In recent years, there has been a lot of interest in a generalized notion of symmetry, obtained by relaxing the invertibility constraint and/or allowing symmetry operators to act on submanifolds rather than the full space. The mathematical structure underlying these generalized symmetries is provided by (higher) category theory, but it turns out that in the lattice setting, the abstract categorical formulation can be broken down to concrete tensor network operators that realize these generalized symmetries. In a certain sense, these tensor network operators provide the lattice representation theory of these generalized symmetries. As an application, I will explain how this representation theory provides a systematic, constructive theory for duality transformations on the lattice. Additionally, I will explain how dualities and generalized symmetries can be turned into unitary operators by including an ancillary degree of freedom, turning them into completely positive maps.