  Based on the modular functor associated with a -- not necessarily semisimple
-- finite non-degenerate ribbon category $\mathcal D$, we present a definition
of a consistent system of bulk field correlators for a conformal field theory
which comprises invariance under mapping class group actions and compatibility
with the sewing of surfaces. We show that when restricting to surfaces of genus
zero such systems are in bijection with commutative symmetric Frobenius
algebras in $\mathcal D$, while for surfaces of any genus they are in bijection
with modular Frobenius algebras in $\mathcal D$. This provides additional
insight into structures familiar from rational conformal field theories and
extends them to rigid logarithmic conformal field theories.
