We study the category of Reedy diagrams in a $$\mathscr {V}$$V-model category. Explicitly, we show that if K is a small category, $$\mathscr {V}$$V is a closed symmetric monoidal category and $$\mathscr {C}$$C is a closed $$\mathscr {V}$$V-module, then the diagram category $$\mathscr {V}^K$$VK is a closed symmetric monoidal category and the diagram category $$\mathscr {C}^K$$CK is a closed $$\mathscr {V}^K$$VK-module. We then prove that if further K is a Reedy category, $$\mathscr {V}$$V is a monoidal model category and $$\mathscr {C}$$C is a $$\mathscr {V}$$V-model category, then with the Reedy model category structures, $$\mathscr {V}^K$$VK is a monoidal model category and $$\mathscr {C}^K$$CK is a $$\mathscr {V}^K$$VK-model category provided that either the unit 1 of $$\mathscr {V}$$V is cofibrant or $$\mathscr {V}$$V is cofibrantly generated.

中文翻译：

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V-Model 类别中的 Reedy 图

我们在 $$\mathscr {V}$$V 模型类别中研究 Reedy 图的类别。明确地，我们证明如果 K 是一个小范畴，$$\mathscr {V}$$V 是一个闭对称幺半群范畴，而 $$\mathscr {C}$$C 是一个闭 $$\mathscr {V}$ $V-module，则图范畴 $$\mathscr {V}^K$$VK 是闭对称幺半群范畴，图范畴 $$\mathscr {C}^K$$CK 是闭 $$\mathscr {V}^K$$VK-模块。然后我们证明，如果进一步 K 是 Reedy 范畴，则 $$\mathscr {V}$$V 是幺半群模型范畴，而 $$\mathscr {C}$$C 是 $$\mathscr {V}$$V -model 类别，然后使用 Reedy 模型类别结构，$$\mathscr {V}^K$$VK 是幺半群模型类别，而 $$\mathscr {C}^K$$CK 是 $$\mathscr {V }^K$$VK 模型类别，前提是 $$\mathscr {V}$$V 的单元 1 是共纤维的，或者 $$\mathscr {V}$$V 是共纤维生成的。