Uniqueness of the prop structure on E_Λ

Establish that the symmetric monoidal prop structure on the category E_Λ^• defined by Theorem T:main is unique among all prop structures on E_Λ^• whose object-level monoidal product is addition of natural numbers and which satisfy, for every integer m ≥ 0, the identities ρ_(1) ⊙ ρ_(1^m) = ρ_(1^{m+1}) and ρ_(1) ⊙ ρ_(2,1^m) = ρ_(2,1^{m+1}). Here E_Λ^•(m,n) denotes the graded morphism space concentrated in degree m−n and spanned by partitions of m into n parts, and ⊙ is the monoidal product introduced in Theorem T:main.

Background

The paper studies the graded linear prop E = ΩC freely generated by the operadic suspension of the commutative operad C, and then constructs the category E_Λ by passing to a subcategory of the Karoubi envelope associated to the alternating idempotents e_(1^n). The morphism spaces E_Λ^•(m,n) are concentrated in degree m−n and have bases given by partitions of m into n parts.

The monoidal product of E does not restrict to a prop structure on E_Λ, so the authors introduce a new monoidal product ⊙ (Theorem T:main) under which E_Λ becomes a symmetric monoidal category (a prop). This product aggregates morphisms in a uniform “averaging” way across surjections/partitions.

The conjecture asserts a uniqueness characterization of this newly defined prop structure by specifying the object-level monoidal product (sum of integers) and two normalization identities involving the generator ρ_(1) acting on identities and on degree-one elements. Proving this would canonically characterize the prop structure on partitions and, via the established equivalences, the prop governing Ext-groups between exterior power functors.