Applicative

An applicative, in functional programming and category theory, is a structure that generalizes function application to computations. It enables applying a function that is itself wrapped in a context to a value wrapped in a context, without requiring the computations to be dependent. The core operations are pure (lifting a plain value into the context) and ap (also written as <*>), which takes a contextual function and a contextual value and returns a contextual result.

In category theory terms, an applicative functor consists of a functor F equipped with operations pure: a -> F a and ap: F (a -> b) -> F a -> F b, satisfying certain laws: identity, composition, homomorphism, and interchange. In practical programming languages, these laws are reflected in the behavior of lifted values and function application within a context.

Relation to other abstractions: every applicative is a functor, and in many languages every Monad is also an Applicative. However, the converse is not guaranteed; some applicatives do not support the monadic bind operation, so they allow combining effects without sequencing.

Common instances include Maybe, List, IO, and Either in languages like Haskell. For example, in Maybe, pure x is Just x and Just f <*> Just x yields Just (f x), while Nothing <*> _ yields Nothing. List represents nondeterministic computations, applying a function inside the list to elements of another list yields all combinations.

Usage: applicatives allow combining multiple independent computations that may have effects or fail, in parallel, without requiring sequential dependencies. They enable lifting multi-argument functions into the context and applying them to several contextual values, providing a flexible tool for structured effectful programming.

---