Model structures on operads and algebras from a global perspective

Abstract

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.