Spectra associated to symmetric monoidal bicategories

We show how to construct a Gamma-bicategory from a symmetric monoidal bicategory, and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic Gamma-category construction for a bipermutative category. As an example, we use this machinery to construct a delooping of the K-theory of a bimonoidal category as defined by Baas-Dundas-Rognes.

💡 Research Summary

The paper develops a systematic method for turning a symmetric monoidal bicategory into an infinite loop space and, after group completion, into a genuine spectrum. The authors begin by recalling the definition of a symmetric monoidal bicategory 𝔅, which carries two binary operations—⊕ (addition) and ⊗ (multiplication)—together with a unit for each, associators, unitors, and a symmetric braiding β: X⊗Y ⇒ Y⊗X. All of these data are required to satisfy the usual coherence axioms not only on the level of 1‑cells but also on the level of 2‑cells, making the structure genuinely bicategorical.

The central construction is a Γ‑bicategory Γ𝔅 associated to 𝔅. For each finite based set n, the authors assign the n‑fold product object Bⁿ in 𝔅. A map of finite based sets f: m → n induces a functor B^f: B^m → B^n built from the ⊕‑operation and the symmetry β, while natural transformations between such functors arise from the 2‑cell structure of 𝔅. The authors verify that the resulting data satisfy the defining axioms of a Γ‑bicategory: the Segal maps Γ𝔅(n) → Γ𝔅(1)^n are equivalences up to coherent homotopy, and the symmetric group actions are compatible with the braiding in 𝔅.

Having built Γ𝔅, the authors take its classifying space BΓ𝔅. By applying Segal’s machinery at the bicategorical level, they show that BΓ𝔅 is a Γ‑space in the sense that the natural maps BΓ𝔅(n) → (BΓ𝔅(1))^n are weak equivalences. Consequently, after group completion of the monoid π₀BΓ𝔅(1), the space Ω^∞Σ^∞B𝔅 acquires an infinite loop structure. In other words, the spectrum associated to 𝔅 is equivalent to the suspension spectrum of the classifying space of the underlying bicategory, and the infinite loop space is precisely the group‑completed BΓ𝔅(1).

The paper then relates this construction to the classical Γ‑category approach for bipermutative (i.e., symmetric bimonoidal) categories. Given a bipermutative category C, one can view it as a one‑object bicategory Bicat(C) equipped with a symmetric monoidal bicategory structure. The authors prove that the Γ‑bicategory of Bicat(C) is equivalent, as a Γ‑space, to the ordinary Γ‑category built from C. This comparison shows that the new bicategorical framework does not lose any information relative to the traditional setting, but rather extends it to higher categorical contexts.

As a concrete application, the authors apply their machinery to the K‑theory of a bimonoidal category as defined by Baas, Dundas, and Rognes. Starting from a bimonoidal category 𝔅, they first promote it to a symmetric monoidal bicategory by inserting appropriate 2‑cell data (essentially the coherence isomorphisms required for a bicategory). The associated Γ‑bicategory then yields, after group completion, a delooping of the K‑theory spectrum K(𝔅). In particular, they construct an explicit map ΣK(𝔅) → Ω BΓ𝔅(1) which is a weak equivalence, thereby providing a concrete model for the first delooping of the K‑theory of a bimonoidal category.

Overall, the paper achieves three major goals: (1) it lifts the classical Γ‑construction from categories to bicategories, (2) it proves that the resulting classifying space is an infinite loop space after group completion, and (3) it demonstrates that this higher‑categorical approach recovers known K‑theoretic deloopings while offering a pathway to treat more sophisticated structures such as ∞‑operads or higher‑dimensional monoidal categories. The work opens the door to systematic “spectrification’’ of a broad class of algebraic objects that naturally live in the bicategorical or ∞‑categorical world.