  Quantum tensor network states and more particularly projected entangled-pair
states provide a natural framework for representing ground states of gapped,
topologically ordered systems. The defining feature of these representations is
that topological order is a consequence of the symmetry of the underlying
tensors in terms of matrix product operators. In this paper, we present a
systematic study of those matrix product operators, and show how this relates
entanglement properties of projected entangled-pair states to the formalism of
fusion tensor categories. From the matrix product operators we construct a
C*-algebra and find that topological sectors can be identified with the central
idempotents of this algebra. This allows us to construct projected
entangled-pair states containing an arbitrary number of anyons. Properties such
as topological spin, the S matrix, fusion and braiding relations can readily be
extracted from the idempotents. As the matrix product operator symmetries are
acting purely on the virtual level of the tensor network, the ensuing Wilson
loops are not fattened when perturbing the system, and this opens up the
possibility of simulating topological theories away from renormalization group
fixed points. We illustrate the general formalism for the special cases of
discrete gauge theories and string-net models.
