Bimonoidal categories, \(E_n\)-monoidal categories, and algebraic \(K\)-theory. Volume I: symmetric bimonoidal categories and monoidal bicategories (Q6605378)

Bimonoidal categories are categorical analogues of rings without additive inverses, having been actively studied in category theory, homotopy theory, and algebraic \(K\)-theory since around 1970. This endeavor, consisting of three volumes, provides the first 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.\N\NThe synopsis of this volume goes as follows.\N\N\begin{itemize}\N\item[Part 1] is concerned with symmetric bimonoidal categories, consisting of Chapters 1--5.\N\N\item[Chapter 1] reviews the basics of category theory.\N\N\item[Chapter 2] introduces symmetric bimonoidal categories and bimonoidal categories, establishing Laplaza's theorem (Theorem 2.2.13) that says that half of the 24 symmetric bimonoidal category axioms are formal consequences of the other 12 axioms.\N\N\item[Chapter 3] establishes Laplaza's first coherence theorem (Theorem 3.9.1) for symmetric bimonoidal categories abiding by a monomorphism assumption. The analogue of this coherence theorem for bimonoidal categories is Theorem 3.10.7. \S 3.11 discusses the main differences between this chapter and Laplaza's original work [\textit{M. L. Laplaza}, Lect. Notes Math. 281, 29--65 (1972; Zbl 0244.18010)].\N\N\item[Chapter 4] establishes Laplaza's second coherence theorem (Theorem 4.4.3) for symmetric bimonoidal categories abiding by the same monomorphism assumption as in Theorem 3.9.1. The analogue of this coherence theorem for bimonoidal categories is Theorem 4.5.8. \S 4.7 discusses the main differences between this chapter and \textit{M. L. Laplaza}'s original work [Lect. Notes Math. 281, 214--235 (1972; Zbl 0244.18011)].\N\N\item[Chapter 5] establishes May's strictification theorem (Theorem 5.4.6) of tight symmetric bimonoidal categories to right bimonoidal categories. Theorem 5.4.7 is another version of the strictification theorem involving left bipermutative categories. Theorems 5.5.11 and 5.5.12 are the corresponding strictification results for tight bimonoidal categories.\N\N\item[Part 2] is concerned with bicategorical aspects of symmetric bimonoidal categories, consisting of Chapters 6--8.\N\N\item[Chapter 6] reviews the basics of 2-/bicategories, pasting diagrams, lax functors, lax transformations, modifications, and adjunctions in bicategories. Then it reviews multiplicative structures, including monoidal bicategories, their braided, sylleptic, and symmetric analogues, the Gray tensor product for 2-categories, (permutative) Gray monoids, permutative 2-categories.\N\N\item[Chapter 7] establishes Baez's conjecture (Theorems 7.8.1 and 7.8.3).\N\N\item[Chapter 8] establishes Theorem 8.15.4 claiming that, for each tight symmetric bimonoidal category \(\mathsf{C}\), a matrix construction \(\mathrm{Mat}^{\mathsf{C}}\)\ is a symmetric monoidal bicategory, with no strict structures in general.\N\end{itemize}