  In classical thermodynamics the work cost of control can typically be
neglected. On the contrary, in quantum thermodynamics the cost of control
constitutes a fundamental contribution to the total work cost. Here, focusing
on quantum refrigeration, we investigate how the level of control determines
the fundamental limits to cooling and how much work is expended in the
corresponding process. \jona{We compare two extremal levels of control. First
coherent operations, where the entropy of the resource is left unchanged, and
second incoherent operations, where only energy at maximum entropy (i.e. heat)
is extracted from the resource. For minimal machines, we find that the lowest
achievable temperature and associated work cost depend strongly on the type of
control, in both single-cycle and asymptotic regimes. We also extend our
analysis to general machines.} Our work provides a unified picture of the
different approaches to quantum refrigeration developed in the literature,
including algorithmic cooling, autonomous quantum refrigerators, and the
resource theory of quantum thermodynamics.
