Bimonoidal categories, \(E_n\)-monoidal categories, and algebraic \(K\)-theory. Volume II: Braided bimonoidal categories with applications (Q6605379)

This second volume is concerned with braided bimonoidal categories (Part 1 consisting of chapters 1--8) as well as --monoidal categories (Part 2 consisting of chapters 9 and 10).\N\NThe synopsis of the volume goes as follows.\N\N\begin{itemize}\N\item[Chapter 1] addresses the braid groups and braided monoidal categories as preliminaries on braided structures.\N\N\item[Chapter 2] defines braided bimonoidal categories. The first main result is Theorem 2.2.1 claiming that each braided bimonoidal satisfies all 24 Laplaza axioms. The second main result is Theorem 2.4.22 claiming that an abelian category with a compatible braided monoidal structure has the structure of a tight braided bimonoidal category.\N\N\item[Chapter 3] shows that braided bimonoidal categories arise naturally in quantum groups and topological quantum computation (TQC). The first main result is Theorem 3.2.19 claiming that for a (symmetric/braided) bialgebra \(A\), the category \(\mathsf{Mod}(A)\) of left \(A\)-modules equipped with the usual direct sum and tensor product is a tight (symmetric/braided)\ bimonoidal category, which is an extension of the important fact in quantum group theory that, for a braided bialgebra \(A\), \(\mathsf{Mod}(A)\) is a braided monoidal category. The second main result is Theorem 3.4.13 claiming that Fibonacci anyons are tight braided bimonoidal categories. The third main result is Theorem 3.6.14 claiming that Ising anyons are tight braided bimonoidal categories.\N\N\item[Chapter 4] generalizes the Drinfeld center of a monoidal category and the symmetric center of a braided monoidal category to the bimonoidal setting. Generalizing the Drinfeld center, Theorem 4.4.3 claims that, for each tight bimonoidal category \(\mathsf{C}\), the bimonoidal Drinfeld center \(\overline{\mathsf{C}}^{\mathsf{bi}}\) is a tight braided bimonoidal category. Generalizing the symmetric center, Theorem 4.5.3 claims that, for each braided bimonoidal category \(\mathsf{C}\), the bimonoidal symmetric center \(\mathsf{C}^{\mathsf{sym}}\) is a symmetric bimonoidal category.\N\N\item[Chapter 5] establishes the Coherence Theorem 5.4.4 for braided bimonoidal categories abiding by a monomorphism assumption, which is the braided analogue of Laplaza's second Coherence Theorem, confirming the Blass-Gurevich Conjecture [\textit{A. Blass} and \textit{Y. Gurevich}, Theor. Comput. Sci. 807, 73--94 (2020; Zbl 1439.18020)] in the form of commutative formal diagrams.\N\N\item[Chapter 6] establishes two Strictification Theorems 6.3.6 and 6.3.7 for tight braided bimonoidal categories, which are two positive answers to the Blass-Gurevich Conjecture [\textit{A. Blass} and \textit{Y. Gurevich}, Theor. Comput. Sci. 807, 73--94 (2020; Zbl 1439.18020)] in the form of strictification.\N\N\item[Chapter 7] establishes the braided version of Baez's Conjecture [\url{https://ncatlab.org/nlab/show/rig+category}]. The first version of the Braided Baez Conjecture (Theorem 7.3.4) claims that the finite ordinal category \(\Sigma\) is an initial object in the bicategorical sense. Another version is Theorem 7.3.6, which claims that the variant \(\Sigma^{\prime}\) of \(\Sigma \) is also such a lax bicolimit.\N\N\item[Chapter 8] establishes Theorem 8.4.7 claiming that, for each braided bicategory \(\mathsf{C}\), the matrix construction \(\mathsf{Mat}^{\mathsf{C}}\) is a monoidal bicategory.\N\N\item[Chapter 9] discusses ring and bipermutative categories in the sense of \textit{A. D. Elmendorf} and \textit{M. A. Mandell} [Adv. Math. 205, No. 1, 163--228 (2006; Zbl 1117.19001); Algebr. Geom. Topol. 9, No. 4, 2391--2441 (2009; Zbl 1205.19003)] and the braided version.\N\N\item[Chapter 10] addresses the categorical structure for the general \(E_{n}\) cases, keeping in mind the ring-like categories in the previous chapter correspond to \(E_{n}\)-symmetric spectra for \(n\in\left\{ 1,2,\infty\right\} \) via algebraic \(K\)-theory.\N\end{itemize}