This is the first paper of a three-part series in which we develop a theory of conformal blocks for C₂-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted generalized modules of C₂-cofinite VOAs.

We are grateful to Liang Kong and Hao Zheng for many enlightening conversations. In particular, we owe to them the definition of the vector space of a dual fusion product. We would also like to thank Chiara Damiolini, Angela Gibney, Yi-Zhi Huang, Haisheng Li, and Robert McRae for helpful discussions.