In this work we propose a realization of Lurie’s prediction that inner fibrations \(p: X \rightarrow A\) are classified by A-indexed diagrams in a “higher category” whose objects are \(\infty \)-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and \(\infty \)-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.

在这项工作中，我们提出了 Lurie 预测的实现，即内部纤维化\(p: X \rightarrow A\)由A索引图分类在“更高类别”中，其对象是\(\infty \) -类别，态射是它们与高等态射之间的对应是高等对应。我们将得到这个作为更一般结果的推论，该结果以类似的方式对普通单纯集合之间的所有单纯映射进行分类。单纯形集之间的对应关系（和\(\infty \)-类别）是与类别相关的profunctor（或双模）概念的概括。虽然类别、函子和profunctor 被组织在一个双重类别中，但我们将展示单纯集、单纯映射和对应关系作为单纯类别的一部分。这使我们能够做出准确的陈述并提供证据。我们的主要工具是双范畴语言，我们也在单纯范畴的上下文中使用它。