Does anybody understand why Delta-generated spaces are locally presentable? This is of course claimed by Jeff Smith, and there is a paper by Fajstrup and Rosicky

A convenient category for directed homotopy

that proves it. But I can't understand the proof, and also it involves things that really should not be necessary from mathematical logic.

Note that Delta-generated spaces are just colimits of copies of the unit interval I, so they are the same as I-generated spaces. The general claim is that A-generated spaces are locally presentable for any A. The point must be that the topology in an A-generated space is determined by sets of a bounded size, depending on A. For example, in I-generated spaces, a point is in the closure of a subset if and only if you can get to the point by a convergent sequence. This has to be the key to the proof, but I have not been able to make this into a proof.