Indexed Categories And Their Applications

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Indexed Categories And Their Applications

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Author : P.I. Johnstone
language : en
Publisher: Springer
Release Date : 2006-11-15

Indexed Categories And Their Applications written by P.I. Johnstone and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.

Categories For The Working Mathematician

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Author : Saunders Mac Lane
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Categories For The Working Mathematician written by Saunders Mac Lane and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

Categories for the Working Mathematician provides 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. The book 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 characterized 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 two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Theories Sites Toposes

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Author : Olivia Caramello
language : en
Publisher: Oxford University Press
Release Date : 2018-01-19

Theories Sites Toposes written by Olivia Caramello and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-19 with Philosophy categories.

According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.

Applications Of Categories In Computer Science

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Author : M. P. Fourman
language : en
Publisher: Cambridge University Press
Release Date : 1992-06-26

Applications Of Categories In Computer Science written by M. P. Fourman and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-06-26 with Computers categories.

Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the subject which was held at the University of Durham. Participants at the conference were leading computer scientists and mathematicians working in the area and this volume reflects the excitement and importance of the meeting. All the papers have been refereed and represent some of the most important and current ideas. Hence this book will be essential to mathematicians and computer scientists working in the applications of category theory.

Cogroups And Co Rings In Categories Of Associative Rings

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Author : George M. Bergman
language : en
Publisher: American Mathematical Soc.
Release Date : 1996

Cogroups And Co Rings In Categories Of Associative Rings written by George M. Bergman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.

This book studies representable functors among well-known varieties of algebras. All such functors from associative rings over a fixed ring R to each of the categories of abelian groups, associative rings, Lie rings, and to several others are determined. Results are also obtained on representable functors on varieties of groups, semigroups, commutative rings, and Lie algebras. The book includes a "Symbol index", which serves as a glossary of symbols used and a list of the pages where the topics so symbolized are treated, and a "Word and phrase index". The authors have strived -- and succeeded -- in creating a volume that is very user-friendly.

Category Theory 1991 Proceedings Of The 1991 Summer Category Theory Meeting Montreal Canada

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Author : Robert Andrew George Seely
language : en
Publisher: American Mathematical Soc.
Release Date : 1992

Category Theory 1991 Proceedings Of The 1991 Summer Category Theory Meeting Montreal Canada written by Robert Andrew George Seely and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Mathematics categories.

Twenty-seven papers address applications of category theory in new domains as well as in its traditional contexts. Among the topics: a Stone duality for metric spaces; sheaves in cocomplete categories; completeness in continuity spaces; dualities for accessible categories; some problems in descriptive locale theory; modeling homotopy coherence; and functorial selection of morphisms. No index. Annotation copyright by Book News, Inc., Portland, OR

Models Logics And Higher Dimensional Categories

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Author : Bradd T. Hart
language : en
Publisher: American Mathematical Soc.
Release Date :

Models Logics And Higher Dimensional Categories written by Bradd T. Hart and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.

Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.

Basic Category Theory For Computer Scientists

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Author : Benjamin C. Pierce
language : en
Publisher: MIT Press
Release Date : 1991-08-07

Basic Category Theory For Computer Scientists written by Benjamin C. Pierce and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-08-07 with Computers categories.

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

Higher Dimensional Categories From Double To Multiple Categories

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Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2019-09-09

Higher Dimensional Categories From Double To Multiple Categories written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-09 with Mathematics categories.

The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.

Categories In Computer Science And Logic

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Author : John Walker Gray
language : en
Publisher: American Mathematical Soc.
Release Date : 1989

Categories In Computer Science And Logic written by John Walker Gray and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Mathematics categories.

Presents the proceedings of AMS-IMS-SIAM Summer Research Conference on Categories in Computer Science and Logic that was held at the University of Colorado in Boulder. This book discusses the use of category theory in formalizing aspects of computer programming and program design.