Categories For Types

This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.

Category theory is a mathematical subject whose importance in several areas of computer science, most notably the semantics of programming languages and the design of programmes using abstract data types, is widely acknowledged. This book introduces category theory at a level appropriate for computer scientists and provides practical examples in the context of programming language design.

For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas keep inspiring upcoming generations of scholars.

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Category Theory is one of the most abstract branches of mathematics. It is usually taught to graduate students after they have mastered several other branches of mathematics, like algebra, topology, and group theory. It might, therefore, come as a shock that the basic concepts of category theory can be explained in relatively simple terms to anybody with some experience in programming.That's because, just like programming, category theory is about structure. Mathematicians discover structure in mathematical theories, programmers discover structure in computer programs. Well-structured programs are easier to understand and maintain and are less likely to contain bugs. Category theory provides the language to talk about structure and learning it will make you a better programmer.

A wide coverage of topics in category theory and computer science is developed in this text, including introductory treatments of cartesian closed categories, sketches and elementary categorical model theory, and triples. Over 300 exercises are included.

Handbook of Categorization in Cognitive Science, Second Edition presents the study of categories and the process of categorization as viewed through the lens of the founding disciplines of the cognitive sciences, and how the study of categorization has long been at the core of each of these disciplines. The literature on categorization reveals there is a plethora of definitions, theories, models and methods to apprehend this central object of study. The contributions in this handbook reflect this diversity. For example, the notion of category is not uniform across these contributions, and there are multiple definitions of the notion of concept. Furthermore, the study of category and categorization is approached differently within each discipline. For some authors, the categories themselves constitute the object of study, whereas for others, it is the process of categorization, and for others still, it is the technical manipulation of large chunks of information. Finally, yet another contrast has to do with the biological versus artificial nature of agents or categorizers. Defines notions of category and categorization Discusses the nature of categories: discrete, vague, or other Explores the modality effects on categories Bridges the category divide - calling attention to the bridges that have already been built, and avenues for further cross-fertilization between disciplines

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.