The notion of vertex operator coalgebra is presented, which corresponds to the family of correlation functions modeling one string propagating in space-time splitting into n strings in conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have
comultiplicative structures meromorphically induced by conformal equivalence classes of the worldsheets swept out by propagating strings. We then show that this category is isomorphic to the
category of vertex operator coalgebras, which is defined in the language of formal algebra with a generalized Jacobi-like identity. This notion is in some sense dual to the notion of vertex operator algebra. We also prove that any vertex operator algebra equipped with a non-degenerate, Virasoro preserving, bilinear form gives rise to a corresponding vertex operator coalgebra. Finally, we explicitly calculate the vertex operator coalgebra structure and unique bilinear form for the Heisenberg algebra case, which corresponds to considering free bosons in conformal field theory.
