Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.
Author(s): Birgit Richter
Series: Cambridge Studies in Advanced Mathematics 188
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: x+390
Contents
Introduction page
Part I Category Theory
1 Basic Notions in Category Theory
1.1 Definition of a Category and Examples
1.2 EI Categories and Groupoids
1.3 Epi- and Monomorphisms
1.4 Subcategories and Functors
1.5 Terminal and Initial Objects
2 Natural Transformations and the Yoneda Lemma
2.1 Natural Transformations
2.2 The Yoneda Lemma
2.3 Equivalences of Categories
2.4 Adjoint Pairs of Functors
2.5 Equivalences of Categories via Adjoint Functors
2.6 Skeleta of Categories
3 Colimits and Limits
3.1 Diagrams and Their Colimits
3.2 Existence of Colimits and Limits
3.3 Colimits and Limits in Functor Categories
3.4 Adjoint Functors and Colimits and Limits
3.5 Exchange Rules for Colimits and Limits
4 Kan Extensions
4.1 Left Kan Extensions
4.2 Right Kan Extensions
4.3 Functors Preserving Kan Extensions
4.4 Ends
4.5 Coends as Colimits and Ends as Limits
4.6 Calculus Notation
4.7 “All Concepts are Kan Extensions”
5 Comma Categories and the Grothendieck Construction
5.1 Comma Categories: Definition and Special Cases
5.2 Changing Diagrams for Colimits
5.3 Sifted Colimits
5.4 Density Results
5.5 The Grothendieck Construction
6 Monads and Comonads
6.1 Monads
6.2 Algebras over Monads
6.3 Kleisli Category
6.4 Lifting Left Adjoints
6.5 Colimits and Limits of Algebras over a Monad
6.6 Monadicity
6.7 Comonads
7 Abelian Categories
7.1 Preadditive Categories
7.2 Additive Categories
7.3 Abelian Categories
8 Symmetric Monoidal Categories
8.1 Monoidal Categories
8.2 Symmetric Monoidal Categories
8.3 Monoidal Functors
8.4 Closed Symmetric Monoidal Categories
8.5 Compactly Generated Spaces
8.6 Braided Monoidal Categories
9 Enriched Categories
9.1 Basic Notions
9.2 Underlying Category of an Enriched Category
9.3 Enriched Yoneda Lemma
9.4 Cotensored and Tensored Categories
9.5 Categories Enriched in Categories
9.6 Bicategories
9.7 Functor Categories
9.8 Day Convolution Product
Part II From Categories to Homotopy Theory
10 Simplicial Objects
10.1 The Simplicial Category
10.2 Simplicial and Cosimplicial Objects
10.3 Interlude: Joyal’s Category of Intervals
10.4 Bar and Cobar Constructions
10.5 Simplicial Homotopies
10.6 Geometric Realization of a Simplicial Set
10.7 Skeleta of Simplicial Sets
10.8 Geometric Realization of Bisimplicial Sets
10.9 The Fat Realization of a (Semi)Simplicial Set or Space
10.10 The Totalization of a Cosimplicial Space
10.11 Dold–Kan Correspondence
10.12 Kan Condition
10.13 Quasi-Categories and Joins of Simplicial Sets
10.14 Segal Sets
10.15 Symmetric Spectra
11 The Nerve and the Classifying Space of a Small Category
11.1 The Nerve of a Small Category
11.2 The Classifying Space and Some of Its Properties
11.3 π0 and π1 of Small Categories
11.4 The Bousfield Kan Homotopy Colimit
11.5 Coverings of Classifying Spaces
11.6 Fibers and Homotopy Fibers
11.7 Theorems A and B
11.8 Monoidal and Symmetric Monoidal Categories, Revisited
12 A Brief Introduction to Operads
12.1 Definition and Examples
12.2 Algebras Over Operads
12.3 Examples
12.4 E∞-monoidal Functors
13 Classifying Spaces of Symmetric Monoidal Categories
13.1 Commutative H-Space Structure on BC for C Symmetric Monoidal
13.2 Group Completion of Discrete Monoids
13.3 Grayson–Quillen Construction
13.4 Group Completion of H-Spaces
14 Approaches to Iterated Loop Spaces via Diagram Categories
14.1 Diagram Categories Determine Algebraic Structure
14.2 Reduced Simplicial Spaces and Loop Spaces
14.3 Gamma-Spaces
14.4 Segal K-Theory of a Permutative Category
14.5 Injections and Infinite Loop Spaces
14.6 Braided Injections and Double Loop Spaces
14.7 Iterated Monoidal Categories as Models for Iterated Loop Spaces
14.8 The Category n
15 Functor Homology
15.1 Tensor Products
15.2 Tor and Ext
15.3 How Does One Obtain a Functor Homology Description?
15.4 Cyclic Homology as Functor Homology
15.5 The Case of Gamma Homology
15.6 Adjoint Base-Change
16 Homology and Cohomology of Small Categories
16.1 Thomason Cohomology and Homology of Categories
16.2 Quillen’s Definition
16.3 Spectral Sequence for Homotopy Colimits in Chain Complexes
16.4 Baues–Wirsching Cohomology and Homology
16.5 Comparison of Functor Homology and Homology of Small Categories
References
Index
