Grp 

Awodey:

Definition 1.4 A group G is a monoid with an inverse g-1 for every element g. Thus, G is a category with one object, in which every arrow is an isomorphism.

Here’s the typeclass contract of cats.kernel.Monoid:

/**
 * A group is a monoid where each element has an inverse.
 */
trait Group[@sp(Int, Long, Float, Double) A] extends Any with Monoid[A] {

  /**
   * Find the inverse of `a`.
   *
   * `combine(a, inverse(a))` = `combine(inverse(a), a)` = `empty`.
   */
  def inverse(a: A): A
}

This enables inverse method if the syntax is imported.

scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._

scala> 1.inverse
res0: Int = -1

scala> assert((1 |+| 1.inverse) === Monoid[Int].empty)

The category of groups and group homomorphism (functions that preserve the monoid structure) is denoted by Grp.

Forgetful functor 

We’ve seen the term homomorphism a few times, but it’s possible to think of a function that doesn’t preserve the structure. Because every group G is also a monoid we can think of a function f: G => M where f loses the inverse ability from G and returns underlying monoid as M. Since both groups and monoids are categories, f is a functor.

We can extend this to the entire Grp, and think of a functor F: Grp => Mon. These kinds of functors that strips the structure is called forgetful functors. If we try to express this using Scala, you would start with A: Group, and somehow downgrade to A: Monoid as the ruturn value.

That’s it for today.