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Author: Niles Johnson
Publisher: Oxford University Press (UK)
ISBN: 0198871376
Category : Computers
Languages : en
Pages : 636
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Book Description
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 (UK)
ISBN: 0198871376
Category : Computers
Languages : en
Pages : 636
Get Book Here
Book Description
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: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 1475747217
Category : Mathematics
Languages : en
Pages : 320
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Book Description
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: Georgia Beth
Publisher: Triangle Interactive, Inc.
ISBN: 1684525071
Category : Juvenile Nonfiction
Languages : en
Pages : 34
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Book Description
Read about a high-stakes competition for a job as a game designer! Two young game designers-Zak and Posie-both want a job at Phenomtech. They'll need to convince Phenomtech's CEO that their idea for a new game is the better one. It's a battle between old-school 2-D games and up-and-coming 3-D virtual reality games. Which designer will prevail? Let the games begin! Students will be engaged in reading fiction content as they learn 2-D and 3-D shapes. This book seamlessly integrates the teaching of math and reading, and uses real-world examples to teach math concepts. Text features include images, a glossary, an index, captions, and a table of contents to build students' vocabulary and reading comprehension skills as they interact with the text. The rigorous practice problems, math charts and diagrams, and sidebars extend learning and provide multiple opportunities for students to practice what they have learned. Math Talk provides an in-depth problem-solving experience.
Author: Georgia Beth
Publisher: Teacher Created Materials
ISBN: 1425859690
Category : Juvenile Fiction
Languages : en
Pages : 35
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Book Description
Read about a high-stakes competition for a job as a game designer! Two young game designers-Zak and Posie-both want a job at Phenomtech. They'll need to convince Phenomtech's CEO that their idea for a new game is the better one. It's a battle between old-school 2-D games and up-and-coming 3-D virtual reality games. Which designer will prevail? Let the games begin! Students will be engaged in reading fiction content as they learn 2-D and 3-D shapes. This book seamlessly integrates the teaching of math and reading, and uses real-world examples to teach math concepts. Text features include images, a glossary, an index, captions, and a table of contents to build students' vocabulary and reading comprehension skills as they interact with the text. The rigorous practice problems, math charts and diagrams, and sidebars extend learning and provide multiple opportunities for students to practice what they have learned. Math Talk provides an in-depth problem-solving experience.
Author: John C. Baez
Publisher: Springer Science & Business Media
ISBN: 1441915362
Category : Algebra
Languages : en
Pages : 292
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Book Description
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
ISBN: 1107044243
Category : Mathematics
Languages : en
Pages : 193
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Book Description
A short introduction ideal for students learning category theory for the first time.
Author: Marco Grandis
Publisher: World Scientific
ISBN: 9811205124
Category : Mathematics
Languages : en
Pages : 535
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Book Description
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
ISBN: 9780521540315
Category : Mathematics
Languages : en
Pages : 260
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Book Description
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:
Publisher: Teacher Created Materials
ISBN: 1425858465
Category : Juvenile Fiction
Languages : en
Pages : 35
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Book Description
In this exciting math reader, Vikram Patel wants to hire a game designer who will help his company create cutting-edge games. The first contender for the position is Zak, a master videogame designer and architect of shapes. The other competitor is Posie, an artist who is pushing the limits of virtual reality. Students will learn about 2-D and 3-D shapes as they are engaged in reading about this high-stakes competition. This 6-Pack of math readers builds math content knowledge and literacy skills, and uses real-world connections to help students explore math in a meaningful way. Text features such as a glossary, a table of contents, an index, and detailed illustrations will increase understanding and develop academic vocabulary. Let's Explore Math sidebars, the Problem Solving section, and the math charts and diagrams provide extensive opportunities for students to practice what they have learned. The DOK-leveled Math Talk section includes questions that facilitate mathematical discourse, and activities that students can respond to at home or school. This fiction title is sure to captivate readers as they are engaged in learning. This 6-Pack includes six copies of this title and a lesson plan.
Author: Emily Riehl
Publisher: Cambridge University Press
ISBN: 1108952194
Category : Mathematics
Languages : en
Pages : 782
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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.
