Describing model categories througth homotopy tiny objects

TL;DR

This paper characterizes certain model categories using homotopy tiny objects, establishing an equivalence with enriched presheaf categories, thus generalizing key theorems in stable and equivariant homotopy theory.

Contribution

It introduces the concept of homotopy tiny objects in enriched model categories and proves a main theorem linking generated categories to enriched presheaves, extending known results.

Findings

Homotopy category of generated subcategory is equivalent to enriched presheaves.

If the model category is generated by tiny objects, it is Quillen equivalent to presheaf categories.

Special cases recover Schwede-Shipley's and Elmendorf's theorems.

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.