  In this paper, we extend existing results about simulation and intrinsic
universality in a model of tile-based self-assembly. Namely, we work within the
2-Handed Assembly Model (2HAM), which is a model of self-assembly in which
assemblies are formed by square tiles that are allowed to combine, using glues
along their edges, individually or as pairs of arbitrarily large assemblies in
a hierarchical manner, and we explore the abilities of these systems to
simulate each other when the simulating systems have a higher "temperature"
parameter, which is a system wide threshold dictating how many glue bonds must
be formed between two assemblies to allow them to combine. It has previously
been shown that systems with lower temperatures cannot simulate arbitrary
systems with higher temperatures, and also that systems at some higher
temperatures can simulate those at particular lower temperatures, creating an
infinite set of infinite hierarchies of 2HAM systems with strictly increasing
simulation power within each hierarchy. These previous results relied on two
different definitions of simulation, one (strong simulation) seemingly more
restrictive than the other (standard simulation), but which have previously not
been proven to be distinct. Here we prove distinctions between them by first
fully characterizing the set of pairs of temperatures such that the high
temperature systems are intrinsically universal for the lower temperature
systems (i.e. one tile set at the higher temperature can simulate any at the
lower) using strong simulation. This includes the first impossibility result
for simulation downward in temperature. We then show that lower temperature
systems which cannot be simulated by higher temperature systems using the
strong definition, can in fact be simulated using the standard definition,
proving the distinction between the types of simulation.
