Homotopy Theory of Enriched Mackey Functors Closed Multicategories, Permutative Enrichments, and Algebraic Foundations for Spectral Mackey Functors
Book contents
- London Mathematical Society Lecture Note Series
- Frontmatter
- Dedication
- Contents
- List of Main Facts
- List of Notations
- Preface
- 1 Motivations from Equivariant Topology
- Part 1 Background On Multicategories And K –Theory Functors
- Part 2 Homotopy Theory Of Pointed Multicategories, M1 –Modules, And Permutative Categories
- Part 3 Enrichment Of Diagrams And Mackey Functors In Closed Multicategories
- Part 4 Homotopy Theory Of Enriched Diagrams And Mackey Functors
- 12 Homotopy Equivalences between Enriched Diagram and Mackey Functor Categories
- 13 Applications to Multicategories and Permutative Categories
- Appendix A Categories
- Appendix B Enriched Category Theory
- Appendix C Multicategories
- Appendix D Open Questions
- Bibliography
- Index
12 - Homotopy Equivalences between Enriched Diagram and Mackey Functor Categories
from Part 4 - Homotopy Theory Of Enriched Diagrams And Mackey Functors
Published online by Cambridge University Press: 16 January 2025
Book contents
- London Mathematical Society Lecture Note Series
- Frontmatter
- Dedication
- Contents
- List of Main Facts
- List of Notations
- Preface
- 1 Motivations from Equivariant Topology
- Part 1 Background On Multicategories And K –Theory Functors
- Part 2 Homotopy Theory Of Pointed Multicategories, M1 –Modules, And Permutative Categories
- Part 3 Enrichment Of Diagrams And Mackey Functors In Closed Multicategories
- Part 4 Homotopy Theory Of Enriched Diagrams And Mackey Functors
- 12 Homotopy Equivalences between Enriched Diagram and Mackey Functor Categories
- 13 Applications to Multicategories and Permutative Categories
- Appendix A Categories
- Appendix B Enriched Category Theory
- Appendix C Multicategories
- Appendix D Open Questions
- Bibliography
- Index
Summary
This chapter establishes the general theory for a pair of nonsymmetric multifunctors (E,F) to provide inverse equivalences of homotopy theories between enriched diagram categories. The main result is Theorem 11.4.14 and does not require E or F to satisfy the symmetry condition of a multifunctor. A similar result for enriched Mackey functor categories, in Theorem 11.4.24, requires that E, but not necessarily F, is a multifunctor. This is important for the applications, Theorems 12.1.6 and 12.4.6. There, E is an endomorphism multifunctor and F is a corresponding free nonsymmetric multifunctor.
Keywords
- Type
- Chapter
- Information
- Homotopy Theory of Enriched Mackey FunctorsClosed Multicategories, Permutative Enrichments, and Algebraic Foundations for Spectral Mackey Functors, pp. 331 - 367Publisher: Cambridge University PressPrint publication year: 2025