relatively k-compact morphism in an (infinity,1)-category
For some cardinal, say a morphism in is relatively -compact if for all (∞,1)-pullbacks along to -compact objects, , the pulled back object is itself a -compact object.
We may write as an (∞,1)-colimit over itself (see there)
and then use the fact that ∞Grpd – being an (∞,1)-topos – has universal colimits, to obtain the (∞,1)-pullback diagram
exhibiting as an -colimit of -small objects over . By stability of -compact objects under -small colimits (see here) it follows that is -compact if is.
This is due to Charles Rezk. The statement appears as HTT, theorem 220.127.116.11.