Adjoint functors | Lambda calculus

Apply

In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology. The most general setting for apply is in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose internal language is simply typed lambda calculus. (Wikipedia).

Apply
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Related pages

ECMAScript | Scott continuity | Tensor-hom adjunction | Homological algebra | Complete partial order | Function application | Continuous function | Currying | Lambda calculus | Lua (programming language) | Exponential object | Homotopy theory | Adjoint functors | Closed monoidal category | Mathematics | Denotational semantics | Function (mathematics) | R (programming language) | Algebraic geometry | Category theory | Order theory | Cartesian closed category | Simply typed lambda calculus