Homotopical recognition of diagram categories

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.