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What is Category Theory?

What is Category Theory? PDF Author: Giandomenico Sica
Publisher: Polimetrica s.a.s.
ISBN: 8876990313
Category : Mathematics
Languages : en
Pages : 290

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Basic Category Theory for Computer Scientists

Basic Category Theory for Computer Scientists PDF Author: Benjamin C. Pierce
Publisher: MIT Press
ISBN: 9780262660716
Category : Computers
Languages : en
Pages : 116

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Book Description
Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading

What is Category Theory?

What is Category Theory? PDF Author: Giandomenico Sica
Publisher: Polimetrica s.a.s.
ISBN: 8876990313
Category : Mathematics
Languages : en
Pages : 290

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Book Description


Category Theory

Category Theory PDF Author: K. H. Kamps
Publisher: Springer
ISBN: 3540395504
Category : Mathematics
Languages : en
Pages : 326

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Involutive Category Theory

Involutive Category Theory PDF Author: Donald Yau
Publisher: Springer Nature
ISBN: 3030612031
Category : Mathematics
Languages : en
Pages : 243

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Book Description
This monograph introduces involutive categories and involutive operads, featuring applications to the GNS construction and algebraic quantum field theory. The author adopts an accessible approach for readers seeking an overview of involutive category theory, from the basics to cutting-edge applications. Additionally, the author’s own recent advances in the area are featured, never having appeared previously in the literature. The opening chapters offer an introduction to basic category theory, ideal for readers new to the area. Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research. Chapters on coherence of involutive symmetric monoidal categories, and categorical GNS construction follow. The last chapter covers involutive operads and lays important coherence foundations for applications to algebraic quantum field theory. With detailed explanations and exercises throughout, Involutive Category Theory is suitable for graduate seminars and independent study. Mathematicians and mathematical physicists who use involutive objects will also find this a valuable reference.

Category Theory in Context

Category Theory in Context PDF Author: Emily Riehl
Publisher: Courier Dover Publications
ISBN: 048680903X
Category : Mathematics
Languages : en
Pages : 272

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Book Description
Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics. Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. Prerequisites are limited to familiarity with some basic set theory and logic.

Basic Category Theory

Basic Category Theory PDF Author: Tom Leinster
Publisher: Cambridge University Press
ISBN: 1107044243
Category : Mathematics
Languages : en
Pages : 190

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Book Description
A short introduction ideal for students learning category theory for the first time.

Elements of ∞-Category Theory

Elements of ∞-Category Theory PDF Author: Emily Riehl
Publisher: Cambridge University Press
ISBN: 1108952194
Category : Mathematics
Languages : en
Pages :

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Book Description
The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

An Introduction to the Language of Category Theory

An Introduction to the Language of Category Theory PDF Author: Steven Roman
Publisher: Birkhäuser
ISBN: 331941917X
Category : Mathematics
Languages : en
Pages : 169

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Book Description
This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible. In writing about this challenging subject, the author has brought to bear all of the experience he has gained in authoring over 30 books in university-level mathematics. The goal of this book is to present the five major ideas of category theory: categories, functors, natural transformations, universality, and adjoints in as friendly and relaxed a manner as possible while at the same time not sacrificing rigor. These topics are developed in a straightforward, step-by-step manner and are accompanied by numerous examples and exercises, most of which are drawn from abstract algebra. The first chapter of the book introduces the definitions of category and functor and discusses diagrams,duality, initial and terminal objects, special types of morphisms, and some special types of categories,particularly comma categories and hom-set categories. Chapter 2 is devoted to functors and naturaltransformations, concluding with Yoneda's lemma. Chapter 3 presents the concept of universality and Chapter 4 continues this discussion by exploring cones, limits, and the most common categorical constructions – products, equalizers, pullbacks and exponentials (along with their dual constructions). The chapter concludes with a theorem on the existence of limits. Finally, Chapter 5 covers adjoints and adjunctions. Graduate and advanced undergraduates students in mathematics, computer science, physics, or related fields who need to know or use category theory in their work will find An Introduction to Category Theory to be a concise and accessible resource. It will be particularly useful for those looking for a more elementary treatment of the topic before tackling more advanced texts.

Mathematical Applications of Category Theory

Mathematical Applications of Category Theory PDF Author: American Mathematical Society. Meeting
Publisher: American Mathematical Soc.
ISBN: 0821850326
Category : Mathematics
Languages : en
Pages : 307

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Book Description
Mathematicians interested in understanding the directions of current research in set theory will not want to overlook this book, which contains the proceedings of the AMS Summer Research Conference on Axiomatic Set Theory, held in Boulder, Colorado, June 19-25, 1983. This was the first large meeting devoted exclusively to set theory since the legendary 1967 UCLA meeting, and a large majority of the most active research mathematicians in the field participated. All areas of set theory, including constructibility, forcing, combinatorics and descriptive set theory, were represented; many of the papers in the proceedings explore connections between areas. Readers should have a background of graduate-level set theory. There is a paper by S. Shelah applying proper forcing to obtain consistency results on combinatorial cardinal 'invariants' below the continuum, and papers by R. David and S. Freidman on properties of $0^\ No. $.Papers by A. Blass, H.D. Donder, T. Jech and W. Mitchell involve inner models with measurable cardinals and various combinatorial properties. T. Carlson largely solves the pin-up problem, and D. Velleman presents a novel construction of a Souslin tree from a morass. S. Todorcevic obtains the strong failure of the \qedprinciple from the Proper Forcing Axiom and A. Miller discusses properties of a new species of perfect-set forcing. H. Becker and A. Kechris attack the third Victoria Delfino problem while W. Zwicker looks at combinatorics on $P_\kappa(\lambda)$ and J. Henle studies infinite-exponent partition relations. A. Blass shows that if every vector space has a basis then $AC$ holds. I. Anellis treats the history of set theory, and W. Fleissner presents set-theoretical axioms of use in general topology.

An Invitation to Applied Category Theory

An Invitation to Applied Category Theory PDF Author: Brendan Fong
Publisher: Cambridge University Press
ISBN: 1108482295
Category : Computers
Languages : en
Pages : 350

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Book Description
Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.