# Invariant Monoidal

`InvariantMonoidal`

combines `Invariant`

and `Semigroupal`

with the addition of a `unit`

methods, defined in isolation the `InvariantMonoidal`

type class could be defined as follows:

```
trait InvariantMonoidal[F[_]] {
def unit: F[Unit]
def imap[A, B](fa: F[A])(f: A => B)(g: B => A): F[B]
def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
}
```

Practical uses of `InvariantMonoidal`

appear in the context of codecs, that is interfaces to capture both serialization and deserialization for a given format. Another notable examples is `Semigroup`

.

This tutorial first shows how `Semigroup`

is `InvariantMonoidal`

, and how this can be used create `Semigroup`

instances by combining other `Semigroup`

instances. Secondly, we present a complete example of `Codec`

for the CSV format, and show how it is `InvariantMonoidal`

. Lastly, we present an alternative definition of `InvariantMonoidal`

as a generalization of `Invariant`

, and show that both definitions are equivalent.

`Semigroup`

is `InvariantMonoidal`

**

As explained in the `Invariant`

tutorial, `Semigroup`

forms an invariant functor. Indeed, given a `Semigroup[A]`

and two functions `A => B`

and `B => A`

, one can construct a `Semigroup[B]`

by transforming two values from type `B`

to type `A`

, combining these using the `Semigroup[A]`

, and transforming the result back to type `B`

. Thus to define an `InvariantMonoidal[Semigroup]`

we need implementations for `unit`

and `product`

.

To construct a `Semigroup`

from a single value, we can define a trivial `Semigroup`

with a combine that always outputs the given value. A `Semigroup[(A, B)]`

can be obtained from two `Semigroup`

s for type `A`

and `B`

by deconstructing two pairs into elements of type `A`

and `B`

, combining these element using their respective `Semigroup`

s, and reconstructing a pair from the results:

```
import cats.Semigroup
def unit: Semigroup[Unit] = (_, _) => ()
def product[A, B](fa: Semigroup[A], fb: Semigroup[B]): Semigroup[(A, B)] = {
case ((xa, xb), (ya, yb)) => fa.combine(xa, ya) -> fb.combine(xb, yb)
}
```

Given an instance of `InvariantMonoidal`

for `Semigroup`

, we are able to combine existing `Semigroup`

instances to form a new `Semigroup`

by using the `Semigroupal`

syntax:

```
import cats.implicits._
// Let's build a Semigroup for this case class
case class Foo(a: String, c: List[Double])
implicit val fooSemigroup: Semigroup[Foo] = (
(implicitly[Semigroup[String]], implicitly[Semigroup[List[Double]]])
.imapN(Foo.apply)(Function.unlift(Foo.unapply))
)
```

Our new Semigroup in action:

```
Foo("Hello", List(0.0)) |+| Foo("World", Nil) |+| Foo("!", List(1.1, 2.2))
// res1: Foo = Foo(a = "HelloWorld!", c = List(0.0, 1.1, 2.2))
```

`CsvCodec`

is `InvariantMonoidal`

**

We define `CsvCodec`

, a type class for serialization and deserialization of CSV rows:

```
type CSV = List[String]
trait CsvCodec[A] {
def read(s: CSV): (Option[A], CSV)
def write(a: A): CSV
}
```

The `read`

method consumes columns from a CSV row and returns an optional value and the remaining CSV. The `write`

method produces the CSV representation of a given value.

Beside the composition capabilities illustrated later in this tutorial, grouping both serialization and deserialization in a single type class has the advantage to allows the definition of a law to capture the fact that both operations play nicely together:

`forAll { (c: CsvCodec[A], a: A) => c.read(c.write(a)) == ((Some(a), List()))`

Let's now see how we could define an `InvariantMonoidal`

instance for `CsvCodec`

. Lifting a single value into a `CsvCodec`

can be done "the trivial way" by consuming nothing from CSV and producing that value, and writing this value as the empty CSV:

```
trait CCUnit {
def unit: CsvCodec[Unit] = new CsvCodec[Unit] {
def read(s: CSV): (Option[Unit], CSV) = (Some(()), s)
def write(u: Unit): CSV = List.empty
}
}
```

Combining two `CsvCodec`

s could be done by reading and writing each value of a pair sequentially, where reading succeeds if both read operations succeed:

```
trait CCProduct {
def product[A, B](fa: CsvCodec[A], fb: CsvCodec[B]): CsvCodec[(A, B)] =
new CsvCodec[(A, B)] {
def read(s: CSV): (Option[(A, B)], CSV) = {
val (a1, s1) = fa.read(s)
val (a2, s2) = fb.read(s1)
((a1, a2).mapN(_ -> _), s2)
}
def write(a: (A, B)): CSV =
fa.write(a._1) ++ fb.write(a._2)
}
}
```

Changing a `CsvCodec[A]`

to `CsvCodec[B]`

requires two functions of type `A => B`

and `B => A`

to transform a value from `A`

to `B`

after deserialized, and from `B`

to `A`

before serialization:

```
trait CCImap {
def imap[A, B](fa: CsvCodec[A])(f: A => B)(g: B => A): CsvCodec[B] =
new CsvCodec[B] {
def read(s: CSV): (Option[B], CSV) = {
val (a1, s1) = fa.read(s)
(a1.map(f), s1)
}
def write(a: B): CSV =
fa.write(g(a))
}
}
```

