Tannaka Reconstruction and the Monoid of Matrices

2025/04/04 by Baez, John C., Trimble, Todd · 1 citation
#Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2504.03094

Abstract

Settling a conjecture from an earlier paper, we prove that the monoid M(n,k) of n × n matrices in a field k of characteristic zero is the "walking monoid with an n-dimensional representation". More precisely, if we treat M(n,k) as a monoid in affine schemes, the 2-rig Rep(M(n,k)) of algebraic representations of M(n,k) is the free 2-rig on an object x with Λⁿ⁺¹(x) ≅ 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.

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