We show that, for each symmetry class based on the tenfold way classification, the effective Dirac operator obtained by integrating out the additional bulk direction takes a value in the corresponding classifying space, from which we obtain the flat band Hamiltonian. We then obtain the overlap Dirac operator for each symmetry class and establish the Ginsparg--Wilson relation associated with $\mathcal{C}$ and $\mathcal{T}$ symmetries, and also the mod-two index theorem.