# Definition of Topological Order in terms of categories

I have a question regarding the definition of topological order as defined in Wen's review article http://www.hindawi.com/journals/isrn/2013/198710/.

Is the distinction between boundary-gapped topological orders in 2+1 dimensions in terms of LU transformations or in terms of spherical fusion categories equivalent?

If so, is there a similar correspondence between categories for the general case?

My paper with Liang Kong, arXiv:1405.5858, presents the following conjecture: Bosonic topological orders in $n$-space-time dimensions (after quotient out the invertible topological orders) are described/classified by modular unitary $n$-categories with one object.
The reason that modular unitary $n$-categories fail to classify topological orders is because modular unitary $n$-categories with one object describe/classify topological excitations. The invertible topological orders has no non-trivial topological excitations. Thus all invertible topological orders correspond to the same trivial unitary $n$-category with one object. But after we quotient out the invertible topological orders, modular unitary $n$-categories with one object classify bosonic topological orders in $n$-space-time dimensions.