I have come across the idea of emergent properties. It seems this subject saw some increase in activity in the beginning of this century (ie. what articles I have found seems to be from around 2006), and much of it is centered around theoretical physics. Often cited examples are things like temperature, pressure, phase and similar properties that seem to emerge from large assemblies of particles, but are somewhat independent of the nature of the paricles involved. A different example that is somewhat fashionable is the idea that space-time emerges in some way from quantum mechanis (I'm not convinced). The fundamental problem seems to be that there is no consensus about how to model emergence in general, and even some suggestion that it is impossible to apply mathematics to it (which I think is silly). So...
I am interested in the question of whether, say, the topology, the differentialble structure or the dimension of a manifold can be considered an emerging property, and if so, how? And in extension, is there any category theoretical definition of what emergent properties are and how they emerge from more 'atomic' properties?