  A natural definition for instanton density operator in lattice QCD has been
long desired. We show this problem is, and has to be, resolved by higher
category theory. The problem is resolved by refining at a conceptual level the
Yang-Mills theory on lattice, in order to recover the homotopy information in
the continuum, which would have been lost if we put the theory on lattice in
the traditional way.
  The refinement needed is a generalization -- through the lens of higher
category theory -- of the familiar process of Villainization that captures
winding in lattice XY model and Dirac quantization in lattice Maxwell theory.
The apparent difference is that Villainization is in the end described by
principal bundles, hence familiar, but more general topological operators can
only be captured on the lattice by more flexible structures beyond the usual
group theory and fibre bundles, hence the language of categories becomes
natural and necessary. The key structure we need for our particular problem is
called multiplicative bundle gerbe, based upon which we can construct suitable
structures to naturally define the 2d Wess-Zumino-Witten term, 3d skyrmion
density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua)
non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density
operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills theory.
  In a broader perspective, higher category theory enables us to rethink more
systematically the relation between continuum quantum field theory and lattice
quantum field theory. We sketch a proposal towards a general machinery that
constructs the suitably refined lattice degrees of freedom for a given
non-linear sigma model or gauge theory in the continuum, realizing the desired
topological operators on the lattice.
