  Many interesting classes of maps from homotopical algebra can be
characterised as those maps with the right lifting property against certain
sets of maps (such classes are sometimes referred to as cofibrantly generated).
In a more sophisticated notion due to Garner (referred to as algebraically
cofibrantly generated) the set of maps is replaced with a diagram over a small
category.
  We give a yet more general definition where the set or diagram of maps is
replaced with a vertical map in a Grothendieck fibration. In addition to an
interesting new view of the existing examples above, we get new notions, such
as computable lifting problems in presheaf assemblies, and internal lifting
problems in a topos.
  We show that under reasonable conditions one can define a notion of universal
lifting problem and carry out step-one of Garner's small object argument. We
give explicit descriptions of what the general construction looks like in some
examples.
