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 E?-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 1?3, 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.

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. 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. Chapter 9. The Cat-Multifunctor A. Chapter 10. Inverse K-Theory is a Pseudo Symmetric Cat-Multifunctor.

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.