicative [] where
pure x = [x]
fs <*> xs = [ f x | f <- fs, x <- xs ]
```

Thus, in the *Maybe* case, a *Nothing* as a function makes us return a *Nothing* as result, but if the function is available then it is fmapped over the argument. In the list case, we use a list comprehension to apply all available functions too all available values.

The instances can be exercised at the Haskell prompt as follows:

```
> Just odd <*> Just 2
Just False
> [odd, even] <*> [1,2,3,4]
[True,False,True,False,False,True,False,True]
```

To see that applicative functors facilitate function application for functorial values pretty well, consider the following functorial variation on plain function application.

```
(<$>) :: Functor f => (a -> b) -> f a -> f b
f <$> x = fmap f x
```

Consider the following application.

```
> (+) <$> [1,2] <*> [3,4]
[4,5,5,6]
```

Thus, the applicative operator "<*>" is used to line up (any number of) functorial arguments and *fmap* is used for the "rest" of the application.

## Metadata

### There are no revisions for this page.

## User contributions

## User edits

## Syntax for editing wiki

For you are available next options:will make text bold.

will make text italic.

will make text underlined.

will make text striked.

will allow you to paste code headline into the page.

will allow you to link into the page.

will allow you to paste code with syntax highlight into the page. You will need to define used programming language.

will allow you to paste image into the page.

is list with bullets.

is list with numbers.

will allow your to insert slideshare presentation into the page. You need to copy link to presentation and insert it as parameter in this tag.

will allow your to insert youtube video into the page. You need to copy link to youtube page with video and insert it as parameter in this tag.

will allow your to insert code snippets from
**
@worker.
**

#### 10 most similar pages:

- Concept:Functor - 0.9096736
- Concept:Maybe monad - 0.8577724
- Concept:State monad - 0.7541657
- Concept:Writer monad - 0.6388609
- Concept:Maybe type - 0.57206786
- Concept:Fold_function - 0.53619516
- Concept:Function composition - 0.52711916
- Concept:Rose tree - 0.5129073
- Concept:Type class - 0.49407142
- Concept:Higher-order function - 0.48038262

## Syntax for editing wiki

For you are available next options:will make text bold.

will make text italic.

will make text underlined.

will make text striked.

will allow you to paste code headline into the page.

will allow you to link into the page.

will allow you to paste code with syntax highlight into the page. You will need to define used programming language.

will allow you to paste image into the page.

is list with bullets.

is list with numbers.

will allow your to insert slideshare presentation into the page. You need to copy link to presentation and insert it as parameter in this tag.

will allow your to insert youtube video into the page. You need to copy link to youtube page with video and insert it as parameter in this tag.

will allow your to insert code snippets from
**
@worker.
**