Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial * Applications * Further Reading
Abstract: "Category theory is a branch of pure mathematics that more and more frequently touches the daily work of computer scientists, especially those with an interest in programming languages or formal specifications. This survey is an 'introduction to the introductions' to category theory--a brief answer to the questions, 'What is category theory?' 'What are its basic concepts?' 'What are computer scientists using it for?' and 'Where can I learn more?' The first section introduces the most common category-theoretic terms and idioms, assuming as little specific mathematical background as possible. the second section presents four case studies from the recent research literature applying category theory to the semantics of computation. A reading list in the third section suggests pathways into the existing literature, including textbooks, standard reference works, and selected research papers."
This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of Category Theory understandable to this broad readership. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads.
The purpose of the book is to advance in the understanding of brain function by defining a general framework for representation based on category theory. The idea is to bring this mathematical formalism into the domain of neural representation of physical spaces, setting the basis for a theory of mental representation, able to relate empirical findings, uniting them into a sound theoretical corpus. The innovative approach presented in the book provides a horizon of interdisciplinary collaboration that aims to set up a common agenda that synthesizes mathematical formalization and empirical procedures in a systemic way. Category theory has been successfully applied to qualitative analysis, mainly in theoretical computer science to deal with programming language semantics. Nevertheless, the potential of category theoretic tools for quantitative analysis of networks has not been tackled so far. Statistical methods to investigate graph structure typically rely on network parameters. Category theory can be seen as an abstraction of graph theory. Thus, new categorical properties can be added into network analysis and graph theoretic constructs can be accordingly extended in more fundamental basis. By generalizing networks using category theory we can address questions and elaborate answers in a more fundamental way without waiving graph theoretic tools. The vital issue is to establish a new framework for quantitative analysis of networks using the theory of categories, in which computational neuroscientists and network theorists may tackle in more efficient ways the dynamics of brain cognitive networks. The intended audience of the book is researchers who wish to explore the validity of mathematical principles in the understanding of cognitive systems. All the actors in cognitive science: philosophers, engineers, neurobiologists, cognitive psychologists, computer scientists etc. are akin to discover along its pages new unforeseen connections through the development of concepts and formal theories described in the book. Practitioners of both pure and applied mathematics e.g., network theorists, will be delighted with the mapping of abstract mathematical concepts in the terra incognita of cognition.
7th International Conference, Beijing China, May 27-30, 2007, Proceedings
Author: Yong Shi
Part of a four-volume set, this book constitutes the refereed proceedings of the 7th International Conference on Computational Science, ICCS 2007, held in Beijing, China in May 2007. The papers cover a large volume of topics in computational science and related areas, from multiscale physics to wireless networks, and from graph theory to tools for program development.
Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: IntroductionCategories, Functors and Natural TransformationsLimits and ColimitsAdjunctions and MonadsApplications in AlgebraApplications in Topology and Algebraic TopologyApplications in Homological AlgebraHints at Higher Dimensional Category TheoryReferencesIndices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and TopologyThe theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hintsThree chapters and some extra sections are devoted to applications