  The notion of vertex operator coalgebra is presented and motivated via the
geometry of 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
worldsheets. We then show this category is isomorphic to the category of vertex
operator coalgebras, which is defined in the language of formal algebra. The
latter has several characteristics which give it the flavor of a coalgebra with
respect to the structure of a vertex operator algebra and several
characteristics that distinguish it from a standard dual -- both will be
mentioned.
