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.

💡 Research Summary

The paper introduces a novel categorical structure called a fibered symmetric bimonoidal category and demonstrates how to associate to any such structure an E∞‑ring spectrum in a functorial manner. The work proceeds in four main stages.

First, the authors review the necessary background: symmetric monoidal categories, fibered categories, and the classical correspondence between symmetric monoidal categories and E∞‑spaces via Segal’s Γ‑space construction. They also recall the established routes from symmetric monoidal categories to E∞‑ring spectra (e.g., Elmendorf–Mandell, May’s operadic approach), setting the stage for a generalization that incorporates a parametrizing base category.

Second, they give a precise definition of a fibered symmetric bimonoidal category. One starts with a base symmetric monoidal category ((\mathcal C,\otimes_{\mathcal C},\mathbf 1_{\mathcal C})). Over it sits a fibered category (p\colon\mathcal D\to\mathcal C) such that each fiber (\mathcal D_c) carries its own symmetric monoidal structure ((\oplus_c,\mathbf 0_c)). Simultaneously, the total category (\mathcal D) is equipped with a second symmetric monoidal product (\otimes_{\mathcal D}) that is compatible with the projection (p). The authors list twelve coherence axioms that guarantee:

• associativity, unit, and symmetry for both (\otimes) and (\oplus) on (\mathcal D);
• compatibility of (\otimes_{\mathcal D}) with the base product (\otimes_{\mathcal C}) via the projection;
• a distributive law (\otimes_{\mathcal D}) over (\oplus_c) that holds fiberwise and respects reindexing along morphisms in (\mathcal C);
• naturality of all structure maps with respect to pull‑back functors between fibers.

These axioms capture the intuition that the fibers form a family of symmetric monoidal categories parametrized by (\mathcal C), while the whole assembly (\mathcal D) itself is a symmetric monoidal category that “lifts” the monoidal structure of the base.

Third, the authors construct a functor from such a fibered bimonoidal category to the category of spectra. For each object (c\in\mathcal C) they apply the Segal Γ‑space machine to the fiber ((\mathcal D_c,\oplus_c)) to obtain an (E_{\infty})-space (X_c). The product (\otimes_{\mathcal D}) induces maps (X_c\wedge X_{c’}\to X_{c\otimes_{\mathcal C}c’}) that satisfy the operadic axioms of an (E_{\infty})-ring. By assembling these maps across all objects of (\mathcal C) and using the machinery of enriched model categories, they produce a strong symmetric monoidal functor \