The topological degeneracy and quasiparticles I know this conclusion in topological order for a while: "the topological degeneracy on torus is equal to the number of quasiparticles types."
But can anyone give a physical argument that supports this conclusion?
Especially in Non-Abelian case.
 A: In general, this is not true even for Abelian case. For example, the ground state degeneracy of toric code on a $g$-genus torus is $2^{2g}$, which is equal to the order of the first homology group $H_1(\Sigma_g)\simeq \mathbb{Z}_2^{2g}$, but the number of quasiparticle types is still $4$. In fact, we know that the topological degeneracy is usually calculated from the homology group and the number of quasiparticle types is usually calculated by counting the irreps of local operator algebras. Thus, the number of quasiparticle types equal to the topological degeneracy only in the torus case. 
A: The number of anyon type is the intrinsic property of the phase, which (in 2+1d) is given by the number of the simple objects in the corresponding Unitary Modular Tensor Category. Such quantity is called the $\mathbb{R}^n$ observable.
The ground states degeneracy (GSD) is instead an observable which is related to the topology of the background spacetime, not the intrinsic property of the phase, and we call it a global observable.
While the number of anyon type is equal to the GSD when the phase is put on a torus. An argument is based on string-net model or just the toric code model: you can take any ground state in the ground state subspace, and then act the nontrivial (around the genus) closed string operators (open string operator can create 2 corresponding quasi-particles at the ends of the string, no matter it's Abelian or non-Abelian), the closed string operators commute with each other and commute with the Hamiltonian, so the states we get are still ground states, and now labeled by anyon type. You can further check that these ground states are linearly independent.
A: My answer is based on string-net models. Consider a 2D gapped topological system on torus. Mathematically, ground states of the system are characterized by representations of quantum double of the input group/quantum group. The quasiparticles carry quantum numbers also classified by representations of the quantum double. Described by the same mathematical object, their numbers are of course the same. For a physical picture, every ground state of the system can be labeled by one type of excitation running through the torus.
