Characterizing braided tensor categories associated to logarithmic vertex operator algebras

Given a non-semisimple braided tensor category, with oplax tensor functors from known braided tensor categories, we ask : How does this knowledge characterize the tensor product and the braiding? We develop tools that address this question. In particular we prove that the associator is fixed by the oplax tensor functors, and we show that a distinguished role is played by the coalgebra structure on the image of theses tensor functors. Our setup constrains the form of quasi bialgebras appearing in the logarithmic Kazhdan-Lusztig conjecture and it applies in particular to the representation categories of the triplet vertex algebras. Here the two oplax tensor functors are determined by two free field realizations, and the coalgebras mentioned above are the Nichols algebras of type $\mathfrak{sl}_2$. We demonstrate in the case of $p=2$ that our setup completely determines the braided tensor category and the realizing quasi-triangular quasi-Hopf algebra is as anticipated in \cite{FGR2}. This proves the logarithmic Kazhdan-Lusztig conjecture for $p=2$, while for general $p$ it only remains to establish that our characterization provides a unique braided tensor category.