论文标题
丰富的$ \ infty $ - 类别与$ \ infty $ - 类别之间的等价性
An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action
论文作者
论文摘要
我们表明，$ \ infty $ -2CATEMORY $ \ MATHCAL {M} $，其封闭的左左右动作是单型$ \ infty $ -Category $ \ Mathcal {V} $完全由$ \ Mathcal {V} $ - 配备了$ \ Mathcal的结构的$ \ Mathcal {v} $ - MATHCAL} $ \ MATHCAL CALCHCAL} $ { Gepner-Haugseng。当$ \ Mathcal {m} $是$ \ Mathcal {V} $ - 富集$ \ infty $ - 类别时，我们证明了类似的结果，从lurie的意义上讲，$ \ infty $ - 类别的概念是封闭左右的操作。确切地说，我们证明，在lurie的意义上，发送$ \ nathcal {v} $ - 丰富$ \ infty $ - 类别，以$ \ nathcal {v} $ - 形态学对象的有价值图精制到等价$χ$χ$χ$χ$ qualty $ category $ catemention $ unfty $ \ quanty $ \ n n $ \ nifty-mathcal- patementy $ cagementy unfty $ cagementy $ cagemention $ cagity- patection $} $} v}从Lurie和Gepner-Haugseng的意义上讲。此外，如果$ \ MATHCAL {V} $是$ \ Mathbb {e} _ {\ Mathrm {k+1}} $ - monoidal $ \ infty $ -category-infty $ -category，$ 1 \ leq k \ leq k \ leq \ leq \ leq \ infty $ $ \ mathbb {张量产品，以及$ \ MATHCAL {V} $的张量产品 - 富含Gepner-Haugseng的$ \ infty $ - 类别。作为我们理论的应用，我们将$ \ infty $ - 小稳定$ \ infty $ - 类别的$ \ infty $类别的宽松对称单嵌入到$ \ infty $ - 小频谱$ \ infty $ - 类别中。作为第二个应用程序，我们为Lurie的富含$ \ infty $类别的概念生成了Yoneda插入。
We show that an $\infty$-category $\mathcal{M}$ with a closed left action of a monoidal $\infty$-category $\mathcal{V}$ is completely determined by the $\mathcal{V}$-valued graph of morphism objects equipped with the structure of a $\mathcal{V}$-enrichment in the sense of Gepner-Haugseng. We prove a similar result when $\mathcal{M}$ is a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie, an operadic generalization of the notion of $\infty$-category with closed left action. Precisely, we prove that sending a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie to the $\mathcal{V}$-valued graph of morphism objects refines to an equivalence $χ$ between the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories in the sense of Lurie and of Gepner-Haugseng. Moreover if $\mathcal{V}$ is a presentably $\mathbb{E}_{\mathrm{k+1}}$-monoidal $\infty$-category for $1 \leq k \leq \infty$, we prove that $χ$ restricts to a lax $\mathbb{E}_{\mathrm{k}}$-monoidal functor between the $\infty$-category of left $\mathcal{V}$-modules in $\mathrm{Pr}^L$, the symmetric monoidal $\infty$-category of presentable $\infty$-categories, endowed with the relative tensor product, and the tensor product of $\mathcal{V}$-enriched $\infty$-categories of Gepner-Haugseng. As an application of our theory we construct a lax symmetric monoidal embedding of the $\infty$-category of small stable $\infty$-categories into the $\infty$-category of small spectral $\infty$-categories. As a second application we produce a Yoneda-embedding for Lurie's notion of enriched $\infty$-categories.
