Fix $n\in \Z_{>0}$ and let $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ for $M$ a compact, connected PL manifold embedded in $\R^N$. Take $\widetilde h\colon X\to (B,\leqslant)$ the conical stratifying map from a previous post (``Conical stratifications via semialgebraic sets," 2018-04-16) compatible with the natural stratification $h\colon X\to SC$. The goal of this post is to construct a functor $F\colon \Sing_B(X) \to N(SC)$ from the $\infty$-category of entry paths that encodes the structure of $X$.
Recall that a simplicial set is a functor, an element of $\text{Fun}(\Delta^{op},\Set)$. A simplicial set $S$ is defined by its collection of $n$-simplices $S_n$, its face maps $s_i:S_{n-1}\to S_n$, and degeneracy maps $d_i:S_{n+1}\to S_k$, for all $i=0,\dots,n$. For the first simplicial set of interest in this post, we have
\begin{align*}
\Sing_B(X)_n & = \Hom_{\Top}^B(|\Delta^n|,X), \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \begin{array}{c}
\left(|\Delta^{n-1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{collapses $i$th with $(i+1)$th vertex, then maps as source}
\end{array}\right)\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{c}
\left(|\Delta^{n+1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{maps as $i$th face of source map}
\end{array}\right)
\end{align*}
We write $\Hom^B_{\Top}$ for the subset of $\Hom_{\Top}$ that respects the stratification $B$ in the context of entry paths. For the second simplicial set, the nerve, we have
\begin{align*}
N(SC)_n & = \{(S_0\tov{f_1} \cdots \tov{f_n} S_n)\ :\ S_i\in SC,\ f_i\ \text{are simplicial maps}\}, \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \left(S_0\tov{f_1}\cdots \tov{f_{n-1}} S_{n-1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_i} S_i \tov{\id} S_i\tov{f_{i+1}} \cdots \tov{f_{n-1}} S_{n-1}\right)\right),\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{r l}
i=0: & \left(S_0\cdots S_{n+1} \right) \mapsto \left(S_1\tov{f_2}\cdots \tov{f_{n+1}} S_{n+1} \right) \\
0<i<n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_{i-1}} S_{i-1} \tov{f_{i+1}\circ f_i} S_{i+1} \tov{f_{i+2}} \cdots \tov{f_{n+1}} S_{n+1}\right) \\
i=n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1}\cdots \tov{f_n} S_n \right)
\end{array} \right).
\end{align*}
Define $F$ on $k$-simplices as \[ F\left(\gamma\colon |\Delta^k|\to \Ran^{\leqslant n}(M)\times \R_{>0}\right) = \left(\widetilde h(\gamma(1,0,\dots,0)) \tov{\left(\widetilde h\circ \gamma \circ s_k\circ \cdots \circ s_2\right)\left( |\Delta^1|\right)} \cdots \tov{\left(\widetilde h\circ \gamma \circ s_{k-2}\circ \cdots \circ s_0\right)\left(|\Delta^1|\right)} \widetilde h(\gamma(0,\dots,0,1))\right). \] A morphism in $\Sing_B(X)$ is a composition of face maps $s_i$ and degeneracy maps $d_i$, so $F$ must satisfy the commutative diagrams
for all $s_i$, $d_i$. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.
Commutativity of the diagram on the left is immediate, as considering a simplex $|\Delta^{n-1}|$ as the $i$th face of a larger simplex $|\Delta^n|$ is the same as adding a step that is the identity map in the Hamiltonian path of vertices of $|\Delta^{n-1}|$. Similarly, observing that the image of the shortest path $v_{i-1}\to v_i\to v_{i+1}$ in $|\Delta^{n+1}|$, for $v_i = (0,\dots,0,1,0,\dots,0)$ the $i$th standard basis vector, induced by an element $\gamma\colon |\Delta^{n+1}|\to X$ in $\Sing_B(X)_{n+1}$, is homotopic to the image of the shortest path $v_{i-1} \to v_{i+1}$ shows that the diagram on the right commutes. Since $F$ is a natural transformation between the two functors $\Sing_B(X)$ and $N(SC)$, it is a functor on the functors as simplicial sets.
