It's well know by the Gabriel-Rosenberg reconstruction theorem that a (quasi-separated) scheme $X$ is completely determined by its category of quasicoherent sheaves $\mathbf{QCoh}(X)$. The latter is always a symmetric monoidal category, so one can develop the viewpoint that identifies a symmetric monoidal category with a scheme or generalization thereof.
One can also, by way of this viewpoint, define non-symmetric monoidal categories (with some extra structure) as non-commutative schemes., see, e.g., this paper. But what if we don't go all the way to remove all symmetry conditions, but instead allow the category to have a non-symmetric braiding?
Has any work been done in this direction? I can expect potential connections to TQFT and knot theory.
