论文标题
合成$（\ infty，1）$ - 类别的两侧笛卡尔纤维
Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories
论文作者
论文摘要
在Riehl-Shulman的合成$（\ Infty，1）$类别理论的框架内，我们提出了双面笛卡尔纤维的理论。中心结果是雪瓦利，灰色，街道和Riehl-Verity（双面Yoneda引理）的两边性条件的几个特征，以及几种闭合性能的证明。 在此过程中，我们还定义和研究了纤维或切片纤维的概念，后来用于以模块化方式开发双面情况。我们还简要讨论了在这种情况下的离散双面笛卡尔振动，对应于$（\ infty，1）$ - 分销商。 我们的定义和结果的系统学紧密遵循Riehl-Verity的$ \ infty $ -COSMOS理论，但在内部符合Riehl-Shulman的简单范围扩展同质型理论。该框架中的所有构造和证明都是在同型等效性下设计不变的。从语义上讲，合成$（\ infty，1）$ - 类别对应于内部$（\ infty，1）$ - 类别在任意给定$（\ infty，1）$ - topos中实现为rezk对象的类别。
Within the framework of Riehl-Shulman's synthetic $(\infty,1)$-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl-Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to $(\infty,1)$-distributors. The systematics of our definitions and results closely follows Riehl-Verity's $\infty$-cosmos theory, but formulated internally to Riehl-Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic $(\infty,1)$-categories correspond to internal $(\infty,1)$-categories implemented as Rezk objects in an arbitrary given $(\infty,1)$-topos.
