We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers ±1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.

This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.

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.

We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.

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.

We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence of ho-motopy theories. We describe an application to ring categories.

We show that each of the three K-theory multifunctors from small permutative categories to $$\mathcal {G}_*$$ -categories, $$\mathcal {G}_*$$ -simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these K-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann–Osorno $$\mathcal {E}_*$$ -categories is equivalent to the homotopy theory of pointed simplicial categories.

Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments along $K$-theory multifunctors provides an important tool for constructing spectral Mackey functors from Mackey functors enriched in algebraic structures such as permutative categories. This work gives a detailed development of diagrams, presheaves, and Mackey functors enriched over closed multicategories. Change of enrichment, including the relevant compositionality, is treated with care. This framework is applied to the homotopy theory of enriched diagram and Mackey functor categories, including equivalences of homotopy theories induced by $K$-theory multifunctors. Particular applications of interest include diagrams and Mackey functors enriched in pointed multicategories, permutative categories, and symmetric spectra.

We show that Mandell's inverse K-theory functor is a categorically enriched nonsymmetric multifunctor. In particular, it preserves algebraic structures parametrized by nonsymmetric operads. As applications, we describe how ring categories arise as the images of inverse K-theory.

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 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.

This book 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, adjunctions and 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, 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, 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.

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.

In this chapter, the tricategory of bicategories is presented in full detail. After a preliminary discussion of the whiskerings of a lax transformation with a lax functor, the chapter goes on to define a tricategory. The rest of the chapter proves in detail the existence of a tricategory with small bicategories as objects (i.e. a tricategory of bicategories), pseudofunctors as 1-cells, strong transformations as 2-cells, and modifications as 3-cells.

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, 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.

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. This work provides 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.

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.