Title: On bi-enriched $\infty$-categories Show Chinese title

Title: 关于双富足的$\infty$-范畴

Abstract: We extend Lurie's definition of enriched $\infty$-categories to notions of left enriched, right enriched and bienriched $\infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $\infty$-categories and share many desirable features with them. We use bienriched $\infty$-categories to endow the $\infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $\infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we construct enriched Kan-extensions from operadic Kan-extensions, compute the monad for enriched functors, prove an end formula for morphism objects of enriched $\infty$-categories of enriched functors and a coend formula for the relative tensor product of enriched profunctors and construct transfer of enrichment from scalar extension of presentably bitensored $\infty$-categories. In particular, we develop an independent theory of enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories. Show Chinese abstract

Abstract: 我们扩展了Lurie对富化$\infty$-范畴的定义，得到左富化、右富化和双富化的$\infty$-范畴的概念，这些概念推广了闭左张量、右张量和双张量$\infty$-范畴的概念，并与它们有许多理想的特性。我们使用双富化的$\infty$-范畴来赋予富化函子的$\infty$-范畴一种富化，这种富化既推广了当后者存在时富化$\infty$-范畴的张量积的内部同态，又推广了在余极限和张量下的自由完备性。 作为应用，我们从操作符Kan-extensions构造丰富的Kan-extensions，计算丰富函子的单子，证明丰富函子的丰富$\infty$-范畴的态射对象的端公式以及丰富广义函子的相对张量积的余端公式，并构造从可呈现双张量$\infty$-范畴的标量扩张中传递的丰富性。特别是，我们为Lurie的丰富$\infty$-范畴模型发展了一个独立的丰富$\infty$-范畴理论。