2 Dimensional Categories PDF Download
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Author | : Niles Johnson |
Publisher | : Oxford University Press |
Total Pages | : 476 |
Release | : 2021-01-31 |
Genre | : Science |
ISBN | : 0192645676 |
Download 2-Dimensional Categories Book in PDF, ePub and Kindle
Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
Author | : Niles Johnson |
Publisher | : Oxford University Press, USA |
Total Pages | : 636 |
Release | : 2021-01-31 |
Genre | : Mathematics |
ISBN | : 0198871376 |
Download 2-Dimensional Categories Book in PDF, ePub and Kindle
2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory.
Author | : Niles Johnson |
Publisher | : |
Total Pages | : |
Release | : 2021 |
Genre | : Categories (Mathematics) |
ISBN | : 9780191914850 |
Download 2-dimensional Categories Book in PDF, ePub and Kindle
2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
Author | : Grandis Marco |
Publisher | : World Scientific |
Total Pages | : 536 |
Release | : 2019-09-09 |
Genre | : Mathematics |
ISBN | : 9811205124 |
Download Higher Dimensional Categories: From Double To Multiple Categories Book in PDF, ePub and Kindle
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 | : Joachim Kock |
Publisher | : Cambridge University Press |
Total Pages | : 260 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 9780521540315 |
Download Frobenius Algebras and 2-D Topological Quantum Field Theories Book in PDF, ePub and Kindle
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. |
Total Pages | : 440 |
Release | : |
Genre | : Mathematics |
ISBN | : 0821883828 |
Download Models, Logics, and Higher-dimensional Categories Book in PDF, ePub and Kindle
Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.
Author | : Saunders Mac Lane |
Publisher | : Springer Science & Business Media |
Total Pages | : 320 |
Release | : 2013-04-17 |
Genre | : Mathematics |
ISBN | : 1475747217 |
Download Categories for the Working Mathematician Book in PDF, ePub and Kindle
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 | : John C. Baez |
Publisher | : Springer Science & Business Media |
Total Pages | : 292 |
Release | : 2009-09-24 |
Genre | : Algebra |
ISBN | : 1441915362 |
Download Towards Higher Categories Book in PDF, ePub and Kindle
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 | : Tom Leinster |
Publisher | : Cambridge University Press |
Total Pages | : 193 |
Release | : 2014-07-24 |
Genre | : Mathematics |
ISBN | : 1107044243 |
Download Basic Category Theory Book in PDF, ePub and Kindle
A short introduction ideal for students learning category theory for the first time.
Author | : Nick Gurski |
Publisher | : Cambridge University Press |
Total Pages | : 287 |
Release | : 2013-03-21 |
Genre | : Mathematics |
ISBN | : 1107034892 |
Download Coherence in Three-Dimensional Category Theory Book in PDF, ePub and Kindle
Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.