structural:summarization
Applications

In Cartesian closed categories, a "function of two variables" (a morphism f : X × Y → Z) can always be represented as a "function of one variable" (the morphism λ f : X → Z). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any Cartesian closed category.

The Curry–Howard–Lambek correspondence provides a deep isomorphism between intuitionistic logic, simply-typed lambda calculus and Cartesian closed categories.

Certain Cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory.

Computer scientist John Backus has advocated a variable-free notation, or Function-level programming, which in retrospect bears some similarity to the internal language of Cartesian closed categories. CAML is more consciously modelled on Cartesian closed categories.