Download three volumes (pdf, 7Mb).

Update 2024-02-11:
1. Preface reorganized and rewritten.
2. Removed Chapter I.9, about matrix permutative Gray monoids, along with two related questions from Appendix A. This does not impact the rest of the text.
3. Added Note III.10.8.10 about Quillen K-Theory.

See change log for further details.

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.