Scala Cats

Semigroup

A semigroup is any set A with an associative operation (combine). --Scala Cats

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

Monoid

A monoid is a semigroup with an identity. A monoid is a specialization of a semigroup, so its operation must be associative. Additionally, combine(x, empty) == combine(empty, x) == x. For example, if we have Monoid[String], with combine as string concatenation, then empty = "". --Scala Cats

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

Functors

Functor.
The name is shorts for "covariant functor".
Must obey the laws defined in cats.laws.FunctorLaws.--Scala Cats

trait Functor[F[_]] {
  def map[A, B](fa: F[A])(f: A => B): F[B]
}

Monads

Monad.
Allows composition of dependent effectful functions.
See: Monads for functional programming
Must obey the laws defined in cats.laws.MonadLaws.

trait Monad[F[_]] extends Functor[F] {
  def pure[A](value: A): F[A]
  def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
}