Homotopical recognition of diagram categories

TL;DR

This paper characterizes when a model category is Quillen equivalent to a presheaf category using homotopy atoms, extending previous orbit models and applying to various homotopy theories and polynomial functors.

Contribution

It introduces homotopy atoms to generalize orbit models and provides a classification of certain polynomial functors in homotopy theory.

Findings

Characterization of model categories as presheaf categories via homotopy atoms

Extension of orbit models to new homotopy theories

Classification of polynomial functors in Goodwillie calculus

Abstract

Building on work of Marta Bunge in the one-categorical case, we characterize when a given model category is Quillen equivalent to a presheaf category with the projective model structure. This involves introducing a notion of homotopy atoms, generalizing the orbits of Dwyer and Kan. Apart from the orbit model structures of Dwyer and Kan, our examples include the classification of stable model categories after Schwede and Shipley, isovariant homotopy theory after Yeakel, and Cat-enriched homotopy theory after Gu. As an application, we give a classification of polynomial functors (in the sense of Goodwillie calculus) from finite pointed simplicial sets to spectra, and compare it to the previous work by Arone and Ching.