  We propose a unified mathematical framework for classifying phases of matter.
The framework is based on different types of combinatorial structures with a
notion of locality called lattices. A tensor lattice is a local prescription
that associates tensor networks to those lattices. Different lattices are
related by local operations called moves. Those local operations define
consistency conditions for the tensors of the tensor network, the solutions to
which yield exactly solvable models for all kinds of phases. We implement the
framework to obtain models for symmetry-breaking and topological phases in up
to three space-time dimensions, their boundaries, defects, domain walls and
symmetries, as well as their anyons for 2+1-dimensional systems. We also
deliver ideas of how other kinds of phases, like SPT/SET, fermionic,
free-fermionic, chiral, and critical phases, can be described within our
framework. We also define another structure called contracted tensor lattices
which generalize tensor lattices: The former associate tensors instead of
tensor networks to lattices, and the consistency conditions for those tensors
are defined by another kind of local operation called gluings. Using this
generalization, our framework also covers mathematical structures like
axiomatic (non-fully extended or defective) TQFTs, that do not directly
describe phases on a microscopic physical level, but formalize certain aspects
of potential phases, like the anyon statistics of 2+1-dimensional phases. We
also introduce the very powerful concept of (contracted) tensor lattice
mapping, unifying a lots of different operations, such as stacking, anyon
fusion, anyon condensation, equivalence of different fixed point models, taking
the Drinfel'd centre, trivial defects or interpreting a bosonic model as a
fermionic model.
