Ring completion of rig categories
arXiv:0706.0531 · doi:10.1515/crelle.2012.024
Abstract
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 topological K-theory spectrum ku is equivalent to the algebraic K-theory of the rig category V of complex vector spaces.
There was a mathematical error in arXiv:0706.0531v2: the map T in the purported proof of Lemma 3.7(2) is not well defined. Version 4 has been edited for notational consistency
Cited by in corpus (7)
- Universality of multiplicative infinite loop space machines
- Stable bundles over rig categories
- Connections on non-abelian Gerbes and their Holonomy
- Fluxes, bundle gerbes and 2-Hilbert spaces
- Higher topological cyclic homology and the Segal conjecture for tori
- Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings
- Generalizations of Loday's assembly maps for Lawvere's algebraic theories