On Infinite Order Simple Current Extensions of Vertex Operator Algebras

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We construct a direct sum completion C⊕ of a given braided monoidal category C which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in CKL. As an example, we construct the vertex operator algebra Vₗ associated to an even lattice L as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.