In this post we continue on a previous topic ("Exit paths, part 1," 2017-08-31) and try to define a constructible sheaf via universality. Let $X$ be an $A$-stratified space, that is, a topological space $X$ and a poset $(A,\leqslant)$ with a continuous map $f:X\to A$, where $A$ is given the upset topology relative to its ordering $\leqslant$. Recall the full subcategory $\Sing^A(X)\subseteq \Sing(X)$ of exit paths on an $A$-stratified space $X$.

Briefly, a stratification $f:X\to A$ is

This proposition, given as part of Theorem A.6.4 in Lurie, has a very long proof, so is not repeated here. Lurie actually proves that the natural functor $\Sing^A(X)\to N(A)$ described below is a (inner) fibration, which implies the unique lifting property of $\Sing^A(X)$ via the unique lifting property of $N(A)$ (and we already know nerves are $\infty$-categories).

It is immediate that every 3-horn can only be filled in one unique way (as there is only one element of $N(A)_3$), as well as that every 2-horn can be filled in one unique way (as every sequence of two composable morphisms appears as a horn of exactly one element of $N(A)_2$).

In Appendix A.9 of Higher Algebra, Lurie says that there is an equivalence of categories \[(A\text{-constructible sheaves on }X) \cong \left[(A\text{-exit paths on }X),\mathcal S\right],\] given that $X$ is conically stratified, and for $\mathcal S$ the $\infty$-category of spaces (equivalently $N(Kan)$, the nerve of all the simplicial sets that are Kan complexes). So, instead of trying to define a particular constructible sheaf on $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, (as in previous posts "Stratifying correctly," 2017-09-17 and "A constructible sheaf over the Ran space," 2017-06-24) we will try to make a functor that takes an exit path of $X$ and gives back a space.

Fix $n\in \Z_{>0}$ and set $X = \Ran^{\leqslant n}\times \R_{\geqslant 0}$. Let $SC$ be the category of simplicial complexes and simplicial maps, with $SC_n$ the full subcategory of simplicial complexes with at most $n$ vertices. There is a map

\[\begin{array}{r c l}

g\ :\ X & \to & SC_n \\

(P,t) & \mapsto & VR(P,t),

\end{array}\]

allowing us to say

\[X = \bigcup_{S\in SC_n}g^{-1}(S).\]

Here we consider that two elements $P_i,P_j\in P$ give an edge of $VR(P,t)$ whenever $t>d(P_i,P_j)$ (this is chosen instead of $t\geqslant d(P_i,P_j)$ so that the boundaries of the strata ``facing downward," with respect to the poset ordering, are open). Now we define a stratifying poset $A$ for $X$.

\[ \left(a_S\leqslant a_T\right)\ \ \Longleftarrow\ \ \left(

\begin{array}{c}

\exists\ \sigma\in \Sing(X)_1\ \text{such that}\\

g(\sigma(0))=S,\ g(\sigma(t>0))=T.

\end{array}

\right)\]

Let $(A,\leqslant)$ be the poset generated by relations of the type given above.

We claim that $f:X\to A$ given by $f(P,t)=a_{g(P,t)}$ is a stratifying map, that is, continuous in the upset topology on $A$. To see this, take the open set $U_S = \{a_T\in A\ :\ a_S\leqslant a_T\}$ in the basis of the upset topology of $A$, for any $S\in SC_n$, and consider $x\in f^{-1}(U_S)$. If for all $\epsilon>0$ we have $B_X(x,\epsilon)\cap f^{-1}(U_S)^C\neq \emptyset$, then there exists $T_\epsilon\in SC_n$ with $B_X(x,\epsilon)\cap f^{-1}(a_{T_\epsilon})\neq\emptyset$, for $S\not\leqslant T_\epsilon$ (as $T_\epsilon\not\in U_S$). This means there exists $\sigma\in \Sing(X)_1$ with $\sigma(0)=x$ and $\sigma(t>0)\in f^{-1}(a_{T_\epsilon})$, which in turn implies $S\leqslant T_\epsilon$, a contradiction. Hence $f$ is continuous, so $f:X\to A$ is a stratification.

