  We elaborate that for topological insulators and topological superconductors
described by Dirac models in any dimension and symmetry class, the topological
order can be mapped to lattice sites by a universal topological marker.
Deriving from a recently discovered momentum-space universal topological
invariant, we introduce a topological operator that consists of alternating
projectors to filled and empty lattice eigenstates and the position operators,
multiplied by the Dirac matrices that are omitted in the Hamiltonian. The
topological operator projected to lattice sites yields the topological marker,
whose form is explicitly constructed for every topologically nontrivial
symmetry class from 1D to 3D. The off-diagonal elements of the topological
operator yields a nonlocal topological marker, which decays with a correlation
length that diverges at topological phase transitions, and represents a Wannier
state correlation function. Various prototype examples, including
Su-Schrieffer-Heeger model, Majorana chain, Chern insulators,
Bernevig-Hughes-Zhang model, 2D chiral and helical $p$-wave superconductors,
lattice model of $^{3}$He B-phase, and 3D time-reversal symmetric topological
insulators, etc, are employed to demonstrate the ubiquity of our formalism.