Putting it all together:

```
import cats.InvariantMonoidal
implicit val csvCodecIsInvariantMonoidal: InvariantMonoidal[CsvCodec] =
new InvariantMonoidal[CsvCodec] with CCUnit with CCProduct with CCImap
```

We can now define a few `CsvCodec`

instances and use the methods provided by `InvariantMonoidal`

to define `CsvCodec`

from existing `CsvCodec`

s:

```
val stringCodec: CsvCodec[String] =
new CsvCodec[String] {
def read(s: CSV): (Option[String], CSV) = (s.headOption, s.drop(1))
def write(a: String): CSV = List(a)
}
def numericSystemCodec(base: Int): CsvCodec[Int] =
new CsvCodec[Int] {
def read(s: CSV): (Option[Int], CSV) =
(s.headOption.flatMap(head => scala.util.Try(Integer.parseInt(head, base)).toOption), s.drop(1))
def write(a: Int): CSV =
List(Integer.toString(a, base))
}
```

```
case class BinDec(binary: Int, decimal: Int)
val binDecCodec: CsvCodec[BinDec] = (
(numericSystemCodec(2), numericSystemCodec(10))
.imapN(BinDec.apply)(Function.unlift(BinDec.unapply))
)
case class Foo(name: String, bd1: BinDec, bd2: BinDec)
val fooCodec: CsvCodec[Foo] = (
(stringCodec, binDecCodec, binDecCodec)
.imapN(Foo.apply)(Function.unlift(Foo.unapply))
)
```

Finally let's verify out CsvCodec law with an example:

```
val foo = Foo("foo", BinDec(10, 10), BinDec(20, 20))
// foo: Foo = Foo(
// name = "foo",
// bd1 = BinDec(binary = 10, decimal = 10),
// bd2 = BinDec(binary = 20, decimal = 20)
// )
val fooCsv = fooCodec.write(foo)
// fooCsv: CSV = List("foo", "1010", "10", "10100", "20")
fooCodec.read(fooCsv)
// res2: (Option[Foo], CSV) = (
// Some(
// value = Foo(
// name = "foo",
// bd1 = BinDec(binary = 10, decimal = 10),
// bd2 = BinDec(binary = 20, decimal = 20)
// )
// ),
// List()
// )
fooCodec.read(fooCodec.write(foo)) == ((Some(foo), List()))
// res3: Boolean = true
```

`InvariantMonoidal`

as a generalization of `Invariant`

**

To better understand the motivations behind the `InvariantMonoidal`

type class, we show how one could naturally arrive to its definition by generalizing the concept of `Invariant`

functor. This reflection is analogous to the one presented in Free Applicative Functors by Paolo Capriotti to show how `Applicative`

are a generalization of `Functor`

.

Given an `Invariant[F]`

instance for a certain *context* `F[_]`

, its `imap`

method gives a way to lift two *unary* pure functions `A => B`

and `B => A`

into *contextualized* functions `F[A] => F[B]`

. But what about functions of other arity?

For instance, a value `a`

of type `A`

can be seen as a pair of nullary functions, one than given no input returns `a`

, and the other than give `a`

return no output, which we might want to lift them into a *contextualized* `F[A]`

. Similarly, given two functions of type `(A, B) => C`

and `C => (A, B)`

, we might want to *contextualize* them as functions of type `(F[A], F[B]) => F[C]`

.

The `Invariant`

instance alone does not provide either of these lifting, and it is therefore natural to define define a type class for generalizing `Invariant`

s for functions of arbitrary arity:

```
trait MultiInvariant[F[_]] {
def imap0[A](a: A): F[A]
def imap1[A, B](f: A => B)(g: B => A)(fa: F[A]): F[B]
def imap2[A, B, C](f: ((A, B)) => C)(g: C => (A, B))(fa: F[A], fb: F[B]): F[C]
}
```

Higher-arity `imapN`

can be defined in terms of `imap2`

, for example for `N = 3`

:

```
trait MultiInvariantImap3[F[_]] extends MultiInvariant[F] {
def imap3[A, B, C, D](
f: ((A, B, C)) => D,
g: D => (A, B, C),
fa: F[A],
fb: F[B],
fc: F[C]
): F[D] = (
imap2[A, (B, C), D]
(f compose { case (a, (b, c)) => (a, b, c) })
(g andThen { case (a, b, c) => (a, (b, c)) })
(fa, imap2[B, C, (B, C)](identity)(identity)(fb, fc))
)
}
```

We can observe that `MultiInvariant`

is none other than an alternative formulation for `InvariantMonoidal`

. Indeed, `imap1`

and `imap`

only differ by the order of their argument, and `imap2`

can easily be defined in terms of `imap`

and `product`

:

```
trait Imap2FromImapProduct[F[_]] extends cats.InvariantMonoidal[F] {
def imap2[A, B, C](f: ((A, B)) => C)(g: C => (A, B))(fa: F[A], fb: F[B]): F[C] =
imap(product(fa, fb))(f)(g)
}
```