Lincoln's Notes

##Type Classes, Monoids, and Functors with Scala's Cats Library Accompanying code can be found on GitHub

No Encapsulation in Functional Programming

In OOP, functions and data are bundled together in classes.

In Functional Programming, data and functions are kept separate. And a multitude of programming techniques are available to create, combine, and edit functions.

We will study a use case of an "Order Processing System" to understand how the FP style of Type Classes varies from the traditional OOP style.

OOP Style

OOP can emulate the Functional Programming style of separating data and functions.

For e.g. in an OOP Order Processing System, there could be an Order class with order information. And, all functions that process Order objects could be kept in a separate OrderProcessing class.

Here is a dummy implementation of such an approach: It consists of

  • Data Definition
  • Functions To Process The Data

Data For Order Processing

Accompanying code can be found on GitHub

case class PlainOrder(val orderId:Int, 
                      val itemId:Int, 
                      val amount:Int)

// Specialized Order derived from PlainOrder
case class ExpressOrder(override val orderId:Int, 
                        override val itemId:Int, 
                        override val amount:Int, expressCharges:Int) 
    extends PlainOrder(orderId, itemId, amount)

// Order Container. 
//List is Co Variant so orders can be List[PlainOrder] or List[ExpressOrder]
class OrderBasket(val orders:List[PlainOrder]);

// Another Order class, but unrelated to the PlainOrder inheritance hierarchy
class UnrelatedMarsOder(info:String);

Functions For Order Processing

Accompanying code can be found on GitHub

The order processing functionality can be enclosed in a class/object, keeping it separate from order data:

object Status {
  val Success = 1;
  val Failure = 0;

object OOOrderProcessor {

  def processOrder(plainOrder: PlainOrder) = Status.Success

  def processOrder(expressOrder: ExpressOrder) =  Status.Success

  def processOrderBasket(orderBasket: OrderBasket):List[Int]  = => processOrder(o))

  def processUnrelatedMarsOrder(marsOder: UnrelatedMarsOder):Int = Status.Success

Functional Style Using Type Classes

Lets see how we can use Type Classes to create an order processing system similar to the OOP one we defined above.

A Type Class defines some functionality just like an Interface in Java.

// This is a "Type Class"
trait OrderProcessor[A] {

  def processOrder(value:A) :Int


Members of the Type Class do not "implement" the Type Class Interface. Rather, to make PlainOrder a member of the OrderProcessor Type Class, OrderProcessor[PlainOrder] has to be defined.

The reason each of the instancesis marked implicit will become clear in later sections.

Accompanying code can be found on GitHub

object OrderProcessorInstances {

  // adding PlainOrder to the OrderProcessor Type Class.
  implicit val plainOrderProcessor:OrderProcessor[PlainOrder] = (value: PlainOrder) => Status.Success

  // adding ExpressOrder to the OrderProcessor Type Class
  implicit val expressOrderProcessor:OrderProcessor[ExpressOrder] = (value: ExpressOrder) => Status.Success

  // adding List[PlainOrder] to the OrderProcessor Type Class
  implicit val orderBasketProcessor:OrderProcessor[List[PlainOrder]] = (orderBasket: List[PlainOrder]) => Status.Success

  // adding UnrelatedMarsOrder to the OrderProcessor Type Class
  implicit val marsOrderProcessor:OrderProcessor[UnrelatedMarsOder] = (value: UnrelatedMarsOder) => Status.Success


Then, the Type Class instances have to be "connected" to the Types. There are two ways of doing it

  • Interface Objects
  • Interface Syntax

Interface Objects

Interface Objects simply define a function with an instance as an implicit parameter.

object InterfaceObjectForOrderProcessing {

  def processOrder[A](a:A)(implicit orderProcessorInstance:OrderProcessor[A]) = orderProcessorInstance.processOrder(a)


When Interface Object functions are invoked, the instances have to be in scope.

object TestOrderProcessingTypeClass {

  import OrderProcessingInstances._

  val order = Order(1,2,3)


Interface Syntax

The second way to "connect" Type Class instances to Classes is by defining Interface Syntax

object InterfaceSyntaxForOrderProcessing {

    implicit class OrderProcessorImplicitClass[A](value:A) {

      def processOrder(implicit processr:OrderProcessor[A]) = processr.processOrder(value)


The Interface Syntax invocation is more elegant than Interface Objects

object TestOrderProcessingTypeClassInterfaceSyntax {

  import OrderProcessingInstances._

  val order = Order(1,2,3)


We have seen how the syntax for OOP differs from that of FP. At first glance, the Type Class syntax seems to be more onerous than the plain and straightforward OOP syntax.

