Finiteness and finite domination in stratified homotopy theory

The part of the lemma concerning compactness is proven in [HPT24, Lemma A.3.10], so we focus on finiteness.

We first observe that the functor $\text{Str}_{{P}}\rightarrow\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$ preserves finite objects. Indeed, by [HPT24, Observation A.3.10], the inclusion $\text{Str}_{{P}}\hookrightarrow\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}{{}_{/P}}$ preserves colimits. Since the functor $\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}{{}_{/P}}\rightarrow\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$ preserves colimits, we deduce that $\text{Str}_{{P}}\rightarrow\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$ does too. The claim about preservation of finite objects then follows by observing that objects of the form $\{p_{1}<\cdots<p_{n}\}\rightarrow P$, with $n\leq 2$, are sent to $\infty$-categories equivalent to either $[0]$ or $[1]$.

Assume now that $\mathscr{C}$ is finite as an object of $\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$. Consider the composite

$${\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}}_{/\mathscr{C}}\rightarrow{\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}}_{/P}\xrightarrow{\text{Env}_{P}}\text{Str}_{{P}},$$

where $\text{Env}_{P}$ is the left adjoint to the inclusion $\text{Str}_{{P}}\hookrightarrow{\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}}_{/P}$ (see for example [HPT24, Observation A.3.3]). Both functors preserve colimits, and the composite sends objects of the form $[n]\rightarrow\mathscr{C}$ to conservative functors $[m]\rightarrow P$, with $m\leq n$. Indeed, this holds because any localization of $[n]$ is equivalent to $[m]$, for some $m\leq n$. The desired conclusion then follows by observing that $\mathscr{C}$, as the terminal object of ${\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}}_{/\mathscr{C}}$, belongs to the smallest full subcategory of ${\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}}_{/\mathscr{C}}$ closed under finite colimits and containing objects of the type $[n]\rightarrow\mathscr{C}$. ∎

The statement about compact objects follows from [Aok23, Proposition 2.8]. Now let $i$ be any object of $I$, and consider the evaluation functor $\text{ev}_{i}:\operatorname{Fun}(I^{op},\mathscr{S})\rightarrow\mathscr{S}$. Since $I$ is strongly finite, we know that $\text{ev}_{i}$ sends representable presheaves to finite spaces. Thus, since $\text{ev}_{i}$ preserves pushouts, we see that it has to preserve finite objects.

Now assume that $F\in\operatorname{Fun}(I^{op},\mathscr{S})$ takes values in finite spaces. Consider the left fibration $p:I_{/F}\rightarrow I$ given by the category of elements of $F$. Since $I$ is finite and by assumption the fibers of $p$ are also finite, by [CDH⁺23, Remark 6.5.4] we see that $I_{/F}$ must be finite. Therefore, writing $F$ as colimit of representable functors indexed by $I_{/F}$, we get that $F$ is finite, and so the proof is concluded. ∎

By Theorem 2.9, there exists $F\in\operatorname{Fun}(\text{Sd}({P})^{\operatorname{op}},\mathscr{S})$ such that $L_{P}(F)\simeq(\mathscr{C}\rightarrow P)$. Moreover, the assumptions on $\mathscr{C}\rightarrow P$ imply that we can assume that $F$ takes values in compact (respectively finite) spaces. Note that, when $P$ is finite, $\text{Sd}(P)$ is a finite poset as well. Therefore, by Lemma 2.5, it follows that $F$ is a finite object in $\operatorname{Fun}(\text{Sd}(P)^{op},\mathscr{S})$. But $(\mathscr{C}\rightarrow P)\simeq L_{P}(F)$, and so by Remark 2.10 we see that $\mathscr{C}\rightarrow P$ is compact (respectively finite). ∎

This is proven in [HPT24, Proposition A.3.17]. ∎

Let $x$ be any object in $\mathscr{C}$, and let $\mathscr{C}_{p}$ be the unique stratum to which $x$ belongs. By the pointwise formula for right Kan extensions, to show that $j_{\ast}$ preserves filtered colimits it suffices to prove that the slice $(\mathscr{C}_{q})_{x/}$ is a compact object in $\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$.

If $p$ is not less or equal to $q$, the slice $(\mathscr{C}_{q})_{x/}$ is empty, and when $p=q$, it has an initial object. Therefore, we can assume that $p<q$. We show that $(\mathscr{C}_{q})_{x/}$ fits in a pullback square

Since we assumed that $\mathscr{C}$ has compact local links, this will conclude the proof.

