论文标题
描述模型类别througth同型微小对象
Describing model categories througth homotopy tiny objects
论文作者
论文摘要
令$ \ Mathcal c $为$ \ Mathcal V $ - 富含模型类别。我们说，如果$ \ MATHCAL C（x， - ）的总右派函数$ \ MATHCAL C $的对象$ x $是同质的，那么\ Mathcal {c} \ rightArrow {\ Mathcal v} $保留同质加权colimits。令$ \ MATHCAL C_0 $为$ \ MATHCAL C $的完整子类别，其所有对象都是同质的。 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 $。如果$ \ MATHCAL C $由$ \ MATHCAL C_0 $生成，则$ \ Mathcal C $是quillen等于$ \ Mathcal V^{\ Mathcal C_0^{op}}} $。我们定理的两个特殊案例是Schwede-Shipley的稳定模型类别定理和Elmendorf的模棱两可的定理。
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.
