Homotopy Tiny Objects in Enriched Diagram Categories

Abstract

The reconstruction theorem of \[Mon22], which gives a homotopical rendering of the density theorem of the nerve-realization adjunction \[Kel82], provides the left adjoint given by the subcategory generated under weak equivalences and homotopy weighted colimits of what are called *homotopy tiny objects* in order to identify a model category enriched over a closed symmetric monoidal model category $\mathcal{V}$ with a category of enriched presheaves generated by the small full subcategory of the homotopy tiny objects by Quillen equivalence. We extend the theorem to the case where the base of enrichment is a diagram category $\mathcal{V}^{\mathcal{D}}$, for $\mathcal{D}$ a small $\mathcal{V}$-category, equipped with the projective model structure. Such diagram bases arise naturally in equivariant operad or infinite loop space theory \[Bar22, BH15, GMMO19], global homotopy theory \[Sch18, Sch20], and related forms of indexed homotopy theory \[MS06, BT14], where homotopical information is organized simultaneously across families of subgroups, isotropy types, or other indexing data. We define $\mathcal{V}^{\mathcal{D}}$-homotopy tiny objects and prove the extended reconstruction theorem, from which two applications follow: a reproof of the classical Elmendorf theorem using these methods which recovers its modern formulation due to Stephan \[Ste16], as well as a generalization of the Schwede–Shipley theorem, first given in \[SS03], identifying a stable $\mathcal{V}^{\mathcal{D}}$-model category with compact generators $\mathcal{G}$ with $(Sp^{\Sigma})^{\mathcal{G}^{op}\times\mathcal{D}}$ for $\mathcal{V}=Sp^{\Sigma}$.