Topology: A Categorical Approach

Description

A graduate-level textbook that presents basic topology from the perspective of category theory. This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics. After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, with examples; looks in detail at adjunctions in topology, particularly in mapping spaces; and examines additional adjunctions, presenting ideas from homotopy theory, the fundamental groupoid, and the Seifert van Kampen theorem. End-of-chapter exercises allow students to apply what they have learned. The book expertly guides students of topology through the important transition from undergraduate student with a solid background in analysis or point-set topology to graduate student preparing to work on contemporary problems in mathematics. Cover Preface Preliminaries Basic Topology Basic Category Theory Categories Functors Basic Set Theory Functions The Empty Set and One-Point Set Products and Coproducts in Set Products and Coproducts in Any Category Exponentiation in Set Partially Ordered Sets Exercises Examples and Constructions Examples and Terminology Examples of Spaces Examples of Continuous Functions The Subspace Topology The First Characterization The Second Characterization The Quotient Topology The First Characterization The Second Characterization The Product Topology The First Characterization The Second Characterization The Coproduct Topology The First Characterization The Second Characterization Homotopy and the Homotopy Category Exercises Connectedness and Compactness Connectedness Definitions, Theorems, and Examples The Functor π₀ Constructions and Connectedness Local (Path) Connectedness Hausdorff Spaces Compactness Definitions, Theorems, and Examples Constructions and Compactness Local Compactness Exercises Limits of Sequences and Filters Closure and Interior Sequences Filters and Convergence Tychonoff’s Theorem Ultrafilters and Compactness A Proof of Tychonoff’s Theorem A Little Set Theory Exercises Categorical Limits and Colimits Diagrams Are Functors Limits and Colimits Examples Terminal and Initial Objects Products and Coproducts Pullbacks and Pushouts Inverse and Direct Limits Equalizers and Coequalizers Completeness and Cocompleteness Exercises Adjunctions and the Compact-Open Topology Adjunctions The Unit and Counit of an Adjunction Free-Forgetful Adjunction in Algebra The Forgetful Functor U: Top → Set and Its Adjoints Adjoint Functor Theorems Compactifications The One-Point Compactification The Stone-Čech Compactification The Exponential Topology The Compact-Open Topology The Theorems of Ascoli and Arzela Enrich the Product-Hom Adjunction in Top Compactly Generated Weakly Hausdorff Spaces Exercises Paths, Loops, Cylinders, Suspensions, . . . Cylinder-Free Path Adjunction The Fundamental Groupoid and Fundamental Group The Categories of Pairs and Pointed Spaces The Smash-Hom Adjunction The Suspension-Loop Adjunction Fibrations and Based Path Spaces Mapping Path Space and Mapping Cylinder Examples and Results Applications of π₁S¹ The Seifert van Kampen Theorem Examples Exercises Glossary of Symbols Bibliography Index