In this paper, we present a new approach to abstract syntax called the duplication comonoid over E : Set. This concept is designed to help us reason about the contextual information that is carried by wires in functional programming. We view these wires as carrying "contextual" information, which we represent graphically in red. Our goal is to make this information more manageable and easier to understand by organizing it into a comonoid structure.
Definition 3.1: The duplication comonoid over E : Set
The duplication comonoid is defined by two operations: del and ∆. Del deletes an element from the original set, while ∆ creates a new element by duplicating the current element and adding the deleted element as its second part. We represent these wires graphically in red to emphasize their contextual nature.
del : E → 1
del(e) = ⋆
∆ : E → E × E
∆(e) = (e, e)
The duplication comonoid induces a comonad on (E × −) known as the environment or reader comonad. This comonad is useful for reasoning about contextual information in functional programming.
Definition 3.2: The environment comonad over E : Set
The environment comonad is defined by the product functor (E × −) and two operations of extraction and duplication. Extraction extracts an element from the original set, while duplication creates a new element by duplicating the current element and adding the extracted element as its second part.
extr : ∀ (A : Set) , E × A → A
extr(e, a) = a
dup : ∀ (A : Set) , E × A → E × (E × A)
dup(e, a) = (e, (e, a))
Conclusion
In summary, this paper presents a new approach to abstract syntax called the duplication comonoid over E : Set. This concept is designed to help us reason about contextual information in functional programming by organizing it into a comonoid structure. The duplication comonoid induces a comonad on (E × −) known as the environment or reader comonad, which is useful for reasoning about contextual information. By using this approach, we can better understand and manage the complex contextual relationships that arise in functional programming.
