On the Role of Category Theory in the Area of Algebraic Specification

The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic speciications. In detail we discuss diierent approaches to an abstract theory of spec-iication logics. Further we present a uniform framework for developing particular speciication logics. We make use of`classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular speciication logics in a uniform manner. The speciication logics considered are: equational logic for total algebras , conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has innuenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review the ideas and concepts of category theory and to discuss their role in theoretical computer science. Especially for readers not familiar with, but interested in, category theory this section contains an introductory survey of the main concepts and paradigms of category theory. Further we present three selected problems in the area of algebraic speciications. In the subsequent sections we show how these problems can be solved using concepts, constructions and results of category theory. These sections provide a concise exposition of the application of more advanced category theory in the area of (algebraic) speciications thus they become harder to read for novice readers. But nevertheless interested readers can use these sections as an appropriate guideline for approaching category theory. Abstracting from the examples we summarize in the conclusion the role of category theory in theoretical computer science, and formulate some general concepts. 1.1 Paradigm and Concepts of Category Theory Category theory arose from the fundamental idea of representing a function by an arrow. This idea rst appeared in topology about 1940 (see Mac71]). The original purpose of category theory in the pioneering paper EM45] by Eilenberg and MacLane was to establish a uniform framework to speak about isomor-phisms and natural equivalences appearing in diierent areas of mathematics, in particular algebra and topology. What was new about category theory? Within set theory we can characterize a mathematical object only by describing its inner structure, that is, we can separate it into diierent parts and elements and we can represent the interrelations between these inner …