I've now read lots more about spinors and vector bundles and am feeling far more confident about understanding exactly what Friedrich is up to. I'm aiming to provide a post on each major section (and sometimes subsection) to both help me and maybe you (assuming anyone reads this!).

So the introduction. It makes a lot more sense the second time around. In fact it's surprisingly clear given how opaque I found it the first time. Perhaps the most interesting point is that the conformal structure indicates that space like infinite should be represented as a point but the Weyl tensor indicate that it should be represented as something else. I look forward to reading (or should that be rereading) Friedrich's justification for choosing a cylinder.

Also it occurs to me that it is precisely this kind of situation for which you'd like to use the abstract boundary. This would allow us to use different representations of the boundary when different structures are being considered. An important thing to think about given the topic of the paper.

In any case Friedrich spells out that he is interested in the transition from catchy data on a space like surface to smooth data on a hyperboloidal surface. Once this is done the, now standard, results on hyperboloidal surfaces can be used to evolve the data fore the entire space tine. Thereby effectively solving an outstanding problem in GR.