标题： 分层同伦理论中的有限性与有限支配性 显示英文标题

标题： Finiteness and finite domination in stratified homotopy theory

摘要： 在本文中，我们研究了一个带有到有限偏序集$P$的保守函子的$\infty$-范畴$\mathcal{C}$的紧性和有限性。我们提供了$\mathcal{C}$在以$\mathcal{C}\rightarrow P$的层和同伦链接为条件的情况下为紧的充分条件。也给出了$\mathcal{C}$为有限的类似条件。 从这些可以推导出，如果$X\rightarrow P$是一个具有性质的锥形分层空间，其层以及其局部链环的弱同伦类型是紧致（分别有限）的$\infty$-群范畴，那么$\text{Exit}_P(X)$是紧致（分别有限）的。这给出了对 Porta 和 Teyssier 提出的问题的肯定回答。如果$X\rightarrow P$被赋予一个锥形光滑结构（例如，一个 Whitney 分层），我们证明$\text{Exit}_P(X)$是有限的当且仅当$X\rightarrow P$的层的弱同伦类型是有限的。 上述特征依赖于$\text{Exit}_P(X)$的有限性，当$X\rightarrow P$是紧致且锥光滑的时候。 我们通过证明类似的陈述在拓扑范畴中不成立来结束本文。 更具体地说，我们提供了一个紧致$C^0$-分层空间的例子，其退出路径$\infty$-范畴是紧致的，但不是有限的。 这个分层空间是由 Quinn 构造的。 我们还观察到，这提供了一个非平凡的$C^0$-分层空间的例子，它不承认任何锥光滑结构。 显示英文摘要

摘要： In this paper, we study compactness and finiteness of an $\infty$-category $\mathcal{C}$ equipped with a conservative functor to a finite poset $P$. We provide sufficient conditions for $\mathcal{C}$ to be compact in terms of strata and homotopy links of $\mathcal{C}\rightarrow P$. Analogous conditions for $\mathcal{C}$ to be finite are also given. From these, we deduce that, if $X\rightarrow P$ is a conically stratified space with the property that the weak homotopy type of its strata, and of strata of its local links, are compact (respectively finite) $\infty$-groupoids, then $\text{Exit}_P(X)$ is compact (respectively finite). This gives a positive answer to a question of Porta and Teyssier. If $X\rightarrow P$ is equipped with a conically smooth structure (e.g. a Whitney stratification), we show that $\text{Exit}_P(X)$ is finite if and only the weak homotopy types of the strata of $X\rightarrow P$ are finite. The aforementioned characterization relies on the finiteness of $\text{Exit}_P(X)$, when $X\rightarrow P$ is compact and conically smooth. We conclude our paper by showing that the analogous statement does not hold in the topological category. More explicitly, we provide an example of a compact $C^0$-stratified space whose exit paths $\infty$-category is compact, but not finite. This stratified space was constructed by Quinn. We also observe that this provides a non-trivial example of a $C^0$-stratified space which does not admit any conically smooth structure.