Mathematical Foundation

Karpal is built on category theory — the mathematics of structure and composition.

HKT Encoding

Higher-Kinded Types (HKTs) are types that take other types as parameters: F<A> rather than just A. Rust doesn't natively support HKTs, but GATs (Generic Associated Types, stable since Rust 1.65) provide a zero-dependency encoding:

#![allow(unused)]
fn main() {
pub trait HKT {
    type Of<T>;
}
}

A marker type like OptionF implements HKT with type Of<T> = Option<T>. This lets us write traits that are generic over the "shape" of a container.

The Functor Hierarchy

The core abstraction is the functor hierarchy:

Functor → Apply → Applicative
                 ↓
          Chain → Monad

Each level adds capabilities:

• Functor: map over a container (fmap)
• Apply: combine two containers (ap)
• Applicative: create a pure value (pure)
• Chain: sequence operations (chain / bind)
• Monad: full sequential computation

Algebraic Structure

Beyond the functor hierarchy, Karpal provides algebraic typeclasses:

• Semigroup / Monoid: associative combine + identity
• Group / AbelianGroup: monoid + inverse
• Semiring / Ring / Field: two operations with distributivity
• Lattice / BoundedLattice: join + meet (poset with all suprema/infima)
• HeytingAlgebra: bounded lattice with implication (intuitionistic logic)

The Heyting algebra is the foundation for structured emptiness — the idea that "why something is empty" carries information.