Defining four-vectors in General Relativity?

Nearly every general relativity textbook I look in has a section on four-vectors. However, this section is usually titled something like 'Four vectors in Special Relativity', 'Special Relativity (Again)' etc. I would therefore like to know the formal definition (without ambiguity) of the following four vectors in general relativity: displacement, 4-velocity, 4-wave vector, 4-momentum (of a massive particle) and 4-momentum (of a photon).

I am pretty sure I know some of these¹ since (correct me if I am wrong) at the origin of any 'locally orthogonal' coordinate system these must take the special relativistic values.

_{¹which I will not here reproduce since I want to compare what I think with any answers given without undue influence.}