This textbook provides a first introduction to category theory, a powerful framework and tool for understanding mathematical structures. Designed for students with no previous knowledge of the subject, this book offers a gentle approach to mastering its fundamental principles.
Unlike traditional category theory books, which can often be overwhelming for beginners, this book has been carefully crafted to offer a clear and concise introduction to the subject. It covers all the essential topics, including categories, functors, natural transformations, duality, equivalence, (co)limits, and adjunctions. Abundant fully-worked examples guide readers in understanding the core concepts, while complete proofs and instructive exercises reinforce comprehension and promote self-study. The author also provides background material and references, making the book suitable for those with a basic understanding of groups, rings, modules, topological spaces, and set theory.
Based on the author's course at the Vrije Universiteit Brussel, the book is perfectly suited for classroom use in a first introductory course in category theory. Its clear and concise style, coupled with its detailed coverage of key concepts, makes it equally suited for self-study.
Author(s): Ana Agore
Series: Universitext
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 284
City: Cham
Tags: Categories, Functors, Limits, Colimits, Adjoint Functors
Preface
Contents
Frequently Used Notations
1 Categories and Functors
1.1 Set Theory
1.2 Categories: Definition and First Examples
1.3 Special Objects and Morphisms in a Category
1.4 Some Constructions of Categories
1.5 Functors
1.6 Isomorphisms of Categories
1.7 Natural Transformations: Representable Functors
1.8 The Duality Principle
Comma Categories
1.9 Functor Categories
1.10 Yoneda's Lemma
1.11 Exercises
2 Limits and Colimits
2.1 (Co)products, (Co)equalizers, Pullbacks and Pushouts
2.2 (Co)limit of a Functor. (Co)complete Categories
(Co)limit as a Functor
2.3 (Co)limit as a Representing Pair
2.4 (Co)limits by (Co)equalizers and (Co)products
2.5 (Co)limit Preserving Functors
2.6 (Co)limits in Comma Categories
2.7 (Co)limits in Functor Categories
2.8 Exercises
3 Adjoint Functors
3.1 Definition and Generic Examples
3.2 Adjoints Via Free Objects
3.3 Galois Connections
3.4 More Examples and Properties of Adjoint Functors
3.5 The Unit and Counit of an Adjunction
3.6 Another Characterisation of Adjoint Functors
3.7 (Co)reflective Subcategories
3.8 Equivalence of Categories
3.9 Localization
3.10 (Co)limits as Adjoint Functors
3.11 Freyd's Adjoint Functor Theorem
3.12 Special Adjoint Functor Theorem
3.13 Representable Functors Revisited
3.14 Exercises
4 Solutions to Selected Exercises
4.1 Chapter 1
4.2 Chapter 2
4.3 Chapter 3
References
Index
