  We give a framework to produce constructible functions from natural functors
between categories, without need of a morphism of moduli spaces to model the
functor. We show using the Riemann-Hilbert correspondence that any natural
(derived) functor on constructible sheaves on smooth complex algebraic
varieties can be used to construct a special kind of constructible sets, called
absolute sets, generalizing a notion introduced by Simpson in presence of
moduli. We conjecture that the absolute sets of local systems satisfy a
"special varieties package", among which is an analog of the Manin-Mumford,
Mordell-Lang, and Andr\'e-Oort conjectures. The conjecture gives a simple proof
of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the
Decomposition Theorem for the geometric ones. We prove the conjecture in the
rank one case by showing that the closed absolute sets in this case are finite
unions of torsion-translated affine tori. This extends a structure result of
the authors for cohomology jump loci to any other natural jump loci. For
example, to jump loci of intersection cohomology and Leray filtrations. We also
show that the Leray spectral sequence for the open embedding in a good
compactification degenerates for all rank one local systems at the usual page,
not just for unitary local systems.
