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 topological K-theory spectrum ku is equivalent to the algebraic K-theory of the rig category V of complex vector spaces.

💡 Research Summary

The paper tackles a long‑standing gap in higher algebra: how to perform an additive group completion of a rig (bimonoidal) category while preserving its multiplicative structure. A rig category R carries two monoidal operations, a “sum’’ ⊕ and a “product’’ ⊗, together with distributivity, but classical Grothendieck‑type completions only address the additive monoid and typically destroy the product. Existing attempts either require extra symmetry, impose model‑category machinery, or give up the multiplicative coherence.

The authors introduce a construction that works entirely inside the categorical framework and yields a new category R′ that is simultaneously a group‑completion of the additive part and a genuine ring category. The key steps are:

1. Pair‑object model – Objects of R′ are formal pairs (a,b) with a,b∈Ob(R). An equivalence relation (a,b)∼(c,d) is imposed precisely when a⊕d ≅ b⊕c in R. This mirrors the usual Grothendieck relation a−b = c−d but is lifted to the level of objects and 2‑cells.

2. Additive structure – The sum on pairs is defined component‑wise: (a,b)⊕(c,d) = (a⊕c, b⊕d). The equivalence relation is stable under this operation, giving R′ a well‑defined symmetric monoidal ⊕ with unit (0,0).

3. Multiplicative lift – The product is defined by a distributive formula: \