Abstract
We introduce regular stratified piecewise linear manifolds to describe lattices and investigate the lattice model approach to topological quantum field theory in all dimensions. We introduce the unitary n+1 alterfold TQFT and construct it from a linear functional on an n-dimensional lattice model on an n-sphere satisfying three conditions: reflection positivity, homeomorphic invariance and complete finiteness. A unitary spherical n-category is mathematically defined and emerges as the local quantum symmetry of the lattice model. The alterfold construction unifies various constructions of n+1 TQFT from n-dimensional lattice models and n-categories. In particular, we construct a non-invertible unitary 3+1 alterfold TQFT from a linear functional and derive its local quantum symmetry as a unitary spherical 3-category of Ising type with explicit 20j-symbols, so that the scalar invariant of 2-knots in piecewise linear 4-manifolds could be computed explicitly.