Category Theory laws in scala

In Scala, we can express the laws of Category Theory using various constructs and concepts. Category Theory provides a framework for understanding and formalizing relationships between different mathematical structures.

To define a category in Scala, we can use traits or classes to represent objects and morphisms. An object represents a mathematical entity, while a morphism represents a relationship between two objects.

Here are the three fundamental laws of Category Theory and how they can be expressed in Scala:

Identity Law:

The identity law states that for every object A in a category, there exists an identity morphism idA that behaves as an identity element with respect to function composition.

In Scala, we can define the identity morphism for an object A as follows:

trait Category[A] {
  def id: A => A
}

Associativity Law:

The associativity law states that for any three morphisms f, g, and h in a category, the composition of (f ∘ g) ∘ h is equivalent to f ∘ (g ∘ h).

In Scala, we can express this law using the composition operator ∘:

trait Category[A] {
  def compose(f: A => A, g: A => A): A => A
}

// Associativity law: (f ∘ g) ∘ h == f ∘ (g ∘ h)
def associativityLaw[A](f: A => A, g: A => A, h: A => A)(implicit category: Category[A]): Boolean = {
  val lhs = category.compose(category.compose(f, g), h)
  val rhs = category.compose(f, category.compose(g, h))
  lhs == rhs
}

Composition with Identity Law:

The composition with identity law states that for any morphism f in a category, the composition of f with the identity morphism id is equivalent to f.

In Scala, we can represent this law as follows:

trait Category[A] {
  def id: A => A
  def compose(f: A => A, g: A => A): A => A
}

// Composition with identity law: f ∘ id == f and id ∘ f == f
def compositionWithIdentityLaw[A](f: A => A)(implicit category: Category[A]): Boolean = {
  val lhs = category.compose(f, category.id)
  val rhs = category.compose(category.id, f)
  lhs == f && rhs == f
}

These are the basic laws of Category Theory that can be expressed in Scala using traits or classes. By adhering to these laws, we can ensure that our code follows the principles and relationships defined by Category Theory.