We must note that this syntax makes it possible to extend membership of the Type Class in ways that would not have been possible without this new syntax.

Consider a Type Class called Russian, with just one method translateToRussian():String

Ordinarily, we could not have been able to invoke "Hello Word".translateToRussian() on a String instance , but with the Type Class syntax we just discussed, it is possible to make that happen. Similarly, there are libraries that add all kinds of additional functionality to existing Scala classes.

Cats Is A Type Class Library

We will take a look at the Cats Library. It is a collection of Type Class definitions and instances that work with most of the Scala Standard Library classes.

Adding Cats Dependency To sbt

The following dependency needs to to be added to the sbt build file to use Cats

libraryDependencies += "org.typelevel" %% "cats-core" % "2.0.0-M4"

Adding Cats Dependency To Scala REPL

To use Cats in the Scala REPL, create a file 'build.sbt', add the following

name := "start_cats"
version := "0.1"
scalaVersion := "2.13.1"
libraryDependencies += "org.typelevel" %% "cats-core" % "2.0.0-M4"

Then, run sbt in the same directory


Once in sbt, type

import cats._
import cats.implicits._

Cats defines more than 25 Type Classes A lot of functionality is available for application programmers.

Lets take a look at a few of the simple ones first:

  • Show
  • Eq

Later on, in this article, we can explore:

  • Monoid
  • Functor

A future post will be dedicated to :

Show Type Class

Accompanying code can be found on GitHub The Show Type Class defines functionality similar to the toString() Object method in Java. It is a good introduction to the Type Classes provided in the Cats Library.

The Show Type Class defines a single function

trait Show[A] {

    def show(a: A): String

Let us see all the ways a Cats Type Class can be used in our application code. Cats provides Show instances/implementation for many of the Scala Standard Library Classes.

  import cats._
  import cats.implicits._

  Map(1->2, 3->4).show

All the above are using Interface Syntax.

Interface Definition can also be used


The Show instance for a custom class Order can be defined as

  case class Order(id:Int, price:Int)

  import cats._
  import cats.implicits._

  implicit val orderShower:Show[Order] = (t: Order) => f"price is ${t.price} id is ${}"

Above is the same as

  implicit val orderShower:Show[Order] = new Show[Order] {

      override def show(t: Order): String = f"price is ${t.price} id is ${}"

Then show can be invoked on Order instances

  val newOrder = Order(id=1, price = 100)

A function that accepts any Show Type Class members can be defined as

  def processAnyShow[A](a:A)(implicit shower:Show[A]) =

  processAnyShow(Map(1->2, 3->4))

A function that accepts Orders that are Show Type Class members can be defined as

  def processOrdersHavingShow[A>:Order](order:A)(implicit shower:Show[A]) = f"${} ${}"

Eq Type Class

Accompanying code can be found on GitHub Eq Type Class does type safe equality comparison.

Scala already has the == method. So, why does Cats library provide another Type Class for equality checking?

To examine some of the drawbacks of the == method, lets consider a code snippet. Suppose we erroneously type in the following piece of code to filter out non 1 elements in a list. We want our final list to only have 1

  val allOnes = List(1, 1, 0, 1 ).filter(_ == "1") //bug in code ... should be 1, not "1"
                                                   //compiler will not detect error

Because of the bug in our code, allOnes will be an empty list. We expected to get List(1,1,1). The compiler did not warn us that we are comparing a String "1" with the Ints in the list.

If Eq Type Class was used instead, the compiler would detect the error

  import cats._
  import cats.implicits._

  List(1,2,3).filter(x => Eq[Int].eqv(x, "2")) //error caught. Using Interface Definition.
  List(1,2,3).filter(x => x === "2") //error caught. Using Interface Syntax.

Cats provides Eq instancesfor many of the Scala Standard Library Classes. The negation of === is =!=

Monoid Type Class

Accompanying code can be found on GitHub

In Discreet Maths, A Monoid consists of a Set, a Binary Operation, and an Identity element from the Set. Two rules have to be satisfied:

  • The result of the Binary Operation on any two elements of the Set should result in another element of the Set.

  • And the Binary Operation applied to any element of the Set with the Identity element as the second operand should yield the the original element.

For e.g., Set of Integers, the operation of addition, and the identity element of 0 form a Monoid. Adding any two integers yields another integer which satisfies the first rule, and adding an integer to the identity element of 0 yields the original integer which satisfies the second rule.

