Donald Yau

This work develops techniques and basic results concerning the homotopy theory of enriched diagrams and enriched Mackey functors. Presentation of a category of interest as a diagram category has become a standard and powerful technique in a range of applications. Diagrams that carry enriched structures provide deeper and more robust applications. With an eye to such applications, this work provides further development of both the categorical algebra of enriched diagrams, and the homotopy theoretic applications in K-theory spectra. The title refers to certain enriched presheaves, known as Mackey functors, whose homotopy theory classifies that of equivariant spectra. More generally, certain stable model categories are classified as modules - in the form of enriched presheaves - over categories of generating objects. This text contains complete definitions, detailed proofs, and all the background material needed to understand the topic. It will be indispensable for graduate students and researchers alike.

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the general title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories-this book, Volume II: Braided Bimonoidal Categories with Applications, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book proves in detail Laplaza's two coherence theorems and May's strictification theorem of symmetric bimonoidal categories, as well as their bimonoidal analogues. This part includes detailed corrections to several inaccurate statements and proofs found in the literature. Part 2 proves Baez's Conjecture on the existence of a bi-initial object in a 2-category of symmetric bimonoidal categories. The next main theorem states that a matrix construction, involving the matrix product and the matrix tensor product, sends a symmetric bimonoidal category with invertible distributivity morphisms to a symmetric monoidal bicategory, with no strict structure morphisms in general.

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications-this book, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book studies braided bimonoidal categories, with applications to quantum groups and topological quantum computation. It is proved that the categories of modules over a braided bialgebra, of Fibonacci anyons, and of Ising anyons form braided bimonoidal categories. Two coherence theorems for braided bimonoidal categories are proved, confirming the Blass-Gurevich Conjecture. The rest of this part discusses braided analogues of Baez's Conjecture and the monoidal bicategorical matrix construction in Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories. Part 2 studies ring and bipermutative categories in the sense of Elmendorf-Mandell, braided ring categories, and $E_n$-monoidal categories, which combine $n$-fold monoidal categories with ring categories.

The three-volume set published by the AMS offers a comprehensive exploration of bimonoidal categories, their applications to quantum groups, topological quantum computation, and algebraic $K$-theory. With a focus on both established results and new discoveries, this work is an essential resource for graduate students and researchers in these fields.

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 E8-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 proofsSelf-contained background. Parts of Chapters 1–3, 7, 9, and 10 contain background material from the research literatureExtensive cross-referencesConnections between chapters. Each chapter has its own introduction discussing not only the topics of that chapter but also its connection with other chaptersOpen questions. Appendix A contains open questions that arise from the material in the text and are suitable for graduate studentsThis 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.

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.

Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

This monograph introduces involutive categories and involutive operads, featuring applications to the GNS construction and algebraic quantum field theory. The author adopts an accessible approach for readers seeking an overview of involutive category theory, from the basics to cutting-edge applications. Additionally, the author’s own recent advances in the area are featured, never having appeared previously in the literature. The opening chapters offer an introduction to basic category theory, ideal for readers new to the area. Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research. Chapters on coherence of involutive symmetric monoidal categories, and categorical GNS construction follow. The last chapter covers involutive operads and lays important coherence foundations for applications to algebraic quantum field theory. With detailed explanations and exercises throughout, Involutive Category Theory is suitable for graduate seminars and independent study. Mathematicians and mathematical physicists who use involutive objects will also find this a valuable reference.

This book provides a general and powerful definition of homotopy algebraic quantum field theory and homotopy prefactorization algebra using a new coend definition of the Boardman-Vogt construction for a colored operad. All of their homotopy coherent structures are explained in details, along with a comparison between the two approaches at the operad level. With chapters on basic category theory, trees, and operads, this book is self-contained and is accessible to graduate students.

Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads.The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of therelational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images. Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.

The topic of this book sits at the interface of the theory of higher categories (in the guise of (8,1)-categories) and the theory of properads. Properads are devices more general than operads and enable one to encode bialgebraic, rather than just (co)algebraic, structures.The text extends both the Joyal-Lurie approach to higher categories and the Cisinski-Moerdijk-Weiss approach to higher operads, and provides a foundation for a broad study of the homotopy theory of properads. This work also serves as a complete guide to the generalised graphs which are pervasive in the study of operads and properads. A preliminary list of potential applications and extensions comprises the final chapter.Infinity Properads and Infinity Wheeled Properads is written for mathematicians in the fields of topology, algebra, category theory, and related areas. It is written roughly at the second year graduate level, and assumes a basic knowledge of category theory.

This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.

This book gives a self-contained introduction to the theory of lambda-rings and closely related topics, including Witt vectors, integer-valued polynomials, and binomial rings. Many of the purely algebraic results about lambda-rings presented in this book have never appeared in book form before. This book concludes with a chapter on open problems related to lambda-rings.