  We explain how it is possible to study $\mathrm{U}\!\left(1\right)$ BF theory
over a connected closed oriented smooth $3$-manifold in the formalism of path
integral thanks to Deligne-Beilinson cohomology. We show how we can
straightforwardly extend the definition to families of theories in any
dimension. We extend then the definition of the Turaev-Viro invariant of a
connected closed oriented smooth $3$-manifold in an Abelian framework to a
family of invariants in any dimension. We show that those invariants can be
written as discrete BF theories. We explain how the extensions of
$\mathrm{U}\!\left(1\right)$ BF theory we defined can be related to the
extensions of Turaev-Viro invariant we constructed.