Similarly, the Set of Integers, the multiplication operation,and the identity element of 1 also forms a Monoid.

In Cats, the Monoid Type Class is defined as follows:

trait Semigroup[A] {
  def combine(x: A, y: A): A

trait Monoid[A] extends Semigroup[A] {
  def empty: A

Like for other Type Classes, Cats provides Monoid instances for many of the Scala Standard Library Classes.

  2 |+| 3  // Interface Syntax
  Monoid[Int].combine(2,3) //Interface Definition
  3 |+| 2 + Monoid[Int].empty // empty is the Identity Element
  Option(2) |+| Option(3) |+| Option(6)
  Map(1->2) |+| Map(2->3) |+| Map.empty
  List(1,2,3) |+| List(4,5,6) |+| List(7)

|+| is the Interface Syntax for Monoid.

The instance provided by Cats is quite useful in many daily programming tasks. For e.g. a List of Maps could be combined like below

  val listOfMaps = List(Map(1->1, 2->2), Map(3->3, 4->4), Map(5->5)
  listOfMaps.reduce( _ |+| _)

Type Classes For Generics

Before we discuss the next Type Class , it would be helpful to have a short discussion about Generics. Generics are classes that take a Type parameters. Most of us are familiar with the Java and Scala collection classes. Collection classes mostly employ Generics as Containers. However, as we will see later in this post, there are other ways in which Generics can be used.

###Producer And Consumer Methods In Generics The methods in a Generic class can be classified as:

  • Consumer Methods: Methods that have a parameter of Type T ... methods that consume T
  • Producer Methods: Methods that return instances of Type T ... methods that produce T This classification is critical for understanding of the next type class: Functor

Generic As Containers

We usually think of Generics as Containers. For e.g. List[T], and Array[T] are Generics that are Containers In general, Containers of Type T store data of type T and have methods to return stored data. for e.g.

class Container[+A](val item:A)

The reason for use +A instead of just A will become clear later in this section. Also note taht the parameter is a val, not a var.

Containers Are Covariant(+)

It makes sense for Containers to be Covariant. i.e. A Container[Apple] should be treated as Container[Fruit]. Any instance of Container[Fruit] should be replaceable by an instance of Container[Apple]. This feature wherein Container[A] is a subclass of Container[B] if A is a subclass of B is called Covariance We defined Conatiner[+A] with the + sign to indicate covariance. Lets take a look at some code to see how this works.

  class Fruit

  class Apple extends Fruit

  // this is allowed because Container is covariant i.e. it is defined as Conatiner[+A]
  val appleContainer:Container[Apple] = new Container[Apple](new Apple())
  val fruitContainer:Container[Fruit] = appleContainer

The fruitContainer now points to an appleContainer. This is fine since both appleContainer and fruitContaier can produce Fruit.

Can we put a Fruit in the fruitContainer? Suppose we had such a Consumer Method defined on Container:

def put(newItem:A) {item = newItem}

then the following invocation would result in puttting a Fruit in an Apple container because fruitContainer actually points to an appleContainer

    fruitContainer.put(new Fruit()) // trying to change appleContainer item to Fruit

and this would corrupt the appleContainer. If we then called get() on the appleContainer, it would return a Fruit, not Apple.

Fortunately,Consumer Methods are not allowed on types that are declared as Covariant(+A). Only Producer Methods are allowed. So Covariant Types work well with the inheritance hierarchy but do not allowConsumer Methods.

Covariant Functors

Say we have a Container[Apple] and we want Container[AppleSauce].

To convert Container[Apple] to Container[AppleSauce], all we need is a function that converts Apple to AppleSauce i.e. MappingFunction:Apple=>AppleSauce

This seems obvious! However, the above statement cannot be generalized to all Generic Types. It is only applicable to Generics that are Covariant.

In general, if we have any Covariant Type F[+A] and a function f:A=>B, we can create F[B]

Let us see how we can define a method to convert F[A] to F[B] given a function f:A=>B:

abstract class CovariantType[+A]() { self =>

  def someProducerMethod:A

  def changeContainerAtoContainerB[B](f:A=>B): CovariantType[B] =  new CovariantType[B] {

    override def som`Producer`Method: B = f(self.someProducerMethod)


The utility method changeContainerAtoContainerB, that takes in f:A=>B and returns F[B] is usually called map. All it has to do is subclass the Container and enclose any original Producer Methods within f

The Container Type together with the Function forms Covariant Functor. The Functor Type Class is defined for Generics.

