(Co)end Calculus

The language of ends and (co)ends is a powerful tool in category theory, but is typically only used by category theorists. This book is the first systematic treatment of the theory of (co)ends, presenting it as a powerful tool to clarify and simplify definitions and results and export them for use in diverse areas of mathematics and computer science.

Format: Paperback / softback
Length: 330 pages
Publication date: 22 July 2021
Publisher: Cambridge University Press

The language of ends and (co)ends offers a natural and versatile approach to expressing a wide range of phenomena in category theory, both in the abstract and in practical applications. While category-theoretic methods have gained widespread use among mathematicians, due to the complexity of (co)ends, they are primarily utilized by category theorists, who consider them a secret weapon in their arsenal. This book represents the first comprehensive treatment of the theory of (co)ends, aimed at a broad audience. It presents the (co)end calculus as a powerful tool for clarifying and simplifying definitions and results in category theory, enabling their export for use in various fields of mathematics and computer science. Organized as an easy-to-cite reference manual, this book will appeal to category theorists and users of category theory alike.

Category theory is a branch of mathematics that deals with the structure and properties of categories, which are mathematical structures that represent objects and their relationships. Categories are used to organize and classify mathematical objects, and they have applications in fields such as algebra, topology, geometry, and computer science.

One of the key concepts in category theory is the notion of an end. An end is a mathematical object that represents a category. It is a kind of "black box" that takes an object of a given category and produces another object of the same category. Ends are used to simplify and clarify definitions and results in category theory, and they are also used to construct new objects and theories.

Another important concept in category theory is the notion of a (co)end. A (co)end is a pair of ends that are related to each other in a specific way. It is a kind of "double black box" that takes two objects of a given category and produces another object of the same category. (Co)ends are used to construct new objects and theories, and they are also used to study the properties of categories and their objects.

The theory of (co)ends is a relatively new area of research in category theory, and it has already had a significant impact on the field. This book is the first comprehensive treatment of the theory of (co)ends, aimed at a wide audience. It presents the (co)end calculus as a powerful tool for clarifying and simplifying definitions and results in category theory, enabling their export for use in various fields of mathematics and computer science.

The book is organized into four chapters. In the first chapter, the authors introduce the basic concepts of category theory and ends, and they discuss the importance of (co)ends in the theory. In the second chapter, the authors introduce the (co)end calculus, and they discuss its properties and applications. In the third chapter, the authors discuss the construction of (co)ends, and they introduce some of the main results in the theory. In the fourth chapter, the authors discuss the applications of (co)ends in various fields of mathematics and computer science, and they introduce some of the main open problems in the theory.

One of the key features of this book is its accessibility. The authors have written the book in a clear and concise manner, and they have included numerous examples and exercises to help the reader understand the concepts and apply the techniques. The book is also well-illustrated, with numerous diagrams and figures to help the reader visualize the ideas.

Another important feature of this book is its relevance to a wide range of fields. Category theory is used in many areas of mathematics and computer science, including algebra, topology, geometry, and computer science. The (co)end calculus is a powerful tool that can be used to clarify and simplify definitions and results in these fields, and it can also be used to construct new objects and theories.

In conclusion, this book is a valuable resource for anyone interested in category theory and its applications. It presents the (co)end calculus as a powerful tool for clarifying and simplifying definitions and results in category theory, enabling their export for use in various fields of mathematics and computer science. The book is well-written, accessible, and relevant to a wide range of fields, and it will be of interest to category theorists and users of category theory alike.

Weight: 492g
Dimension: 153 x 227 x 22 (mm)
ISBN-13: 9781108746120