A long standing problem in lattice QCD is to construct an explicit instanton density operator on the lattice. We introduce such a construction, after suitably refining the lattice Yang-Mills action, so that it captures the continuum Yang-Mills theory better than the traditional lattice Yang-Mills action does. This refinement needed follows naturally from higher category theory considerations, and in this sense it can be viewed as a generalization of the Villain model familiar in condensed matter physics. More broadly, we sketch a general program towards how to use the power of higher category theory to "most faithfully" put a generic continuum quantum field theory, with all of its topological aspects, onto the lattice.