  Given a collection of modules of a vertex algebra parametrized by an abelian
group, together with one dimensional spaces of composable intertwining
operators, we assign a canonical element of the cohomology of an Eilenberg-Mac
Lane space. This element describes the obstruction to locality, as the
vanishing of this element is equivalent to the existence of a vertex algebra
structure with multiplication given by our intertwining operators, and given
existence, the structure is unique up to isomorphism. The homological
obstruction reduces to an "evenness" problem that naturally vanishes for
2-divisible groups, so simple currents organized into odd order abelian groups
always produce vertex algebras. Furthermore, in cases most relevant to
conformal field theory (i.e., when we have well-behaved contragradients and
tensor products), we obtain our spaces of intertwining operators naturally, and
the evenness obstruction reduces to the question of whether the contragradient
bilinear form on certain order 2 currents is symmetric or skew-symmetric. We
show that if we are given a simple regular VOA and integral-weight modules
parametrized by a group of even units in the fusion ring, then the direct sum
admits the structure of a simple regular VOA, called the simple current
extension, and this structure is unique up to isomorphism.
