  Topological orders are a class of phases of matter that beyond the Landau
symmetry breaking paradigm. The two (spatial) dimensional (2d) topological
orders have been thoroughly studied. It is known that they can be fully
classified by a unitary modular tensor category (UMTC) and a chiral central
charge c. And a class of 2d topological orders whose boundary are gappable can
be systematically constructed by Levin-Wen model whose ground states are
string-net condensed states. Previously, the three spatial dimensional
topological orders have been classified based on their canonical boundary
described by some special unitary fusion 2-category, $2\mathcal{V}ec_G^\omega$
or an EF 2-category. However, a lattice realization of a three spatial
dimensional topological orders with both canonical boundary and arbitrary
boundaries are still lacking. In this paper, we construct a 3d membrane-net
model based on spherical fusion 2-category, which can be used to systematically
study all general 3d topological order with gapped boundary. The partition
function and lattice Hamiltonian of the membrane-net model is constructed based
on state sum of spherical fusion 2-category. We also construct the 3d tube
algebra of the membrane-net model to study excitations in the model. We
conjecture all intrinsic excitations in membrane-net model have a one-to-one
correspondence with the irreducible central idempotents (ICI) of the 3d tube
algebra. We also provide a universal framework to study mutual statistics of
all excitations in 3d topological order through 3d tube algebra. Our approach
can be straightforwardly generalized to arbitrary dimension.
