"Set" Category

def. $Set$ is a category where its objects are sets, and morphisms are standard functions. Consider the following objects:

• Empty set: $\text{Void}$. You can have morphism $\text{Absurd}::\text{Void}\to \text{Int}$.
  • You can define morphism from void to any object.
• Singleton set: $\text{Unit}$. You can have morphism from unit to anything, and anything to unit.
  • $\text{unitmorph}::a\to \text{Unit}$
  • $\text{one}::\text{Unit}\to \text{Int}$, always returns $1$.
    • ! This is like generating elements without looking at the elements of the codomain set. Great abstraction!

Category Theory Definition §

def. Terminal (=Unit) Object. ”$()$” is a unit object iff both:

• $\forall a.\exists f::a\to f$
• $\forall a.\forall f::a\to ().g.::a\to ()⟹f=g$ …
• i.e. there is one unique arrow arriving from every object to ”$()$”, then $()$ is a terminal.
• This is terminal because all arrows end that the terminal object.
• This is equivalent to a singleton set in set theory. def. Initial (=Void) Object. $v$ is a initial object iff:
• $\forall a.\exists f::v\to a$
• $\forall a.\forall f::v\to a.g::v\to a⟹f=g$
• This is initial because all arrows start from that initial object.
• This is equivalent to empty set in set theory.