  This paper studies the homotopy theory of the Grothendieck construction using
model categories and semi-model categories, provides a unifying framework for
the homotopy theory of operads and their algebras and modules, and uses this
framework to produce model structures, rectification results, and properness
results in new settings. In contrast to previous authors, we begin with a
global (semi-)model structure on the Grothendieck and induce (semi-)model
structures on the base and fibers. In a companion paper, we show how to produce
such global model structures in general settings. Applications include numerous
flavors of operads encoded by polynomial monads and substitudes (symmetric,
non-symmetric, cyclic, modular, higher operads, dioperads, properads, and
PROPs), (commutative) monoids and their modules, and twisted modular operads.
We also prove a general result for upgrading a semi-model structure to a full
model structure.
