Abstract

This paper studies the homotopy theory of the Grothendieck construction using model categories and semimodel 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 construction and induce (semi)model structures on the base and fibers. In our companion paper Batanin et al. (2025) published in Theory and Applications of Categories, we show how to produce such global model structures in general settings. Applications include (commutative) monoids and their modules, numerous flavors of operads encoded by polynomial monads and substitudes (symmetric, nonsymmetric, cyclic, modular, higher operads, dioperads, properads, and PROPs), and twisted modular operads. We also prove a general result for upgrading a semimodel structure to a model structure.

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