Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

TL;DR

This paper offers a comprehensive overview of bimonoidal and higher ring-like categories, exploring their connections to algebraic K-theory, homotopy theory, and applications in quantum groups and topological quantum computation.

Contribution

It unifies the treatment of bimonoidal and higher ring-like categories, linking them to algebraic K-theory and homotopy theory, and discusses new applications and open questions.

Findings

Unified framework for bimonoidal and higher ring-like categories

Connections established with algebraic K-theory and homotopy theory

Applications demonstrated in quantum groups and topological quantum computation

Abstract

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.