Towards Higher Categories

Author : John C. Baez,J. Peter May
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 43,6 Mb
Release : 2009-09-24
Category : Algebra
ISBN : 9781441915368

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Towards Higher Categories by John C. Baez,J. Peter May Pdf

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 : Gijs Heuts
Publisher : American Mathematical Society
Page : 108 pages
File Size : 47,8 Mb
Release : 2021-11-16
Category : Mathematics
ISBN : 9781470448936

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Goodwillie Approximations to Higher Categories by Gijs Heuts Pdf

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Author : Tom Leinster
Publisher : Cambridge University Press
Page : 451 pages
File Size : 41,6 Mb
Release : 2004-07-22
Category : Mathematics
ISBN : 9780521532150

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Higher Operads, Higher Categories by Tom Leinster Pdf

Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics.

Author : Simona Paoli
Publisher : Springer
Page : 343 pages
File Size : 55,6 Mb
Release : 2019-06-03
Category : Mathematics
ISBN : 9783030056742

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Simplicial Methods for Higher Categories by Simona Paoli Pdf

This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic. While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, is one of the simplest known algebraic structures yielding a model of weak n-categories. The central result is the equivalence of this model to one of the existing models, due to Tamsamani and further studied by Simpson. This theory has intended applications to homotopy theory, mathematical physics and to long-standing open questions in category theory. As the theory is described in elementary terms and the book is largely self-contained, it is accessible to beginning graduate students and to mathematicians from a wide range of disciplines well beyond higher category theory. The new model makes a transparent connection between higher category theory and homotopy theory, rendering it particularly suitable for category theorists and algebraic topologists. Although the results are complex, readers are guided with an intuitive explanation before each concept is introduced, and with diagrams showing the interconnections between the main ideas and results.

Author : Carlos Simpson
Publisher : Cambridge University Press
Page : 653 pages
File Size : 55,5 Mb
Release : 2011-10-20
Category : Mathematics
ISBN : 9781139502191

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Homotopy Theory of Higher Categories by Carlos Simpson Pdf

The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others.

Author : Denis-Charles Cisinski
Publisher : Cambridge University Press
Page : 449 pages
File Size : 54,8 Mb
Release : 2019-05-02
Category : Mathematics
ISBN : 9781108473200

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Higher Categories and Homotopical Algebra by Denis-Charles Cisinski Pdf

At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.

Author : Ezra Getzler,Mikhail M. Kapranov
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 48,9 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821810569

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Higher Category Theory by Ezra Getzler,Mikhail M. Kapranov Pdf

Comprises six presentations on new developments in category theory from the March 1997 workshop. The topics are categorification, computads for finitary monads on globular sets, braided n- categories and a-structures, categories of vector bundles and Yang- Mills equations, the role of Michael Batanin's monoidal globular categories, and braided deformations of monoidal categories and Vassiliev invariants. No index. Annotation copyrighted by Book News, Inc., Portland, OR.

Author : Niles Johnson,Donald Yau
Publisher : Oxford University Press, USA
Page : 636 pages
File Size : 43,8 Mb
Release : 2021-01-31
Category : Mathematics
ISBN : 9780198871378

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2-Dimensional Categories by Niles Johnson,Donald Yau Pdf

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 : 51,5 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475747218

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Categories for the Working Mathematician by Saunders Mac Lane Pdf

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 : Jacob Lurie
Publisher : Princeton University Press
Page : 944 pages
File Size : 51,9 Mb
Release : 2009-07-06
Category : Mathematics
ISBN : 9781400830558

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Higher Topos Theory (AM-170) by Jacob Lurie Pdf

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Author : Grandis Marco
Publisher : World Scientific
Page : 536 pages
File Size : 54,6 Mb
Release : 2019-09-09
Category : Mathematics
ISBN : 9789811205125

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Higher Dimensional Categories: From Double To Multiple Categories by Grandis Marco Pdf

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 : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 53,5 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

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Category Theory in Context by Emily Riehl Pdf

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 : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 52,9 Mb
Release : 2024-07-03
Category : Electronic
ISBN : 8210379456XXX

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Homotopy Type Theory: Univalent Foundations of Mathematics by Anonim Pdf

Author : Emily Riehl
Publisher : Cambridge University Press
Page : 371 pages
File Size : 49,5 Mb
Release : 2014-05-26
Category : Mathematics
ISBN : 9781107048454

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Categorical Homotopy Theory by Emily Riehl Pdf

This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others.

Author : Benjamin C. Pierce
Publisher : MIT Press
Page : 117 pages
File Size : 49,7 Mb
Release : 1991-08-07
Category : Computers
ISBN : 9780262326452

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Basic Category Theory for Computer Scientists by Benjamin C. Pierce Pdf

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