Looking for a Refined Monster

Abstract: We discuss some categorical aspects of the objects that appear in the construction of the Monster and other sporadic simple groups. We define the basic representation of the categorical torus $\mathcal T$ classified by an even symmetric bilinear form $I$ and of the semi-direct product of $\mathcal T$ with its canonical involution. We compute the centraliser of the basic representation of $\mathcal T\rtimes{\pm1}$ and find it to be a categorical extension of the extraspecial $2$-group with commutator $I\mod 2$. We study the inertia groupoid of a categorical torus and find that it is given by the torsor of the topological Looijenga line bundle, so that $2$-class functions on $\mathcal T$ are canonically theta-functions. We discuss how discontinuity of the categorical character in our formalism means that the character of the basic representation fails to be a categorical class function. We compute the automorphisms of $\mathcal T$ and of $\mathcal T\rtimes{\pm1}$ and relate these to the Conway groups.