By definition of the slice, we have a pullback square

But the square above can be factored as

where the square on the right is a pullback by definition. Therefore, also the square on left is a pullback. By composing it with the pullback square

we then get the desired conclusion. ∎

By Proposition 2.11, it will suffice to show that our assumptions imply that for each sequence $\{p_{1}<\dots<p_{n}\}$, ${\mathscr{C}}[{p_{1}<\dots<p_{n}}]\in\mathscr{S}$ is compact (finite). We have a pullback square

By assumption, we know that the fibers of the right vertical map are compact (finite). Therefore, it will suffice to show that ${\mathscr{C}}[{p_{1}<\dots<p_{n-1}}]$ is compact (finite). Considering a finite number of pullback squares as above, we can reduce the question to proving that ${\mathscr{C}}[{p_{1}<p_{2}}]$ is compact (finite). But we know that $\mathscr{C}_{p_{1}}$ is compact (finite), and the fibers of ${\mathscr{C}}[{p_{1}<p_{2}}]\rightarrow\mathscr{C}_{p_{1}}$ are compact (finite) as well. Therefore, we can conclude that ${\mathscr{C}}[{p_{1}<p_{2}}]$ is compact (finite). ∎

The functor (3.1) is essentially surjective. The cone point is in the essential image, and since any point $[z,t]\in\mathbb{R}_{>0}\times Z$ is connected to $[z,1]$ by a path which doesn’t leave the stratum of $[z,t]$, we get the claim. Thus, it remains to prove that (3.1) is fully faithful. Arguing as above, one sees that it is sufficient to prove that, for any $z\in Z$, the map

$$\ast\simeq\text{Hom}_{\scalebox{1.0}{$\scriptscriptstyle[0]\ast\operatorname{Exit}_{{Q}}({Z})$}}(0,z)\rightarrow A\coloneqq\text{Hom}_{\scalebox{1.0}{$\scriptscriptstyle\operatorname{Exit}_{{C(Q)}}({C(Z)})$}}(\ast,[z,1])$$

in an isomorphism in $\mathscr{S}$. We now prove this by providing an homotopy between the identity of $A$ and the constant map with value the path $s\mapsto[z,s]$. For any exit path $\gamma:I\rightarrow C(Z)$ starting at the cone point and ending at $[z,1]$ and any $s\in I$, we denote $[\gamma^{Z}(s),\gamma^{I}(s)]\coloneqq\gamma(s)$. One checks that the homotopy

does the job. ∎

Since the inclusion $\mathscr{S}\hookrightarrow\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$ preserves limits, it will suffice to prove that (3.3) is a pullback in $\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$. By [Lur17, Proposition A.7.9], the right rectangle is a pullback, and so it suffices to prove that the left one is a pullback. Notice that

$${\operatorname{Exit}_{{P_{\geq p}}}({U\times C(Z)})}[{p\leq q}]\simeq\operatorname{Sing}({U})\times{\operatorname{Exit}_{{P_{\geq p}}}({C(Z)})}[{p\leq q}]$$

and that the cospan

$$[0]\xrightarrow{x}\operatorname{Sing}({U})\xleftarrow{\text{ev}_{0}}{\operatorname{Exit}_{{P_{\geq p}}}({U\times C(Z)})}[{p\leq q}]$$

is obtained as a product of the two cospans

$$[0]\xrightarrow{x}\operatorname{Sing}({U})\xleftarrow{\text{id}}\operatorname{Sing}({U}),[0]\xrightarrow{\text{id}}[0]\leftarrow{\operatorname{Exit}_{{P_{\geq p}}}({C(Z)})}[{p\leq q}].$$

Thus, it suffices to prove that the map

$$\operatorname{Sing}({Z_{q}})\xrightarrow{c_{x}}{\operatorname{Exit}_{{P_{\geq p}}}({C(Z)})}[{p\leq q}]$$

is an equivalence, where $x\in C(Z)$ now denotes the cone point. The map $c_{x}$ is induced by the commutativity of the outer rectangle in the diagram

and thus it suffices to prove that the outer rectangle is a pullback in $\mathscr{C}\mathrm{at}_{\scalebox{1.0}{$\scriptscriptstyle\infty$}}$. The left triangle is evidently a pullback, the middle one is a pullback because by Lemma 3.2 $x$ and $p$ are initial objects of $\operatorname{Exit}_{{P_{\geq p}}}({C(Z)})$ and $N(P_{\geq p})$ respectively, and thus the horizontal maps are equivalences. We are then only left to show that the right rectangle is a pullback. Consider the following commutative diagram

We want to show that the upper horizontal square is a pullback. The front vertical square is a pullback, and so it suffices to show that the composition of the upper horizontal square and the front vertical square is a pullback. But this coincides with the composition of the back vertical square and the lower horizontal square, which are both evidently pullbacks, and so we may conclude. ∎

By Proposition 2.16, it suffices to show that $\operatorname{Exit}_{{P}}({X})$ has compact (finite) local links and strata. After Proposition 3.4, one sees that these two conditions are equivalent to (i) and (ii) in the statement of the theorem. ∎

