Invariant Monoidal

InvariantMonoidal combines Invariant and Monoidal with the addition of a pure methods, defined in isolation the InvariantMonoidal type class could be defined as follows:

trait InvariantMonoidal[F[_]] {
  def pure[A](x: A): F[A]
  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. Other notable examples are Semigroup and Monoid.

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 pure 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 Semigroups for type A and B by deconstructing two pairs into elements of type A and B, combining these element using their respective Semigroups, and reconstructing a pair from the results:

import cats.Semigroup

def pure[A](a: A): Semigroup[A] =
  new Semigroup[A] {
    def combine(x: A, y: A): A = a

def product[A, B](fa: Semigroup[A], fb: Semigroup[B]): Semigroup[(A, B)] =
  new Semigroup[(A, B)] {
    def combine(x: (A, B), y: (A, B)): (A, B) = (x, y) match {
      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 Cartesian 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]]])

Our new Semigroup in action:

Foo("Hello", List(0.0)) |+| Foo("World", Nil) |+| Foo("!", List(1.1, 2.2))
// res5: Foo = Foo(HelloWorld!,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) => == ((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 CCPure {
  def pure[A](a: A): CsvCodec[A] = new CsvCodec[A] {
    def read(s: CSV): (Option[A], CSV) = (Some(a), s)
    def write(a: A): CSV = List.empty

Combining two CsvCodecs 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) =
        val (a2, s2) =
        ((a1 |@| a2).map(_ -> _), 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) =
        (, s1)

      def write(a: B): CSV =

Putting it all together:

import cats.InvariantMonoidal

implicit val csvCodecIsInvariantMonoidal: InvariantMonoidal[CsvCodec] =
  new InvariantMonoidal[CsvCodec] with CCPure with CCProduct with CCImap

We can now define a few CsvCodec instances and use the methods provided by InvariantMonoidal to define CsvCodec from existing CsvCodecs:

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))

case class Foo(name: String, bd1: BinDec, bd2: BinDec)

val fooCodec: CsvCodec[Foo] = (
  (stringCodec |@| binDecCodec |@| binDecCodec)

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

val foo = Foo("foo", BinDec(10, 10), BinDec(20, 20))
// foo: Foo = Foo(foo,BinDec(10,10),BinDec(20,20))

val fooCsv = fooCodec.write(foo)
// fooCsv: CSV = List(foo, 1010, 10, 10100, 20)
// res12: (Option[Foo], CSV) = (Some(Foo(foo,BinDec(10,10),BinDec(20,20))),List()) == ((Some(foo), List()))
// res13: 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 it’s 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 Invariants 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, imap0 and pure have exactly the same signature, 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)