# Invariant

The `Invariant`

type class is for functors that define an `imap`

function with the following type:

```
def imap[A, B](fa: F[A])(f: A => B)(g: B => A): F[B]
```

Every covariant (as well as contravariant) functor gives rise to an invariant
functor, by ignoring the `g`

(or in case of contravariance, `f`

) function.

Examples for instances of `Invariant`

are `Semigroup`

and `Monoid`

, in
the following we will explain why this is the case using `Semigroup`

, the
reasoning for `Monoid`

is analogous.

## Invariant instance for Semigroup

Pretend that we have a `Semigroup[Long]`

representing a standard UNIX
timestamp. Let’s say that we want to create a `Semigroup[Date]`

, by
*reusing* `Semigroup[Long]`

.

### Semigroup does not form a covariant functor

If `Semigroup`

had an instance for the standard covariant `Functor`

type class, we could use `map`

to apply a function `longToDate`

:

```
import java.util.Date
def longToDate: Long => Date = new Date(_)
```

But is this enough to give us a `Semigroup[Date]`

? The answer is no,
unfortunately. A `Semigroup[Date]`

should be able to combine two
values of type `Date`

, given a `Semigroup`

that only knows how to
combine `Long`

s! The `longToDate`

function does not help at all,
because it only allows us to convert a `Long`

into a `Date`

. Seems
like we can’t have an `Functor`

instance for `Semigroup`

.

### Semigroup does not form a contravariant functor

On the other side, if `Semigroup`

would form a *contravariant* functor
by having an instance for `Contravariant`

, we could make use of
`contramap`

to apply a function `dateToLong`

:

```
import java.util.Date
def dateToLong: Date => Long = _.getTime
```

Again we are faced with a problem when trying to get a
`Semigroup[Date]`

based on a `Semigroup[Long]`

. As before consider
the case where we have two values of `Date`

at hand. Using
`dateToLong`

we can turn them into `Long`

s and use `Semigroup[Long]`

to combine the two values. We are left with a value of type `Long`

,
but we can’t turn it back into a `Date`

using only `contramap`

!

### Semigroup does form an invariant functor

From the previous discussion we conclude that we need both the `map`

from (covariant) `Functor`

and `contramap`

from `Contravariant`

.
There already is a type class for this and it is called `Invariant`

.
Instances of the `Invariant`

type class provide the `imap`

function:

```
def imap[A, B](fa: F[A])(f: A => B)(g: B => A): F[B]
```

Reusing the example of turning `Semigroup[Long]`

into
`Semigroup[Date]`

, we can use the `g`

parameter to turn `Date`

into a
`Long`

, combine our two values using `Semigroup[Long]`

and then
convert the result back into a `Date`

using the `f`

parameter of
`imap`

:

```
import java.util.Date
// import everything for simplicity:
import cats._
import cats.implicits._
def longToDate: Long => Date = new Date(_)
def dateToLong: Date => Long = _.getTime
implicit val semigroupDate: Semigroup[Date] =
Semigroup[Long].imap(longToDate)(dateToLong)
val today: Date = longToDate(1449088684104l)
val timeLeft: Date = longToDate(1900918893l)
```

```
today |+| timeLeft
// res1: Date = Thu Dec 24 15:40:02 EST 2015
```