Kernel and injectivity of the Grothendieck degree homomorphism

Determine whether the degree homomorphism from the Grothendieck group of the symmetric monoidal category of weighted dual skeletons, \(\overline{\deg}:K_0(SDS)\to\mathbb R_{>0}\), is injective and characterize its kernel, including the contributions of site cardinality, orbit structure, and orbit-ratio coordinates.

Background

The paper constructs a symmetric monoidal category whose objects are weighted dual skeletons and whose tensor product is the Cartesian product. The degree invariant KK is multiplicative under this product, so it induces a surjective group homomorphism from the Grothendieck group K0(SDS)K_0(SDS) to the multiplicative group of positive real numbers.

The unresolved issue is whether the degree captures all information in the Grothendieck group. The potential kernel may retain data concerning the number of sites, the involution’s orbit structure, and the internal orbit-ratio parameters. The paper suggests that additive invariants, such as the logarithm of the number of sites, could help distinguish nontrivial kernel classes, but does not establish whether these invariants descend to the Grothendieck group or provide an independent classification.

References

In Section 6 we proved that the degree \deg induces a surjective homomorphism from the Grothendieck group K_0(SDS) onto \mathbb R_{>0}, but we could not establish injectivity. Indeed, the kernel contains additional information about the number of sites, orbit structure, and coordinates r_O. A complete description of K_0(SDS) is a natural open problem.

— Algebra of spectral duality structures  (2609.04430 - Coso, 3 Sep 2026) in Remark 6, Section 6; Section 10.1, subsection “The Grothendieck group of the monoidal category”