Putting continuum QFT (not just TQFT) onto the lattice is important for both fundamental understanding and practical numerics. The traditional way to do so, based on simple intuitions, however, does not admit natural definitions for general topological operators of continuous-valued fields---one prominent example is the lack of a natural definition for Yang-Mills instantons in lattice quantum chromodynamics.

In this talk, I will explain a more systematic way to relate continuum QFT and lattice QFT, based on higher categories and higher anafunctors, so that the topological operators in the continuum can be naturally defined on the lattice. The idea, though formulated formally, is physically very intuitive---we want to effectively capture the different possibilities of how a lattice field may interpolate into the continuum, so the higher categories that appear in higher homotopy theory are naturally involved. Via this formalism, we solve the long-standing problem of defining instanton and Chern-Simons term in lattice Yang-Mills theory using multiplicative bundle gerbes. Notably, when the continuous-valued fields in our formalism become discrete-valued, our construction reduces to the Dijkgraaf-Witten theory and Turaev-Viro theory, so we hope this formalism to be a good starting point towards (in the very long term) a more comprehensive categorical understanding of QFT that can encompass both continuous and discrete degrees of freedom, applicable both to IR and to UV.