Twisted Modules of a Vertex Operator Algebra and Associators as Classifying Morphisms

The monoidal category of twisted modules of a Vertex Operator Algebra $V$ is defined and reduced to its 2-group of invertible objects $G_\alpha$, which can be described by a 3-cocycle $\alpha$ on its 0-truncation $G$ with values in the group of units $A$ of the field of definition of $V$ serving as its associator. This cocycle also presents the classifying morphism of an $\infty$-group extension of $G$ by the delooping $BA$. Motivated by this, it is proven that the $\infty$-group extension classified by a 3-cocycle $\alpha$ is presented by the skeletal 2-group $G_\alpha$ with associator $\alpha$. The results are discussed in light of current developments in Moonshine and $(\infty,1)$-topos theory.