Why Consumer Methods Will Not Work

If the Container type had a Consumer Method, the above strategy of subclassing(anonymous subclassing) the Container would not work. We will not be able to define new CovariantType[B] using methods of CovariantType[A] like we did above.

The Consumer Methods in new CovariantType[B] would need to process process/consume a B instance. Could we somehow use the Consumer Method definition in CovariantType[A] to redefine the same Consumer Method for CovariantType[B] like we did for the Producer Methods?

No. The Container[A] Consumer Method processes an A instance.

And the subclassed Container[B] Consumer Method would need to consume/process a B instance.

What we would need is a function that converts B to A.

What we have is a function that converts A to B.

Why Do We Care About Covariant Functors

Covariant Functors allow us to compose and build chains


Without the map method, the same code would have to be written as

    val originalList = List(1,2,3)
    val firstChange =  firstMap(originalList)
    val secondChange = secondMap(firstChange)
    val result = thirdMap(secondChange)

There is a important case that we have not yet considered while chaining the map calls. Is the map allowed to fail?

Consider the following code snippet:


The second chain has a problem. What should convertStringToInt do when supplied with an illegal value i?

One solution would be to just throw an Exception.

Or maybe the it should return some kind of error code.

The Functor pattern does not have the ability to handle intermediate errors in a chain. We will have use a slightly different pattern. Mondad. We will examine Mondad in the next post.

Contravariant Functors

In the last section we saw how Generics of Type A could be used as Containers of Type A. And it makes sense to make them Covariant.

There is another way we could use Generic Types: A Generic of Type T may be thought of as a Service Provider or Processor of Type T.

Consider the following snippet:

class SweetnessChecker[-A]() {
    def isSweet(a:A):Boolean = False;

SweetnessChecker[Apple] is not a Container. It does not store any value. It processes or is a ServiceProvider for the Apple Type.

Service Types Are Contravariant

Suppose, we have a hierarchy of classes Green Apple < Apple < Fruit and function that process Type Apple:

def checkApple(apple:Apple):Boolean = {
    val sweetnessChecker = new SweetnessChecker[Apple]();
    return sweetnessChecker.isSweet(apple)

In the function checkApple, could we change

  val sweetnessChecker = new SweetnessChecker[Apple]();


  val sweetnessChecker = new SweetnessChecker[GreenApple]` 

No. Because SweetnessChecker[GreenApple].isSweet() takes in GreenApple. Our function checkApple has an ordinary Apple.

But we can change it to

 val sweetnessChecker = new SweetnessChecker[Fruit]()

That is Fruit.isSweet(Fruit) takes in a Fruit and we can pass in the Apple that we have.

This seems counter intuitive at first: The SweetnessChecker[Fruit] seems to be subclass of SweetnessChecker[Apple] even though Apple is a subclass of Fruit.

Service Types have Consumer Methods and Types with Consumer Methods are Contravariant.

Contravariant Functors

Contravariant Types also have an interesting property.

If we have a contravariant type F[A] and a function Z => A then we can create type F[Z]

Say we have Covariant Type SweetnessChecker[Apple]

class SweetnessChecker[Apple]() {
    def isSweet(a:Apple):Boolean = False;

and a Function which converts Fruit to Apple

    def fruitToAppleConverter(fruit:Fruit):Apple = new Apple()

then we can have create Type SweetnessChecker[Fruit]

class SweetnessChecker[Fruit] () {
    def isSweet(f:Fruit):Boolean = new SweetnessChecker[Apple]().isSweet(fruitToApple(f)))

This Generic Type along with the converter function forms a Contravariant Functor

The utility function that applies the converter function is usually called contramap

How would we define contramap?

abstract class SweetnessChecker[-A]() { self=>
  def isSweet(a: A): Boolean

  def contramap[Z](f:Z=>A) = new SweetnessChecker[Z] {
    override def isSweet(z: Z): Boolean = self.isSweet(f(z))

Can we figure out why Producer Methods cannot be supported by Contravariant Functors? Take a look how the Producer Methods are handled in the map definition of Container class.

Why Do We Care About Contravariant Functors?

Just like Covariant Functors, Contravariant Functors allow us to compose and build chains, by prepending functions:


The first contramap() yields SweetnessChecker[Fruit], and the second contramap converts SweetnessChecker[Fruit] to SweetnessChecker[Eatable] The conversions are happening in reverse() order of the inheritance hierarchy.

