Describing model categories througth homotopy tiny objects

Abstract

Let $\mathcal C$ be a $\mathcal V$-enriched model category. We say that an object $x$ of $\mathcal C$ is homotopy tiny if the total right derived functor of $\mathcal C(x, -) : \mathcal{C} \rightarrow {\mathcal V}$ preserves homotopy weighted colimits. Let $\mathcal C_0$ be a full subcategory of $\mathcal C$ all of whose objects are homotopy tiny. Our main result says that the homotopy category of the category generated by $\mathcal C_0$ under weak equivalences and homotopy weighted colimits is equivalent to the homotopy category of the category $\mathcal V^{\mathcal C_0^{op}}$ of $\mathcal V$-enriched presheaves on $\mathcal C_0$ with values in $\mathcal V$. If $\mathcal C$ is generated by $\mathcal C_0$, then $\mathcal C$ is Quillen equivalent to $\mathcal V^{\mathcal C_0^{op}}$. Two special cases of our theorem are Schwede-Shipley's theorem on stable model categories and Elmendorf's theorem on equivariant spaces.