  We connect two key concepts in quantum information: compatibility and
divisibility of quantum channels. Two channels are compatible if they can be
both obtained via marginalization from a third channel. A channel divides
another channel if it reproduces its action by sequential composition with a
third channel. (In)compatibility is of central importance for studying the
difference between classical and quantum dynamics. The relevance of
divisibility stands in its close relationship with the onset of Markovianity.
We emphasize the simulability character of compatibility and divisibility, and,
despite their structural difference, we find a set of channels --
self-degradable channels -- for which the two notions coincide. We also show
that, for degradable channels, compatibility implies divisibility, and that,
for anti-degradable channels, divisibility implies compatibility. These results
motivate further research on these classes of channels and shed new light on
the meaning of these two largely studied notions.
