  The laws of thermodynamics are usually formulated under the assumption of
infinitely large environments. While this idealization facilitates theoretical
treatments, real physical systems are always finite and their interaction range
is limited. These constraints have consequences for important tasks such as
cooling, not directly captured by the second law of thermodynamics. Here, we
study catalytic transformations that cannot be achieved when a system
exclusively interacts with a finite environment. Our core result consists of
constructive conditions for these transformations, which include the
corresponding global unitary operation and the explicit states of all the
systems involved. From this result we present various findings regarding the
use of catalysts for cooling. First, we show that catalytic cooling is always
possible if the dimension of the catalyst is sufficiently large. In particular,
the cooling of a qubit using a hot qubit can be maximized with a catalyst as
small as a three-level system. We also identify catalytic enhancements for
tasks whose implementation is possible without a catalyst. For example, we find
that in a multiqubit setup catalytic cooling based on a three-body interaction
outperforms standard (non-catalytic) cooling using higher order interactions.
Another advantage is illustrated in a thermometry scenario, where a qubit is
employed to probe the temperature of the environment. In this case, we show
that a catalyst allows to surpass the optimal temperature estimation attained
only with the probe.
