From Categories to Homotopy Theory

Description

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. 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