On finite group global and gauged q-form symmetries in TQFT

We describe a method to implement finite group global and gauged q-form symmetries into the axiomatic structure of d-dimensional Topological Quantum Field Theory (TQFT) in terms of bordisms decorated by cohomology classes. Namely, on a manifold with a boundary, the gauge field is considered as a class in an appropriate relative cohomology group. It is defined in a way that allows self-consistent cutting and gluing of the manifolds and involves a choice of a (d-q-2) -skeleton in the boundary. The method, in a sense, generalizes to arbitrary d and q a method that has been considered in the literature in the case of d=3, \;q=0, 1.