1st Edition
Grothendieck Construction of Bipermutative-Indexed Categories
The Grothendieck construction provides an explicit link between indexed categories and opfibrations. It is a fundamental concept in category theory and related fields with far-reaching applications. Bipermutative categories are categorifications of rings. They play a central role in algebraic K-theory and infinite loop space theory.
This monograph is a detailed study of the Grothendieck construction over a bipermutative category in the context of categorically enriched multicategories, with new and important applications to inverse K-theory and pseudo symmetric Eā-algebras. After carefully recalling preliminaries in enriched categories, bipermutative categories, and enriched multicategories, we show that the Grothendieck construction over a small tight bipermutative category is a pseudo symmetric Cat-multifunctor and generally not a Cat-multifunctor in the symmetric sense.
Pseudo symmetry of Cat-multifunctors is a new concept we introduce in this work.
The following features make it accessible as a graduate text or reference for experts:
- Complete definitions and proofs
- Self-contained background. Parts of Chapters 1ā3, 7, 9, and 10 contain background material from the research literature
- Extensive cross-references
- Connections between chapters. Each chapter has its own introduction discussing not only the topics of that chapter but also its connection with other chapters
- Open questions. Appendix A contains open questions that arise from the material in the text and are suitable for graduate students
This book is suitable for graduate students and researchers with an interest in category theory, algebraic K-theory, homotopy theory, and related fields. The presentation is thorough and self-contained, with complete details and background material for non-expert readers.
Part I: Bipermutative Categories, Enriched Multicategories, and Pseudo Symmetry
Chapter 1: Preliminaries on Enriched Categories and 2-Categories
Chapter 2: Symmetric Bimonoidal and Bipermutative Categories
Chapter 3: Enriched Multicategories and Multiequivalences
Chapter 4: Pseudo Symmetry
Part II: Grothendieck Multiequivalence from Bipermutative-Indexed Categories to Permutative Opfibrations
Chapter 5: Enriched Multicategories of Indexed Categories
Chapter 6: The Grothendieck Construction is a Pseudo Symmetric Cat-Multifunctor
Chapter 7: Permutative Opfibrations from Bipermutative-Indexed Categories
Chapter 8: The Grothendieck Construction is a Cat-Multiequivalence
Part III: Pseudo Symmetric Enriched Multifunctorial Inverse K-Theory
Chapter 9: The Cat-Multifunctor A
Chapter 10: Inverse K-Theory is a Pseudo Symmetric Cat-Multifunctor
Appendix A: Open Questions
Appendix B: List of Main Facts
Bibliography
Index
Biography
Donald Yau is a Professor of Mathematics at The Ohio State University at Newark, specializing in homotopy theory and algebraic K-theory. He obtained his PhD at MIT under the direction of Haynes Miller and held a post-doctoral position at the University of Illinois at Urbana-Champaign. He is the author of over fifty research articles and ten books.
