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arXiv · 2311.07322

Quasi-tame substitudes and the Grothendieck construction

Michael Batanin·Florian De Leger·David White
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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.

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Michael Batanin, Florian De Leger, David White. 2023-11-13. Quasi-tame substitudes and the Grothendieck construction. https://arxiv.org/abs/2311.07322

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