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Categories, Types, and Structures

.least in one of the possible directions, namely the mathematical semantics of data types and programs as objects and morphisms of categories. We were urged to write the general introduction contained in part I, since most available books in category theory are written for the “working mathematician” and, as the subject is greatly indebted to algebraic geometry and related disciplines, the examples and motivations can be understood only by readers with some a | This book is currently out of print. Upon kind permission of the M.l.T.-Press it is available on ftp.ens.fr pub dmi users longo CategTypesStructures All references should be made to the published book. CATEGORIES TYPES AND STRUCTURES An Introduction to Category Theory for the working computer scientist Andrea Asperti Giuseppe Longo FOUNDATIONS OF COMPUTING SERIES M.I.T. PRESS 1991 I INTRODUCTION The main methodological connection between programming language theory and category theory is the fact that both theories are essentially theories of functions. A crucial point though is that the categorical notion of morphism generalizes the set-theoretical description of function in a very broad sense which provides a unified understanding of various aspects of the theory of programs. This is one of the reasons for the increasing role of category theory in the semantic investigation of programs if compared say to the set-theoretic approach. However the influence of this mathematical discipline on computer science goes beyond the methodological issue as the categorical approach to mathematical formalization seems to be suitable for focusing concerns in many different areas of computer science such as software engineering and artificial intelligence as well as automata theory and other theoretical aspects of computation. This book is mostly inspired by this specific methodological connection and its applications to the theory of programming languages. More precisely as expressed by the subtitle it aims at a selfcontained introduction to general category theory part I and at a categorical understanding of the mathematical structures that constituted in the last twenty or so years the theoretical background of relevant areas of language design part II . The impact on functional programming for example of the mathematical tools described in part II is well known as it ranges from the early dialects of Lisp to Edinburgh ML to the current work in polymorphisms and modularity. .