  We investigate the limit of sequences of vertex algebras. We discuss under
what condition the vector space direct limit of such a sequence is again a
vertex algebra. We then apply this framework to permutation orbifolds of vertex
operator algebras and their large N limit. We establish that for any nested
oligomorphic permutation orbifold such a large N limit exists, and we give a
necessary and sufficient condition for that limit to factorize. This helps
clarify the question of what VOAs are candidates for holographic conformal
field theories in physics.
