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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.
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.
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.
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 the relational 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 ( ,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 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.
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.
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.
The subject of this book is the theory of operads and colored
operads, sometimes called symmetric multicategories. A (colored)
operad is an abstract object which encodes operations with multiple
inputs and one output and relations between such operations. The
theory originated in the early 1970s in homotopy theory and quickly
became very important in algebraic topology, algebra, algebraic
geometry, and even theoretical physics (string theory). Topics
covered include basic graph theory, basic category theory, colored
operads, and algebras over colored operads. Free colored operads
are discussed in complete detail and in full generality. The
intended audience of this book includes students and researchers in
mathematics and other sciences where operads and colored operads
are used. The prerequisite for this book is minimal. Every major
concept is thoroughly motivated. There are many graphical
illustrations and about 150 exercises. This book can be used in a
graduate course and for independent study.
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