  We apply concepts and tools from abstract homotopy theory to complex analysis
and geometry, continuing our development of the idea that the Oka Principle is
about fibrancy in suitable model structures. We explicitly factor a holomorphic
map between Stein manifolds through mapping cylinders in three different model
structures and use these factorizations to prove implications between
ostensibly different Oka properties of complex manifolds and holomorphic maps.
We show that for Stein manifolds, several Oka properties coincide and are
characterized by the geometric condition of ellipticity. Going beyond the Stein
case to a study of cofibrant models of arbitrary complex manifolds, using the
Jouanolou Trick, we obtain a geometric characterization of an Oka property for
a large class of manifolds, extending our result for Stein manifolds. Finally,
we prove a converse Oka Principle saying that certain notions of cofibrancy for
manifolds are equivalent to being Stein.
