Author : Niles Johnson
Publisher : Oxford University Press, USA
Page : 636 pages
File Size : 23,7 MB
Release : 2021-01-31
Category : Mathematics
ISBN : 0198871376
2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory.
Author : Saunders Mac Lane
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 40,21 MB
Release : 2013-04-17
Category : Mathematics
ISBN : 1475747217
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 : Marco Grandis
Publisher : World Scientific
Page : 535 pages
File Size : 25,15 MB
Release : 2019-09-09
Category : Mathematics
ISBN : 9811205124
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.
Author : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 24,54 MB
Release : 2014-07-24
Category : Mathematics
ISBN : 1107044243
A short introduction ideal for students learning category theory for the first time.
Author : Joachim Kock
Publisher : Cambridge University Press
Page : 260 pages
File Size : 34,91 MB
Release : 2004
Category : Mathematics
ISBN : 9780521540315
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.
Author : Bradd T. Hart
Publisher : American Mathematical Soc.
Page : 440 pages
File Size : 10,25 MB
Release :
Category : Mathematics
ISBN : 0821883828
Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.
Author : John C. Baez
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 44,33 MB
Release : 2009-09-24
Category : Algebra
ISBN : 1441915362
The purpose of this book is to give background for those who would like to delve into some higher category theory. It is not a primer on higher category theory itself. It begins with a paper by John Baez and Michael Shulman which explores informally, by analogy and direct connection, how cohomology and other tools of algebraic topology are seen through the eyes of n-category theory. The idea is to give some of the motivations behind this subject. There are then two survey articles, by Julie Bergner and Simona Paoli, about (infinity,1) categories and about the algebraic modelling of homotopy n-types. These are areas that are particularly well understood, and where a fully integrated theory exists. The main focus of the book is on the richness to be found in the theory of bicategories, which gives the essential starting point towards the understanding of higher categorical structures. An article by Stephen Lack gives a thorough, but informal, guide to this theory. A paper by Larry Breen on the theory of gerbes shows how such categorical structures appear in differential geometry. This book is dedicated to Max Kelly, the founder of the Australian school of category theory, and an historical paper by Ross Street describes its development.
Author : Emily Riehl
Publisher : Cambridge University Press
Page : 782 pages
File Size : 16,82 MB
Release : 2022-02-10
Category : Mathematics
ISBN : 1108952194
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.
Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 273 pages
File Size : 36,59 MB
Release : 2017-03-09
Category : Mathematics
ISBN : 0486820807
Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
Author : Nick Gurski
Publisher : Cambridge University Press
Page : 287 pages
File Size : 10,49 MB
Release : 2013-03-21
Category : Mathematics
ISBN : 1107034892
Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.
