We present a model allowing to extend an L-topology \(\tau \) on a set X (i.e. \(\tau \subseteq L^X\) ) to a bipolar \(\mathcal{L}\) -fuzzy topology \(\mathcal{T}\) on this set (i.e. \(\mathcal{T}: L^X \rightarrow \mathcal{L}\) ). This model is based on the use of a residuated-type structure on a lattice L, and the derived lattice \(\mathcal{L}\) obtained by “bipolarizing” the original lattice L. The properties of the obtained bipolar \(\mathcal{L}\) -fuzzy topology are studied. We specially consider the case when the original lattice L is enriched with a monoid Girard structure. In this case, the results obtained become especially transparent.