Finiteness and finite domination in stratified homotopy theory

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.