  Let $V$ be a vertex operator algebra satisfying certain reductivity and
finiteness conditions such that $\mathcal{C}_V$, the category of V-modules, is
a modular tensor category. We study open-closed field algebras over V equipped
with nondegenerate invariant bilinear forms for both open and closed sectors.
We show that they give algebras over certain $\C$-extension of the Swiss-cheese
partial dioperad, and we obtain Ishibashi states easily in such algebras. We
formulate Cardy condition algebraically in terms of the action of the modular
transformation $S: \tau \mapsto -\frac{1}{\tau}$ on the space of intertwining
operators. We then derive a graphical representation of S in the modular tensor
category $\mathcal{C}_V$. This result enables us to give a categorical
formulation of Cardy condition and modular invariant conformal full field
algebra over $V\otimes V$. Then we incorporate the modular invariance condition
for genus-one closed theory, Cardy condition and the axioms for open-closed
field algebra over V equipped with nondegenerate invariant bilinear forms into
a tensor-categorical notion called Cardy $\mathcal{C}_V|\mathcal{C}_{V\otimes
V}$-algebra. We also give a categorical construction of Cardy
$\mathcal{C}_V|\mathcal{C}_{V\otimes V}$-algebra in Cardy case.
