  Classically, regular homomorphisms have been defined as a replacement for
Abel--Jacobi maps for smooth varieties over an algebraically closed field. In
this work, we interpret regular homomorphisms as morphisms from the functor of
families of algebraically trivial cycles to abelian varieties and thereby
define regular homomorphisms in the relative setting, e.g., families of schemes
parameterized by a smooth variety over a given field. In that general setting,
we establish the existence of an initial regular homomorphism, going by the
name of algebraic representative, for codimension-2 cycles on a smooth proper
scheme over the base. This extends a result of Murre for codimension-2 cycles
on a smooth projective scheme over an algebraically closed field. In addition,
we prove base change results for algebraic representatives as well as descent
properties for algebraic representatives along separable field extensions. In
the case where the base is a smooth variety over a subfield of the complex
numbers we identify the algebraic representative for relative codimension-2
cycles with a subtorus of the intermediate Jacobian fibration which was
constructed in previous work.
