Some aspects of categories in computer science

Abstract

This chapter examines some aspects of categories in computer science. The close connections of certain closed categories with typed lambda calculi on one hand, and with the proof theory of various logics on the other are discussed. It cannot be overemphasized that modem computer science heavily uses formal syntax. The Curry-Howard isomorphism, which identifies formal proofs with lambda terms, hence with arrows in certain free categories, is the cornerstone of modem programming language semantics and simply cannot be overlooked. For many purposes in computer science, it is often useful to have categories with explicitly given strict structure along with strict functors. Cartesian closed categories (ccc's) equationally, in the spirit of multi-sorted universal algebra are presented. CCCs can themselves be made into a category in many ways. Lambda Calculus is an abstract theory of functions and the β-rule is the foundation of the lambda calculus, fundamental in programming language theory. The Formulas-as-Types view, sometimes called the “Curry-Howard isomorphism,” plays an increasingly influential role in the logical foundations of computing, especially in the foundations of functional programming languages.