Invariant Functors

  • Covariant Types can have Producers but no Consumers
  • Contravariant Types can have Consumers but no Producers

If a Generic Type has Producer as well asConsumer Methods,it is Invariant ...neither Covariant nor Contravariant. Suppose we have a Generic Type F[A] with both Producers and Consumers. How can we convert it to F[B] using the strategy of anonymous subclassing that we used for Covariant and Contravariant Functors?

F[A] will have Producers. i.e. methods that return A. We will need to somehow modify these methods to return B instead. So we will need a function to convert A=>B

F[A] will haveConsumers i.e. methods that consume A. We will need to somehow modify these methods to accept B instead. Se we need a function that will convert B=>A and feed it to the Consumer methods.

So, in order to convert F[A] to F[B] we will need two functions: A=>B and B=>A

Consider a class that has Producer andConsumer Methods. Lets try to creat an imap method to convent a Generic Type with Producers and Consumers from A to B:

abstract class DollarConverterFor[A] { self =>

  /`Consumer Method` consumes A
  def toDollar(v:A):Double

  /`Producer Method` produces A
  def fromDollar(y:Double):A

  def imap[B](f:A=>B)(g:B=>A):DollarConverterFor[B] = new DollarConverterFor[B] {

    // Use B=>A and invoke the toDollar() method from Dollar[A]
    override def toDollar(v: B): Double = self.toDollar(g(v))
    // same as contramap

    // Use A=>B to convert the result of the method from Dollar[A]
    override def fromDollar(y: Double): B =  f(self.fromDollar(y))
    // same as map

The Generic Type along with the two functions form an Invariant functor.

Why Do We Care About Invariant Functors

Invariant Functors allow chaining in both directions. Consider the following code:

case class Euro(euroValue:Double)

case class Yen(yenValue:Double)

case class Pound(poundValue:Double)

val euroToPound = (e:Euro) => Pound(e.euroValue/2)   // A => B

val poundToEuro = (p:Pound) => Euro(p.poundValue *2) // B => A

val euroAndDollar =  new DollarConverterFor[Euro] () {

  override def toDollar(v: Euro): Double = v.euroValue/0.75

  override def fromDollar(y: Double): Euro = new Euro(y * 0.75)

//chaining forward
val euroAndPound = euroAndDollar.imap(euroToPound)(poundToEuro)

chaining backward
val anotherEuroAndDollar = euroAndPound.imap(poundToEuro)(euroToPound)

We have DollarConveterTo[Euro]. How can we get DollarConverterTo[Pound]? All we need is Euro=>Poung and Pound=>Euro

####Summary For Generic Types And Functors

Restriction Gain Use Functor Functor Use
OnlyProducer Methods,Consumer Methods Not Allowed Covariance Containers Forms a Functor with A=>B Chaining
OnlyConsumer Methods,Producer Methods Not Allowed Contravariance Service Types or Processors Forms a Functor with B=>A Chaining
No Covariance or Contravariance Can haveProducer andConsumer Methods Forms a Functor with A=>B and B=>A Chaining

Functors in Cats

We have see that a Type Class is just like an interface. For e.g. the Type Class Show is defined as

trait Show[A] {
     def show(a:A):String

How would we define a Type Class on a Generic Type? A Functor Type Class would need to have a map method which converts the Generic Type from A to B using a function A=>b If we look at the definition in Cats, we see the following:

trait CovariantFunctor[F[_]] {
  def map[A, B](gt:F[A], F:A=>B):F[B]

The F[_] indicates that this Type Class is for Generic Types that has a single Type Parameter. And there is a map method that processes generic type F[A], and a function A=>B to return F[B].

As usual Cats defines Covariant functors for many of the Covariant Types in the Scala Standard Library. Many of the Scala Standard Library Classes already have a map method, so the instances provided by Cats is redundant.

Below code snipped shows how we can make our Container class a member of the Functor Type Class

import cats._

implicit val functorInstanceForContainer = new cats.Functor[Container] {

  override def map[A, B](fa: Container[A])(f: A => B): Container[B] =  new Container[B](f(fa.item))

A meta use case would be to define a function that works with all "mappable" Classes

  import cats.implicits._;

  def changeContainer[F[_], A](container:F[A])(implicit functor:cats.Functor[F])  =>1)

We can then invoke the function for which Functor instances are defined

  changeContainer(new Container("hello"))


The cats(short for Categories) library provides a lot of functionality that can make our Scala Code better. It is worth while to spend time and get to know the various Type Classes and instances that have been made available to us.


Scala With Cats by Noel Welsh and Dave Gurnell

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