We point out that the integrability condition for lattice chiral determinant of overlap Weyl fermion can be reformulated in parallel with the modern understanding of anomaly inflow based on Dai-Freed theorem and topological classification of global anomalies by bordism invariance.
The known relations of the (2n+1)- and (2n+2)-dim domain-wall fermions and (2n)- and (2n+1)-dim overlap fermions, respectively, imply that Dai-Freed theorem and Atiya-Patodi-Singer index theorem.
These relations also hold precisely true on the lattice, where the complex phase of (2n+1)-dim overlap fermion determinant defines the $\eta$-invariant.
This $\eta$-invariant becomes ``bordism invariant", if the local chiral anomaly density of the (2n+2)-dim overlap fermion is classified as ``cohomologically" trivial along with the perturbative condition of gauge anomaly cancellation. Then, the integrability condition is given simply by the fact that the exponentiate of lattice $\eta$-invariant square is strictly unity for any admissible (2n+1)-dim gauge-link fields.