Since $X$ is compact and Hausdorff, by [Lur09, Corollary 7.3.4.12] we know that the global section functor preserves filtered colimits. The lemma is then proven by observing that, via the exodromy equivalence (see [Lur17, Theorem A.9.3]), the global section functor corresponds to taking the limit indexed by $\operatorname{Exit}_{{P}}({X})$. ∎

By Remark 3.7, $\operatorname{Sing}({X_{p}})$ is a compact object in $\mathscr{S}$ if and only the limit functor preserves filtered colimits. If $j:\operatorname{Sing}({X_{p}})\rightarrow\operatorname{Exit}_{{P}}({X})$ is the inclusion, then we can factor the limit functor on $\operatorname{Sing}({X_{p}})$ as

$$\operatorname{Fun}(\operatorname{Sing}({X_{p}}),\mathscr{S})\xrightarrow[]{j_{\ast}}\operatorname{Fun}(\operatorname{Exit}_{{P}}({X}),\mathscr{S})\xrightarrow[]{\text{lim}}\mathscr{S}.$$

But by Lemma 2.15 and Lemma 3.8 both functors preserve filtered colimits, and therefore we may conclude. ∎

By [Lur17, Theorem A.9.3], we know that $X\rightarrow P$ is exodromic. Therefore it makes sense to consider $\operatorname{Exit}_{{P}}({X})$. Notice that the compactness of $X$ forces $P$ to be finite (see [Vol22, Lemma 3.1]). We proceed by induction on the cardinality of $P$.

Denote by $d$ the cardinality of $P$. If $d=1$, then $X$ is a compact topological manifold. Using for example [Wes77], we know that its homotopy type is finite, and therefore compact.

Assume that $d>1$. By [AFT17, Lemma 2.2.2], we know that $X$ has a basis given by open subsets which are isomorphic as stratified spaces to $\mathbb{R}^{n}\times C(Z)$, where $Z$ is a compact $C^{0}$-stratified space whose stratified poset $Q$ has cardinality smaller than $d$. Therefore, by the inductive assumption $\operatorname{Exit}_{{Q}}({Z})$ is compact, and hence all its strata are compact by Proposition 2.12. By Proposition 3.4, we get that $\operatorname{Exit}_{{P}}({X})$ has compact local links. Thus, by Proposition 2.16 it suffices to show that the strata of $X$ have compact homotopy type. But this follows immediately from Corollary 3.9.

Let us now assume that $X$ is equipped with a conically smooth atlas. In this situation, finiteness of $\operatorname{Exit}_{{P}}({X})$ has already been proven in [Vol22, Proposition 2.19]. Here we provide a more direct argument, relying on the results of the previous section. Observe that, arguing as in the $C^{0}$ case, it suffices to prove that any open stratum $U$ in $X$ has the homotopy type of a finite CW-complex. By resolution of singularities (see [AFT17, Proposition 7.3.10]), we know that $U$ is the interior of a compact smooth manifold with corners $Y$. A routine application of the existence of collarings of corners shows that $U$ is homotopy equivalent to $Y$. Therefore, one deduce finiteness of $U$ from the finiteness of $Y$. ∎

By Proposition 2.12, we know that compactness (finiteness) of $\operatorname{Exit}_{{P}}({X})$ implies compactness (finiteness) of $\operatorname{Sing}({X_{p}})$ for all $p\in P$. Therefore, assume that a $C^{0}$-stratified space with the property that $\operatorname{Sing}({X_{p}})$ is compact for all $p\in P$. We want to show that $\operatorname{Exit}_{{P}}({X})$ is compact. By Theorem 3.5, it suffices to prove that $\operatorname{Exit}_{{P}}({X})$ has compact local links. But this follows immediately from Theorem 3.10, since $X$ has conical charts of the form $\mathbb{R}^{n}\times C(Z)$, where $Z\rightarrow Q$ is a compact $C^{0}$-stratified space. The last assertion of the corollary is deduced analogously, by observing that the presence of a conically smooth atlas provides conical charts as above where $Z$ is itself equipped with a conically smooth atlas. ∎

Let $H$ be any closed subgroup of $G$. Unraveling the definition of the Alexandroff topology, one sees that proving the lemma amounts to showing that

$$X_{\leq H}\coloneqq\{x\in X\mid G_{x}\supseteq H\}$$

is a closed subset of $X$. For a fixed $g$, the set

$$\text{eq}(g,e)\coloneqq\{x\in X\mid gx=x\}$$

is the preimage of the diagonal under the continuous map

Since $X$ is Hausdorff, we deduce that $\text{eq}(g,e)$ is a closed subset of $X$. The proof is concluded by observing that we have the equality

$$\{x\in X\mid G_{x}\supseteq H\}=\bigcap_{g\in H}\text{eq}(g,e).$$

∎