  We study a simple lattice model with local symmetry, whose construction is
based on a crossed module of finite groups. Its dynamical degrees of freedom
are associated both to links and faces of a four-dimensional lattice. In
special limits the discussed model reduces to certain known topological quantum
field theories. In this work we focus on its dynamics, which we study both
analytically and using Monte Carlo simulations. We prove a factorization
theorem which reduces computation of correlation functions of local observables
to known, simpler models. This, combined with standard Krammers-Wannier type
dualities, allows us to propose a detailed phase diagram, which form is then
confirmed in numerical simulations. We describe also topological charges
present in the model, its symmetries and symmetry breaking patterns. The
corresponding order parameters are the Polyakov loop and its generalization,
which we call a Polyakov surface. The latter is particularly interesting, as it
is beyond the scope of the factorization theorem. As shown by the numerical
results, expectation value of Polyakov surface may serve to detects all phase
transitions and is sensitive to a value of the topological charge.
