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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. Many illustrations and examples are included. Assuming only basic category theory, this monograph is intended for graduate students and researchers. In addition to being a coherent reference for the topics covered, this book is also suitable for a graduate student seminar and a reading course.

2-Dimensional Categories provides 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; the 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.

In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in bicategory theory, are discussed. In addition to well-known examples, the 2-categories of multicategories and of polycategories are constructed. This chapter ends with a discussion of duality of bicategories.

In this chapter, Grothendieck fibrations are defined, and the Grothendieck Fibration Theorem is proved. After discussing some basic definitions, properties, and examples of fibrations, this chapter constructs a 2-monad and proves in detail that its pseudo algebras are precisely cloven fibrations. Moreover, the strict algebras of this 2-monad are shown to be precisely split fibrations.

In this chapter, adjunctions and monads in 2-/bicategories are defined and discussed, along with their basic properties and examples. This includes the internal theory of mates, adjoint equivalences, and monads in bicategories, along with 2-monads on 2-categories and various concepts of algebras for a 2-monad. A section is devoted to several examples of duality in algebra as adjunctions in the bicategory of bimodules over rings.

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.

In this chapter, the Whitehead Theorem for bicategories is proved in detail. The Whitehead Theorem states that a pseudofunctor between bicategories is a biequivalence if and only if it is surjective up to adjoint equivalences on objects, surjective up to isomorphisms on 1-cells, and bijective on 2-cells. The chapter covers the lax slice bicategory, lax terminal objects, and the Quillen Theorem A for bicategories. A 2-categorical version of the Whitehead Theorem is also discussed.

In this chapter, categories are defined, and basic concepts are reviewed. Starting from the definitions of a category, a functor, and a natural transformation, the chapter reviews limits, adjunctions, equivalences, the Yoneda Lemma, monads, monoidal categories, and Mac Lane's Coherence Theorem. Enriched categories, which provide one characterization of 2-categories, are also discussed. This chapter makes this book self-contained and accessible to beginners.

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, sylleptic monoidal bicategories, and symmetric monoidal bicategories. The rest of the chapter discusses the Gray tensor product on 2-categories, Gray monoids, double categories, and monoidal double categories.

This chapter focuses on bicategorical limits and nerves, which are the 2-/bicategorical analogues of (co)limits and nerves. The chapter proves that lax limits, lax bilimits, pseudo limits, and pseudo bilimits are unique up to an equivalence and an invertible modification. The chapter discusses 2-limits and 2-colimits, as well as the Duskin nerve and the 2-nerve, which are two different generalizations of the 1-categorical Grothendieck nerve. In each case, an explicit description of the simplices is provided.

In this chapter, pasting diagrams are defined, and pasting theorems for 2-/bicategories are proved. Pasting is an essential reasoning tool in 2-dimensional category theory. Each pasting theorem says that a pasting diagram, in a 2-category or a bicategory, has a unique composite. String diagrams, which provide another way to visualize and manipulate pasting diagrams, are also discussed.

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax transformations, modifications, and icons. A section is devoted to representable pseudofunctors, representable transformations, and representable modifications, which will be used in the Bicategorical Yoneda Lemma.

There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra., Comment: 24 pages. To appear in Homology, Homotopy, and Applications. We discuss multifunctoriality of the free construction in arXiv:2202.13659

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted trees $\Omega$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\Xi$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad., Comment: This version has been accepted to AGT. Substantial updates throughout, including an alternative description (suggested by the referee) of the morphisms of $\Xi$, a new appendix, and various other improvements

The purpose of this chapter is to study coherence of involutive monoidal categories. The next chapter will deal with the symmetric case. In Section 5.1 and Section 5.2, we give explicit constructions of the free involutive monoidal category and of the free involutive strict monoidal category generated by a category. We observe that they are equivalent via a strict involutive strict monoidal functor. In Section 5.3 we show that in a small involutive monoidal category, every formal diagram is commutative. Here a formal diagram is defined as in the case of monoidal categories, but with the involutive structure also taken into account. In Section 5.4 we show that every involutive monoidal category can be strictified to an involutive strict monoidal category via an involutive adjoint equivalence involving involutive strong monoidal functors. The remaining two sections contain explicit constructions of the free involutive (strict) monoidal category generated by an involutive category.

This chapter contains a categorical analogue of the Gelfand–Naimark–Segal (GNS) construction, which is originally a functional analytic statement. For a C∗-algebra A, the GNS construction gives a correspondence between states on A and cyclic ∗-representations of A.

The purpose of this chapter is to study coherence of involutive symmetric monoidal categories. Most of the constructions are adapted from Chapter 5, where we studied coherence of involutive monoidal categories. In Section 6.1 and Section 6.2, we give explicit constructions of the free involutive symmetric monoidal category and of the free involutive strict symmetric monoidal category generated by a category. We observe that they are equivalent via a strict involutive strict symmetric monoidal functor. In Section 6.3 we show that in a small involutive symmetric monoidal category, every suitably defined formal diagram is commutative. In Section 6.4 we show that every involutive symmetric monoidal category can be strictified to an involutive strict symmetric monoidal category via an involutive adjoint equivalence involving involutive strong symmetric monoidal functors. The remaining two sections contain explicit constructions of the free involutive (strict) symmetric monoidal category generated by an involutive category.

In this chapter, we develop the involutive versions of (symmetric) monoidal categories, (commutative) monoids, monads, and algebras over a monad. In an involutive symmetric monoidal category, in addition to involutive commutative monoids, there is also a notion of a reversing involutive monoid, which incorporates the symmetry isomorphism in a way that is different from a commutative monoid. Reversing involutive monoids are important because they include complex unital ∗-algebras and their differential graded analogues as examples.

For each type of categories with extra structures, coherence can be understood as the question of describing the free structured category of a category. The purpose of this chapter is to study the coherence of involutive categories and related topics.

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $\Omega$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $\mathbf{U}$., Comment: Minor expository changes throughout, including a new section on "Further directions", Remark 4.3, Example 5.4, Figures 2 and 4

We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andr\'e Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full., Comment: Minor changes

In this chapter we recall the definition of an undirected wiring diagram and give a proof that the collection of undirected wiring diagrams forms an operad.

In this chapter we establish a finite presentation for the operad of undirected wiring diagrams involving 6 operadic generators and 17 relations.

In this chapter we observe that each undirected wiring diagram has a highly structured decomposition in terms of generators, called a stratified presentation.

In this chapter we extend the operad map in the previous chapter, which goes from the operad of normal wiring diagrams to the operad of undirected wiring diagrams, to the operad of all wiring diagrams. Furthermore, we show that the extended operad map is surjective.

The main purpose of this chapter is to provide a finite presentation theorem for algebras over the operad of undirected wiring diagrams. As examples, we apply the finite presentation theorem to the relational algebra and the typed relational algebra.