Classic work in the '70s and '80s showed how many lattice models, both local and geometric, could be treated in a unified framework by expressing the quantum Hamiltonian or 2d classical transfer matrix in terms of an algebra, most prominently that of Temperley and Lieb. Classic work starting with Jones in the '80s showed how said models could be analysed by using fusion categories. All these results are naturally described in the language of tensor networks. Aasen, Mong and I have shown how this setup allows one to construct lattice topological defects that generate symmetries generalising Kramers-Wannier duality of the Ising and Potts models. Such "non-invertible" symmetries allow a number of exact computations to be done directly on the lattice, including the g-factors of boundary CFT, critical exponents and the spectrum of low-lying states. I will give an overview of these results, and briefly describe an application to the Ryberg-blockade ladder.