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Journal für die reine und angewandte Mathematik

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Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Ring completion of rig categories

1Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway

2Department of Mathematics, University of Bergen, 5008 Bergen, Norway

3Department Mathematik der Universität Hamburg, 20146 Hamburg, Germany

4Department of Mathematics, University of Oslo, 0316 Oslo, Norway

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2013, Issue 674, Pages 43–80, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2012.024, March 2012

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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 ℛ̅ 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 ℛ̅ will be a commutative ring category. In an accompanying paper we show how to use this construction to prove the conjecture that the algebraic K-theory of the connective topological K-theory ring spectrum ku is equivalent to the algebraic K-theory of the rig category 𝒱 of complex vector spaces.

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