Universal property of Rep(Sp(n, k)) via an antisymmetric self-duality

Establish that for any field k of characteristic zero and integer n ≥ 1, the 2-rig Rep(Sp(n, k)) is the free 2-rig on a self-dual object x of dimension n (i.e., An(x) invertible) whose counit ε: x ⊗ x → I is antisymmetric, equivalently satisfying ε ∘ Sx,x = −ε, where Sx,x is the symmetry isomorphism.

Background

For the symplectic group, the defining structure is a nondegenerate alternating bilinear form. In the 2-rig framework, this is encoded by giving a self-dual object with an antisymmetric counit map ε: x ⊗ x → I.

The conjecture asserts that Rep(Sp(n, k)) is generated freely by such data, paralleling how symplectic vector spaces are linear algebra’s free objects with a chosen symplectic form.