The celebrated Ptolemy relation plays an important role in various studies of triangulated surfaces including hyperbolic geometry, geometrical applications of cluster algebras and so on. We will discuss a noncommutative version of the relation which can be seen as a "categorification" of the classical one. This leads to new noncommutative topological invariants of the surfaces and provides several examples of the noncommutative Laurent phenomenon. (Joint work with Arkady Berenstein from University of Oregon)