Remark: The particular choice of $X$ did not seem to play a large role in the arguments above. However, the stratifying map $\widetilde h\colon X\to B$ has image sitting inside $SC$, the nerve of which is the target of $F$, and every morphism in $\Sing_B(X)$ can be interpreted as a relation in $B\subseteq SC$ (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor $\Sing_A(X)\to N(A')$ may exist for a stratified space $X\to A\subseteq A'$.
Recall that a simplicial set is a functor, an element of $\text{Fun}(\Delta^{op},\Set)$. A simplicial set $S$ is defined by its collection of $n$-simplices $S_n$, its face maps $s_i:S_{n-1}\to S_n$, and degeneracy maps $d_i:S_{n+1}\to S_k$, for all $i=0,\dots,n$. For the first simplicial set of interest in this post, we have
\begin{align*}
\Sing_B(X)_n & = \Hom_{\Top}^B(|\Delta^n|,X), \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \begin{array}{c}
\left(|\Delta^{n-1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{collapses $i$th with $(i+1)$th vertex, then maps as source}
\end{array}\right)\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{c}
\left(|\Delta^{n+1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{maps as $i$th face of source map}
\end{array}\right)
\end{align*}
We write $\Hom^B_{\Top}$ for the subset of $\Hom_{\Top}$ that respects the stratification $B$ in the context of entry paths. For the second simplicial set, the nerve, we have
\begin{align*}
N(SC)_n & = \{(S_0\tov{f_1} \cdots \tov{f_n} S_n)\ :\ S_i\in SC,\ f_i\ \text{are simplicial maps}\}, \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \left(S_0\tov{f_1}\cdots \tov{f_{n-1}} S_{n-1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_i} S_i \tov{\id} S_i\tov{f_{i+1}} \cdots \tov{f_{n-1}} S_{n-1}\right)\right),\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{r l}
i=0: & \left(S_0\cdots S_{n+1} \right) \mapsto \left(S_1\tov{f_2}\cdots \tov{f_{n+1}} S_{n+1} \right) \\
0<i<n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_{i-1}} S_{i-1} \tov{f_{i+1}\circ f_i} S_{i+1} \tov{f_{i+2}} \cdots \tov{f_{n+1}} S_{n+1}\right) \\
i=n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1}\cdots \tov{f_n} S_n \right)
\end{array} \right).
\end{align*}
Define $F$ on $k$-simplices as \[ F\left(\gamma\colon |\Delta^k|\to \Ran^{\leqslant n}(M)\times \R_{>0}\right) = \left(\widetilde h(\gamma(1,0,\dots,0)) \tov{\left(\widetilde h\circ \gamma \circ s_k\circ \cdots \circ s_2\right)\left( |\Delta^1|\right)} \cdots \tov{\left(\widetilde h\circ \gamma \circ s_{k-2}\circ \cdots \circ s_0\right)\left(|\Delta^1|\right)} \widetilde h(\gamma(0,\dots,0,1))\right). \] A morphism in $\Sing_B(X)$ is a composition of face maps $s_i$ and degeneracy maps $d_i$, so $F$ must satisfy the commutative diagrams
for all $s_i$, $d_i$. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.
Commutativity of the diagram on the left is immediate, as considering a simplex $|\Delta^{n-1}|$ as the $i$th face of a larger simplex $|\Delta^n|$ is the same as adding a step that is the identity map in the Hamiltonian path of vertices of $|\Delta^{n-1}|$. Similarly, observing that the image of the shortest path $v_{i-1}\to v_i\to v_{i+1}$ in $|\Delta^{n+1}|$, for $v_i = (0,\dots,0,1,0,\dots,0)$ the $i$th standard basis vector, induced by an element $\gamma\colon |\Delta^{n+1}|\to X$ in $\Sing_B(X)_{n+1}$, is homotopic to the image of the shortest path $v_{i-1} \to v_{i+1}$ shows that the diagram on the right commutes. Since $F$ is a natural transformation between the two functors $\Sing_B(X)$ and $N(SC)$, it is a functor on the functors as simplicial sets.
Remark: The particular choice of $X$ did not seem to play a large role in the arguments above. However, the stratifying map $\widetilde h\colon X\to B$ has image sitting inside $SC$, the nerve of which is the target of $F$, and every morphism in $\Sing_B(X)$ can be interpreted as a relation in $B\subseteq SC$ (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor $\Sing_A(X)\to N(A')$ may exist for a stratified space $X\to A\subseteq A'$.
