  We derive the collision term relevant for neutrino quantum kinetic equations
in the early universe and compact astrophysical objects, displaying its full
matrix structure in both flavor and spin degrees of freedom. We include in our
analysis neutrino-neutrino processes, scattering and annihilation with
electrons and positrons, and neutrino scattering off nucleons (the latter in
the low-density limit). After presenting the general structure of the collision
terms, we take two instructive limiting cases. The one-flavor limit highlights
the structure in helicity space and allows for a straightforward interpretation
of the off-diagonal entries in terms of the product of scattering amplitudes of
the two helicity states. The isotropic limit is relevant for studies of the
early universe: in this case the terms involving spin coherence vanish and the
collision term can be expressed in terms of two-dimensional integrals, suitable
for computational implementation.
