I've recently posted a pair of papers on the arXiv that might be of interest to some people on this list. Both are on my work on lifting problems in fibrations and the small object argument. An interesting side effect was that a construction used can be seen as a kind of higher inductive type generalising W-types (this is the subject of the second paper).

1. Lifting Problems in Grothendieck Fibrations

2. W-Types with Reductions and the Small Object Argument

Comments, questions, corrections, etc are welcome.

Best,

Andrew