Tannaka Reconstruction and the Monoid of Matrices

Settling a conjecture from an earlier paper, we prove that the monoid $\mathrm{M}(n,k)$ of $n\times n$ matrices in a field $k$ of characteristic zero is the “walking monoid with an $n$-dimensional representation”. More precisely, if we treat $\mathrm{M}(n,k)$ as a monoid in affine schemes, the 2-rig $\mathrm{R}\mathrm{e}\mathrm{p}(\mathrm{M}(n,k))$ of algebraic representations of $\mathrm{M}(n,k)$ is the free 2-rig on an object $x$ with ${\mathrm{\Lambda }}^{n+1}(x)\cong 0$. Here a “2-rig” is a symmetric monoidal $k$-linear category that is Cauchy complete. Our proof uses Tannaka reconstruction and a general theory of quotient 2-rigs and 2-ideals. We conclude with a series of conjectures about the universal properties of representation 2-rigs of classical groups.