From Categories to Homotopy Theory (Cambridge Studies in Advanced Mathematics, Series Number 188)

"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." -- Prové de l'editor 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 "Category theory has at least two important features. The first one is that it allows us to structure our mathematical world. Many constructions that you encounter in your daily life look structurally very similar, like products of sets, products of topological spaces, products of modules and then you might be delighted to learn that there is a notion of a product of objects in a category and all the above examples are actually just instances of such products, here in the category of sets, topological spaces and modules, so you don't have to reprove all the properties products have, because they hold for every such construction. So category theory helps you to recognize things as what they are. It also allows you to express objects in a category by something that looks apparently way larger. For instance the Yoneda Lemma describes a set of the form F(C) (where C is an object of some category and F is a functor from that category to the category of sets) as the set of natural transformations between another nice functor and F. This might look like a bad deal, but in this set of natural transformations you can manipulate things and this reinterpretation for instance gives you cohomology operations as morphisms between the representing objects. Another feature is that you can actually use category theory in order to build topological spaces and to do homotopy theory. A central example is the nerve of a (small) category: You view the objects of your category as points, every morphism gives a 1-simplex, a pair of composable morphisms gives a 2-simplex and so on. Then you build a topological space out of this by associating a topological n-simplex to an n-simplex in the nerve, but you do some non-trivial gluing, for instance identity morphisms don't really give you any information so you shrink the associated edges. In the end you get a CW complex BC for every small category C. Properties of categories and functors translate into properties of this space and continuous maps between such spaces. For instance a natural transformation between two functors gives rise to a homotopy between the induced maps and an equivalence of categories gives a homotopy equivalence of the corresponding classifying spaces. Classifying spaces of categories give rise to classifying spaces of groups but you can also use them and related constructions to build the spaces of the algebraic K-theory spectrum of a ring, you can give models for iterated based loop spaces and you can construct explicit models of homotopy colimits and much more"-- Provided by publisher