Universal property of Rep(SL(n, k)) with trivialized top exterior power

Establish that for any field k of characteristic zero and integer n ≥ 1, the 2-rig Rep(SL(n, k)) is the free 2-rig on an object x equipped with a specified isomorphism An(x) ≅ I, where I denotes the tensor unit, i.e., verify that Rep(SL(n, k)) has the universal property of freely adjoining an object whose nth exterior power is trivialized.

Background

Motivated by the established universal property for Rep(M(n, k)), the authors formulate conjectures for classical groups. For SL(n, k), the determinant-one condition naturally corresponds to trivializing the nth exterior power.

The conjecture posits that Rep(SL(n, k)) is the free 2-rig generated by an object x together with an isomorphism An(x) ≅ I, mirroring how the determinant character is trivial on SL(n, k).