  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.
