  What are the fundamental limits and advantages of using a catalyst to aid
thermodynamic transformations between quantum systems? In this work, we answer
this question by focusing on transformations between energy-incoherent states
under the most general energy-conserving interactions among the system, the
catalyst, and a thermal environment. The sole constraint is that the catalyst
must return unperturbed and uncorrelated with the other subsystems. More
precisely, we first characterise the set of states to which a given initial
state can thermodynamically evolve (the catalysable future) or from which it
can evolve (the catalysable past) with the help of a strict catalyst. Secondly,
we derive lower bounds on the dimensionality required for the existence of
catalysts under thermal process, along with bounds on the catalyst's state
preparation. Finally, we quantify the catalytic advantage in terms of the
volume of the catalysable future and demonstrate its utility in an exemplary
task of generating entanglement using thermal resources.
