The discovery of topological phases of matter in the 1980's brought about a paradigm shift in the study of condensed matter systems and the general quest for a complete phase classification of many-body systems. Topological phases host a whole set of exotic properties such as excitations with anyonic statistics (neither bosonic nor fermionic), gapless (chiral) edge modes and topological ground state degeneracies. Although these phases cannot be understood through the Landau-Ginzburg symmetry breaking principle, they are described by generalized "categorical symmetries". The tensor network formalism, developed in the last two decades, has made these symmetries explicit in concrete lattice systems through their non-local action on the entanglement degrees of freedom of the corresponding quantum states. Furthermore, through a map coined the "strange correlator", the intimate relation between the mathematical description of both (2+1) dimensional topological field theories and (1+1) conformal field theories in terms of fusion categories has been exposed in concrete lattice settings using tensor network methods and algorithms. The situation both on the side of topological phases and conformal field theory is much less understood in three (and higher) dimensions. The goal of my research proposal is to boost our understanding of higher dimensional topological phases through the study of these non-local categorical symmetries.