Quasi-tame substitudes and the Grothendieck construction

TL;DR

This paper introduces quasi-tame polynomial monads to extend the homotopy theory of algebras, producing model structures on Grothendieck categories and applying to operads, modules, and commutative monoids.

Contribution

It generalizes tame polynomial monads to quasi-tame ones, enabling new model structures and a unified approach to operads and their modules.

Findings

Quasi-tame monads are necessary and sufficient for admissibility.

Model structures can be transferred to algebras over quasi-tame monads.

The approach extends to non-polynomial cases like commutative monoids.

Abstract

This paper continues the study of the homotopy theory of algebras over polynomial monads initiated by the first author and Clemens Berger. We introduce the notion of a quasi-tame polynomial monad (generalizing tame ones) and produce transferred model structures (left proper in many settings) on algebras over such a monad. Our motivating application is to produce model structures on Grothendieck categories, which are used in a companion paper to give a unified approach to the study of operads, their algebras, and their modules. We prove a general result regarding when a Grothendieck construction can be realized as a category of algebras over a polynomial monad, examples illustrating that quasi-tameness is necessary as well as sufficient for admissibility, and an extension of classifier methods to a non-polynomial situation, namely the case of commutative monoids.