An involution on the K-theory of bimonoidal categories with anti-involution

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

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 pres…

关于双幺半函子的May猜想与乘法性无限环空间理论

May's Conjecture on Bimonoidal Functors and Multiplicative Infinite Loop Space Theory

本文证明了关于双置换范畴间对称双幺半函子严格化的May猜想的较弱形式，并将其应用于May的乘法性无限环空间机器，实现了乘法强对称双幺半函子到严格对称双幺半函子的替换。

From fibered symmetric bimonoidal categories to symmetric spectra

In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories D->C whose fibers are symmetric monoidal categories parametrized by C and such that both D and C have a further structure of a symmetric monoidal category that satisfy certain coherences that we describe. Our goal is to show that we can correspond to a fibered symmetric bimonoidal category an E_{\infty}-ring spectrum in a functorial way.

环 operad 与对称双幺半范畴

Ring operads and symmetric bimonoidal categories

本研究将经典operad对理论推广为环operad理论，建立其与经典理论的关联，使乘法无限循环机器可应用于任意E∞环operad代数，并证明对称双幺半范畴的分类空间与环operad意义下的E∞环空间同胚，提供了从对称双幺半范畴构造经典E∞环空间的新方案。

Symmetric bimonoidal intermuting categories and $ω\timesω$ reduced bar constructions

A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for $ω\timesω$-indexed family of iterated reduced bar constructions based on such a category.

Ring completion of rig categories

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category R (also known as a symmetric bimonoidal category), the additive group completion R' will be a commutative ring category. In an accompanying paper we show how this can be used to prove the conjecture from [BDR] that the algebraic K-theory of the connective top…