Quasi-tame substitudes and the Grothendieck construction

Michael Batanin, Florian De Leger, and David White

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 \cite{companion} 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.

Keywords: polynomial monads, substitudes, Grothendieck construction, model structure

2020 MSC: 18M65,55P48

Theory and Applications of Categories, Vol. 44, 2025, No. 24, pp 676-730.

Published 2025-08-06.

http://www.tac.mta.ca/tac/volumes/44/24/44-24.pdf

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