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We propose a unified mathematical framework for classifying phases of matter.The framework is based on different types of combinatorial structures with anotion of locality called lattices. A tensor lattice is a local prescriptionthat associates tensor networks to those lattices. Different lattices arerelated by local operations called moves. Those local operations defineconsistency conditions for the tensors of the tensor network, the solutions towhich yield exactly solvable models for all kinds of phases. We implement theframework to obtain models for symmetry-breaking and topological phases in upto three space-time dimensions, their boundaries, defects, domain walls andsymmetries, as well as their anyons for 2+1-dimensional systems. We alsodeliver ideas of how other kinds of phases, like SPT/SET, fermionic,free-fermionic, chiral, and critical phases, can be described within ourframework. We also define another structure called contracted tensor latticeswhich generalize tensor lattices: The former associate tensors instead oftensor networks to lattices, and the consistency conditions for those tensorsare defined by another kind of local operation called gluings. Using thisgeneralization, our framework also covers mathematical structures likeaxiomatic (non-fully extended or defective) TQFTs, that do not directlydescribe phases on a microscopic physical level, but formalize certain aspectsof potential phases, like the anyon statistics of 2+1-dimensional phases. Wealso introduce the very powerful concept of (contracted) tensor latticemapping, unifying a lots of different operations, such as stacking, anyonfusion, anyon condensation, equivalence of different fixed point models, takingthe Drinfel'd centre, trivial defects or interpreting a bosonic model as afermionic model.

Towards a mathematical formalism for classifying phases of matter