  We consider categories of rational maps to algebraic groups and study
existence and construction of universal objects for such categories, using the
duality theory of Laumon 1-motives. In particular, we obtain functorial
descriptions of the Albanese of a singular projective variety X over an
algebraically closed field of characteristic zero, which is the universal
regular quotient of a relative Chow group of points on X, and of its dual.
