  We introduce a systematic mathematical language for describing fixed point
models and apply it to the study to topological phases of matter. The framework
is reminiscent of state-sum models and lattice topological quantum field
theories, but is formalised and unified in terms of tensor networks. In
contrast to existing tensor network ansatzes for the study of ground states of
topologically ordered phases, the tensor networks in our formalism represent
discrete path integrals in Euclidean space-time. This language is more directly
related to the Hamiltonian defining the model than other approaches, via a
Trotterization of the respective imaginary time evolution. We introduce our
formalism by simple examples, and demonstrate its full power by expressing
known families of models in 2+1 dimensions in their most general form, namely
string-net models and Kitaev quantum doubles based on weak Hopf algebras. To
elucidate the versatility of our formalism, we also show how fermionic phases
of matter can be described and provide a framework for topological fixed point
models in 3+1 dimensions.
