Category Theory for the Sciences

Author: David I. Spivak
Publisher: MIT Press
ISBN: 9780262320535
Release Date: 2014-10-17
Genre: Mathematics

Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs -- categories in disguise. After explaining the "big three" concepts of category theory -- categories, functors, and natural transformations -- the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics.

Category Theory for the Sciences

Author: David I. Spivak
Publisher: MIT Press
ISBN: 9780262028134
Release Date: 2014-10-10
Genre: Computers

An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences.

Basic Category Theory for Computer Scientists

Author: Benjamin C. Pierce
Publisher: MIT Press
ISBN: 0262660717
Release Date: 1991
Genre: Computers

Basic Category Theory for Computer Scientists provides a straightforward presentationof the basic constructions and terminology of category theory, including limits, functors, naturaltransformations, adjoints, and cartesian closed categories.

An Introduction to Category Theory

Author: Harold Simmons
Publisher: Cambridge University Press
ISBN: 9781139503327
Release Date: 2011-09-22
Genre: Mathematics

Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.

Category Theory

Author: Steve Awodey
Publisher: OUP Oxford
ISBN: 9780191612558
Release Date: 2010-06-17
Genre: Philosophy

Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. 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 assuming 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. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.

Mathematics of Big Data

Author: Jeremy Kepner
Publisher: MIT Press
ISBN: 9780262347914
Release Date: 2018-07-13
Genre: Computers

The first book to present the common mathematical foundations of big data analysis across a range of applications and technologies. Today, the volume, velocity, and variety of data are increasing rapidly across a range of fields, including Internet search, healthcare, finance, social media, wireless devices, and cybersecurity. Indeed, these data are growing at a rate beyond our capacity to analyze them. The tools—including spreadsheets, databases, matrices, and graphs—developed to address this challenge all reflect the need to store and operate on data as whole sets rather than as individual elements. This book presents the common mathematical foundations of these data sets that apply across many applications and technologies. Associative arrays unify and simplify data, allowing readers to look past the differences among the various tools and leverage their mathematical similarities in order to solve the hardest big data challenges. The book first introduces the concept of the associative array in practical terms, presents the associative array manipulation system D4M (Dynamic Distributed Dimensional Data Model), and describes the application of associative arrays to graph analysis and machine learning. It provides a mathematically rigorous definition of associative arrays and describes the properties of associative arrays that arise from this definition. Finally, the book shows how concepts of linearity can be extended to encompass associative arrays. Mathematics of Big Data can be used as a textbook or reference by engineers, scientists, mathematicians, computer scientists, and software engineers who analyze big data.

Categories and Computer Science

Author: R. F. C. Walters
Publisher: Cambridge University Press
ISBN: 0521422264
Release Date: 1991
Genre: Computers

Category Theory has, in recent years, become increasingly important and popular in computer science, and many universities now introduce Category Theory as part of the curriculum for undergraduate computer science students. Here, the theory is developed in a straightforward way, and is enriched with many examples from computer science.

Tool and Object

Author: Ralph Krömer
Publisher: Springer Science & Business Media
ISBN: 9783764375249
Release Date: 2007-06-25
Genre: Mathematics

Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

The Big Book of Concepts

Author: Gregory Murphy
Publisher: MIT Press
ISBN: 0262250063
Release Date: 2004-01-30
Genre: Psychology

Concepts embody our knowledge of the kinds of things there are in the world. Tying our past experiences to our present interactions with the environment, they enable us to recognize and understand new objects and events. Concepts are also relevant to understanding domains such as social situations, personality types, and even artistic styles. Yet like other phenomenologically simple cognitive processes such as walking or understanding speech, concept formation and use are maddeningly complex.Research since the 1970s and the decline of the "classical view" of concepts have greatly illuminated the psychology of concepts. But persistent theoretical disputes have sometimes obscured this progress. The Big Book of Concepts goes beyond those disputes to reveal the advances that have been made, focusing on the major empirical discoveries. By reviewing and evaluating research on diverse topics such as category learning, word meaning, conceptual development in infants and children, and the basic level of categorization, the book develops a much broader range of criteria than is usual for evaluating theories of concepts.

Conceptual Mathematics

Author: F. William Lawvere
Publisher: Cambridge University Press
ISBN: 9780521894852
Release Date: 2009-07-30
Genre: Mathematics

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.

Categories for the Working Mathematician

Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 9781475747218
Release Date: 2013-04-17
Genre: Mathematics

An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

Types and Programming Languages

Author: Benjamin C. Pierce
Publisher: MIT Press
ISBN: 0262162091
Release Date: 2002
Genre: Computers

A comprehensive introduction to type systems and programming languages.

Sets for Mathematics

Author: F. William Lawvere
Publisher: Cambridge University Press
ISBN: 0521010608
Release Date: 2003-01-27
Genre: Mathematics

In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.