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, or 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. Recent applications, such as CAML, which will be described, use categorical formalization for the purposes of implementation.
In addition to its direct relevance to theoretical knowledge and current applications, category theory is often used as an (implicit) mathematical jargon rather than for its explicit notions and results. Indeed, category theory may prove useful in construction of a sound, unifying mathematical environment, one of the purposes of theoretical investigation. As we have all probably experienced, it is good to know in which “category” one is working, i.e., which are the acceptable morphisms and
constructions, and the language of categories may provide a powerful standardization of methods and language. In other words, many different formalisms and structures may be proposed for what is essentially the same concept; the categorical language and approach may simplify through abstraction, display the generality of concepts, and help to formulate uniform definitions. This has been the case, for example, in the early applications of category theory to algebraic geometry.
The first part of this book should encourage even the reader with no specific interest in programming language theory to acquire at least some familiarity with the categorical way of looking at formal descriptions. The explicit use of deeper facts is a further step, which becomes easier with access to this information. Part II and some chapters in part I are meant to take this further step, at least in one of the possible directions, namely the mathematical semantics of data types and programs as objects and morphisms of categories.
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, or 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. Recent applications, such as CAML, which will be described, use categorical formalization for the purposes of implementation.
In addition to its direct relevance to theoretical knowledge and current applications, category theory is often used as an (implicit) mathematical jargon rather than for its explicit notions and results. Indeed, category theory may prove useful in construction of a sound, unifying mathematical environment, one of the purposes of theoretical investigation. As we have all probably experienced, it is good to know in which “category” one is working, i.e., which are the acceptable morphisms and
constructions, and the language of categories may provide a powerful standardization of methods and language. In other words, many different formalisms and structures may be proposed for what is essentially the same concept; the categorical language and approach may simplify through abstraction, display the generality of concepts, and help to formulate uniform definitions. This has been the case, for example, in the early applications of category theory to algebraic geometry.
The first part of this book should encourage even the reader with no specific interest in programming language theory to acquire at least some familiarity with the categorical way of looking at formal descriptions. The explicit use of deeper facts is a further step, which becomes easier with access to this information. Part II and some chapters in part I are meant to take this further step, at least in one of the possible directions, namely the mathematical semantics of data types and programs as objects and morphisms of categories.
