Author: David I. Spivak
Publisher: MIT Press
Release Date: 2014-10-17
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.
Author: Benjamin C. Pierce
Publisher: MIT Press
Release Date: 1991
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.
Author: Janet H. Murray
Publisher: MIT Press
Release Date: 2011-11-23
Digital artifacts from iPads to databases pervade our lives, and the design decisions that shape them affect how we think, act, communicate, and understand the world. But the pace of change has been so rapid that technical innovation is outstripping design. Interactors are often mystified and frustrated by their enticing but confusing new devices; meanwhile, product design teams struggle to articulate shared and enduring design goals. With Inventing the Medium, Janet Murray provides a unified vocabulary and a common methodology for the design of digital objects and environments. It will be an essential guide for both students and practitioners in this evolving field.Murray explains that innovative interaction designers should think of all objects made with bits -- whether games or Web pages, robots or the latest killer apps -- as belonging to a single new medium: the digital medium. Designers can speed the process of useful and lasting innovation by focusing on the collective cultural task of inventing this new medium. Exploring strategies for maximizing the expressive power of digital artifacts, Murray identifies and examines four representational affordances of digital environments that provide the core palette for designers across applications: computational procedures, user participation, navigable space, and encyclopedic capacity. Each chapter includes a set of Design Explorations -- creative exercises for students and thought experiments for practitioners -- that allow readers to apply the ideas in the chapter to particular design problems. Inventing the Medium also provides more than 200 illustrations of specific design strategies drawn from multiple genres and platforms and a glossary of design concepts.
Author: Harold Simmons
Publisher: Cambridge University Press
Release Date: 2011-09-22
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.
Author: Steve Awodey
Publisher: Oxford University Press
Release Date: 2010-06-17
A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems (with full proofs), as well as numerous examples and exercises.
Author: F. William Lawvere
Publisher: Cambridge University Press
Release Date: 2009-07-30
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.
Author: Noam Chomsky
Publisher: MIT Press
Release Date: 2014-12-26
Genre: Language Arts & Disciplines
In his foundational book, The Minimalist Program, published in 1995, Noam Chomsky offered a significant contribution to the generative tradition in linguistics. This twentieth-anniversary edition reissues this classic work with a new preface by the author. In four essays, Chomsky attempts to situate linguistic theory in the broader cognitive sciences, with the essays formulating and progressively developing the minimalist approach to linguistic theory. Building on the theory of principles and parameters and, in particular, on principles of economy of derivation and representation, the minimalist framework takes Universal Grammar as providing a unique computational system, with derivations driven by morphological properties, to which the syntactic variation of languages is also restricted. Within this theoretical framework, linguistic expressions are generated by optimally efficient derivations that must satisfy the conditions that hold on interface levels, the only levels of linguistic representation. The interface levels provide instructions to two types of performance systems, articulatory-perceptual and conceptual-intentional. All syntactic conditions, then, express properties of these interface levels, reflecting the interpretive requirements of language and keeping to very restricted conceptual resources. In the preface to this edition, Chomsky emphasizes that the minimalist approach developed in the book and in subsequent work "is a program, not a theory." With this book, Chomsky built on pursuits from the earliest days of generative grammar to formulate a new research program that had far-reaching implications for the field.
Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
Release Date: 2013-04-17
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.
Author: Caroline A. Jones
Publisher: Mit Press
Release Date: 2016-01-09
Experience offers a reading experience like no other. A heat-sensitive cover by Olafur Eliasson reveals words, colors, and a drawing when touched by human hands. Endpapers designed by Carsten H�ller are printed in ink containing carefully calibrated quantities of the synthesized human pheromones estratetraenol and androstadienone, evoking the suggestibility of human desire. The margins and edges of the book are designed by Tauba Auerbach in complementary colors that create a dynamically shifting effect when the book is shifted or closed. When the book is opened, bookmarks cascade from the center, emerging from spider web prints by Tom�s Saraceno. Experience produces experience while bringing the concept itself into relief as an object of contemplation. The sensory experience of the book as a physical object resonates with the intellectual experience of the book as a container of ideas. Experience convenes a conversation with artists, musicians, philosophers, anthropologists, historians, and neuroscientists, each of whom explores aspects of sensorial and cultural realms of experience. The texts include new essays written for this volume and classic texts by such figures as William James and Michel Foucault. The first publication from MIT's Center for Art, Science, & Technology, Experience approaches its subject through multiple modes. Publication design by Kimberly Varella with Becca Lofchie, Content Object Design Studio. ContributorsTauba Auerbach, Bevil Conway, John Dewey, Olafur Eliasson, Michel Foucault, Adam Frank, Vittorio Gallese, Ren�e Green, Stefan Helmreich, Carsten H�ller, Edmund Husserl, William James, Caroline A. Jones, Douglas Kahn, Brian Kane, Leah Kelly, Bruno Latour, Alvin Lucier, David Mather, Mara Mills, Alva No�, Jacques Ranci�re, Michael Rossi, Tom�s Saraceno, Natasha Sch�ll, Joan W.Scott, Tino Sehgal, Alma Steingart, Josh Tenenbaum, Rebecca Uchill
In this book, Sanjoy Mahajan shows us that the way to master complexity is through insight rather than precision. Precision can overwhelm us with information, whereas insight connects seemingly disparate pieces of information into a simple picture. Unlike computers, humans depend on insight. Based on the author's fifteen years of teaching at MIT, Cambridge University, and Olin College, The Art of Insight in Science and Engineering shows us how to build insight and find understanding, giving readers tools to help them solve any problem in science and engineering.To master complexity, we can organize it or discard it. The Art of Insight in Science and Engineering first teaches the tools for organizing complexity, then distinguishes the two paths for discarding complexity: with and without loss of information. Questions and problems throughout the text help readers master and apply these groups of tools. Armed with this three-part toolchest, and without complicated mathematics, readers can estimate the flight range of birds and planes and the strength of chemical bonds, understand the physics of pianos and xylophones, and explain why skies are blue and sunsets are red.The Art of Insight in Science and Engineering will appear in print and online under a Creative Commons Noncommercial Share Alike license.