Exit path categories of regular CW complexes

Question

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ordered set of its cells, where the ordering is given by inclusions. I have been wondering about the claim that the $\infty$-category of exit-paths $\operatorname{Sing}^S (X)$ is equivalent to (the nerve of) $S$, but I couldn't find a reference for this - is it true?

As for some more background and indications on why this should hold: First of all, for conically stratified spaces, there always is a conservative functor from the exit-path category into the stratification poset $\operatorname{Sing}^S(X) \to N(S)$. The fibers of this map are, by "Higher Algebra" A.7.5, given by the homotopy types of the strata, which in out case are contractible (they are just open balls and points respectively). In fact, A.6.10 even tells us that for a simplicial simplicial complex stratified by its poset of simplices, the canonical map from the exit-path category into the nerve of the stratification poset is an equivalence. My hope is that the requirement on a CW complex to be regular is similar to the restriction from arbitrary simplicial sets to simplicial complexes (or maybe even semi-simplicial sets).

Finally, the paper "Constructible hypersheaves via exit paths" in 4.2 gives references that show that a regular CW complex is conically stratified, and proves that for a locally countable regular CW complex, exodromy holds: $$ \operatorname{Sh}^{cbl} (X) \simeq \operatorname{Fun}(\operatorname{Sing}^S (X), \mathcal{S})$$ It however does not explicitly describe the exit-path category, which could however be done using their methods if the finite case is understood.

Answer 1

It seems to me like this statement is folklore, since e.g. the paper Stellar Stratifications on Classifying Spaces tries to show a generalization of it and at least hints that my simpler claim is true (see 1.9 there). I think I have found a simple proof, but would be happy about corrections/ comments.

First, note that a regular CW complex $X$ is in particular normal. In particular, the set of cells $S$ carries a partial order where $e_1 \leq e_2$ iff, equivalently,

• $e_1$ is contained in the closure $\bar{e_2}$,
• $e_1 \cap \bar{e_2} \neq \emptyset$,

see 3.1 in the mentioned paper for more. To show that the map $\operatorname{Sing}^S (X) \to S$ is an equivalence, we proceed by showing it is essentiall surjective and fully faithful.

Essentially surjective: As indicated by the post, from HA A.7.5, we know that the fiber of this map over a cell is just the singular simplicial set of the open cell itself (or, in dimension $0$, a point), in particular contractible and non-empty.

Fully faithful: Let $e_1$ and $e_2$ be cells in $X$, and $x \in e_1$, $y \in e_2$. If $e_1 \nleq e_2$, the mapping space $\operatorname{Map}_{\operatorname{Sing}^S(X)}(x,y)$ is also empty since there can't be a path $\gamma : [0,1] \to X$ from $x$ to $y$ that lies over the arrow $e_1 \to e_2$ in $S$, as it would have to somehow jump from $e_1$ to $e_2$ even though $e_1 \cap \bar{e_2} = \emptyset$.

If $e_1 \leq e_2$, so $e_1$ lies in the boundary of $e_2$, we need to show that $\operatorname{Map}_{\operatorname{Sing}^S(X)}(x,y)$ is contractible. As in the proof of HA A.6.10, we can identify this with $\operatorname{Sing}(P)$ with $P$ the space of paths $\gamma: [0,1] \to X$ from $x$ to $y$ such that $\gamma((0,1]) \subseteq e_2$. In particular, $\gamma([0,1]) \subseteq \bar{e_2}$, the image of the gluing map $D^n \to X$ of $e_2$, which by regularity is a homeomorphism onto its image.

Thus, we can identify $P$ with the space of maps $\gamma: [0,1] \to \mathbb{R}^n$ such that $\gamma(0) = y'$ for some fixed $y'$ with $|y'| = 1$ that corresponds to $y$, $\gamma(1) = x'$, and $|\gamma(t)|<1$ for all $0<t\leq 1$. This can clearly be contracted to the linear path, since the open unit ball is convex.