Placement of ⊙-bicharades within known mathematical structures

Determine the mathematical classification or conceptual context of a ⊙-bicharade, defined by an object B in a symmetric monoidal category together with a morphism E: B⊗B → B⊗B satisfying (1⊗E)(E⊗1)=(E⊗1)(1⊗σ)(E⊗1)(1⊗σ)(1⊗E), where σ is the symmetry. Clarify whether such structures correspond to any established algebraic or categorical notions or constitute a new class.

Background

Example 29 treats the terminal category ⊙, which is strict bicharadic. In this case, a ⊙-bicharade reduces to a single object B with a self-map E on B⊗B subject to a nontrivial coherence identity involving the symmetry σ.

The authors explicitly remark that they lack a known framework or classification for this object, thereby posing a conceptual question about its nature and connections with familiar structures (e.g., braidings, Yang–Baxter-type solutions, or other categorical symmetries).

References

We have no idea where to place this object.

— Bivariant operadic categories  (2402.12963 - Markl, 2024) in Example 29, Section 4 (Bicharades)

Whether the resulting quadratic algebra admits a natural coproduct, and whether its appropriate language is that of a bialgebra, a multi-object braided category, a weak Hopf algebra or a bialgebroid, remains open.

— Multiparameter Quantum Affine Spaces and the Scalene Yang--Baxter Equation  (2608.20714 - Padmanabhan et al., 21 Aug 2026) in Section 3, “Scalene character, Yang--Baxter systems and possible RTT structures” (also discussed in Section “Conclusion and outlook”)