David I. Spivak's Books

An Invitation to Applied Category Theory

Seven Sketches in Compositionality

By: Brendan Fong,David I. Spivak

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An Invitation to Applied Category Theory

Seven Sketches in Compositionality

By: Brendan Fong,David I. Spivak

Category theory is unmatched in its ability to organize and layer abstractions and to find commonalities between structures of all sorts. No longer the exclusive preserve of pure mathematicians, it is now proving itself to be a powerful tool in science, informatics, and industry. By facilitating communication between communities and building rigorous bridges between disparate worlds, applied category theory has the potential to be a major organizing force. This book offers a self-contained tour of applied category theory. Each chapter follows a single thread motivated by a real-world application and discussed with category-theoretic tools. We see data migration as an adjoint functor, electrical circuits in terms of monoidal categories and operads, and collaborative design via enriched profunctors. All the relevant category theory, from simple to sophisticated, is introduced in an accessible way with many examples and exercises, making this an ideal guide even for those without experience of university-level mathematics.

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351

Publisher

Cambridge University Press

Published Date

ISBN 10

1108582249

ISBN 13

9781108582247

Temporal Type Theory

A Topos-Theoretic Approach to Systems and Behavior

By: Patrick Schultz,David I. Spivak

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Temporal Type Theory

A Topos-Theoretic Approach to Systems and Behavior

By: Patrick Schultz,David I. Spivak

This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action. Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.

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Page Count

237

Publisher

Springer

Published Date

ISBN 10

3030007049

ISBN 13

9783030007041

Category Theory for the Sciences

By: David I. Spivak

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Category Theory for the Sciences

By: David I. Spivak

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

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Page Count

495

Publisher

MIT Press

Published Date

ISBN 10

0262320533

ISBN 13

9780262320535

An Invitation to Applied Category Theory

Seven Sketches in Compositionality

By: Brendan Fong,David I. Spivak

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An Invitation to Applied Category Theory

Seven Sketches in Compositionality

By: Brendan Fong,David I. Spivak

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

351

Publisher

Cambridge University Press

Published Date

ISBN 10

1108582249

ISBN 13

9781108582247

Category Theory for the Sciences

By: David I. Spivak

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Category Theory for the Sciences

By: David I. Spivak

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N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

495

Publisher

MIT Press

Published Date

ISBN 10

0262320533

ISBN 13

9780262320535

Book's by David I. Spivak

An Invitation to Applied Category Theory

An Invitation to Applied Category Theory

Temporal Type Theory

Temporal Type Theory

Category Theory for the Sciences

Category Theory for the Sciences

An Invitation to Applied Category Theory

An Invitation to Applied Category Theory

Category Theory for the Sciences

Category Theory for the Sciences

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