AI 中文总结
本研究在带有限极限的$(\nfty,1)$-范畴中构造了跨度的$(\nfty,\nfty)$-范畴新模型，验证其与现有模型一致，证明了刻画到跨度$(\nfty,n)$-范畴函子的新泛性质，为后续构造高阶Hall代数奠定基础。
AI 中文摘要
本文在任意给定的具有有限极限的$(\nfty,1)$-范畴$\ncal{C}$中，构造了跨度（也称为对应）的$(\nfty,\nfty)$-范畴$\nmathsf{Span}_\nty(\ncal{C})$。这为$n \n \natnums \n \nfty$的跨度$(\nfty,n)$-范畴提供了新模型。我们刻画了这些$(\nfty,n)$-范畴中的映射$(\nfty, n-1)$-范畴，从而验证了我们的模型与其他跨度模型一致。最后且最重要的是，我们证明了一个新的泛性质，刻画了到跨度$(\nfty,n)$-范畴的函子，该性质可特殊化为1维情况下与扭箭头范畴的熟知关系。这些结果将在后续论文中用于构造经典Hall代数构造的高阶类似物，其中“高阶”既指高阶范畴结构，也指“高阶幺半”结构，即$n, k>1$时$(\nfty, n)$-范畴中的$\nmathsf{E}_k$-代数。
英文摘要
In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.