As all morphisms in $\Sing(X)$ are compsitions of the face maps $s_i$ and degenracy maps $d_i$, so are all morphisms in $\Sing^A(X)$. There is a natural functor $F:\Sing^A(X)\to N(A)$ defined in the following way:

\[\begin{array}{r r c l}

%%

%% L1

%%

\text{objects} & \left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\

a_0\leqslant \cdots \leqslant a_k\subseteq A \\

f(\sigma(t_0,\dots,t_i\neq 0,0,\dots,0)) = a_i

\end{array}

\right) & \mapsto & \left( a_0\to\cdots\to a_k\in N(A)_k\right) \\[20pt]

%%

%% L2

%%

\text{face maps} & \left(

\begin{array}{c}

\left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A

\end{array}

\right)\\[10pt]

\downarrow \\[10pt]

\left(

\begin{array}{c}

\tau:|\Delta^{k+1}|\to X \\ a_0\leqslant \cdots \leqslant a_i\leqslant a_i\leqslant \cdots a_k\subseteq A

\end{array}

\right)

\end{array}

\right) & \mapsto &

\left(\begin{array}{c}

\left(a_0\to\cdots \to a_k\right)\\[10pt]

\downarrow\\[10pt]

\left(a_0\to\cdots \to a_i\xrightarrow{\text{id}}a_i\to\cdots \to a_k\right)

\end{array}\right)\\[40pt]

%%

%% L3

%%

\text{degeneracy maps} & \left(

\begin{array}{c}

\left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A

\end{array}

\right)\\[10pt]

\downarrow \\[10pt]

\left(

\begin{array}{c}

\tau:|\Delta^{k-1}|\to X \\ a_0\leqslant \cdots \leqslant a_{i-1}\leqslant a_{i+1}\leqslant \cdots a_k\subseteq A

\end{array}

\right)

\end{array}

\right) & \mapsto &

\left(\begin{array}{c}

\left(a_0\to\cdots \to a_k\right)\\[10pt]

\downarrow\\[10pt]

\left(a_0\to\cdots \to a_{i-1}\xrightarrow{\circ}a_{i+1}\to\cdots \to a_k\right)

\end{array}\right)

\end{array}\]

As all maps in $\Sing^A(X)$ are generated by compositions of face and degeneracy maps, this completely defines $F$. Naturality of $F$ follows precisely because of this.

A poset (which can be viewed as a directed simple graph) may be naturally viewed as a 1-dimensional simplicial set, moreover an $\infty$-category (by virtue of being a \emph{simple} graph, with no multi-edges or loops). Hence there is a natural map, the inclusion, that takes $N(A)$ into $N(\mathcal Kan) = \mathcal S$. Finally, Construction A.9.2 of Lurie describes a map that takes a functor from $A$-exit paths into spaces and gives back an $A$-constructible sheaf over $X$, which Theorem A.9.3 shows to be an equivalence, given the following conditions:

**Proposition:**If $X\to A$ is conically stratified, $\Sing^A(X)$ is an $\infty$-category.Briefly, a stratification $f:X\to A$ is

*conical*if for every stratum there exists a particular embedding from a stratified cone into $X$ (see Lurie for "conical stratification" and Ayala, Francis, Tanaka for "conically smooth stratified space," which seem to be the same). We will leave confirming the described stratification as conical to a later post.This proposition, given as part of Theorem A.6.4 in Lurie, has a very long proof, so is not repeated here. Lurie actually proves that the natural functor $\Sing^A(X)\to N(A)$ described below is a (inner) fibration, which implies the unique lifting property of $\Sing^A(X)$ via the unique lifting property of $N(A)$ (and we already know nerves are $\infty$-categories).

**Example:**The nerve of a poset is an $\infty$-category. Being a nerve, it is already immediate, but it is worthwhile to consider the actual construction. For example, if $A = \{a\leqslant b\leqslant c \leqslant d\}$ is the poset with the ordering $\leqslant$, then the pieces $N(A)_i$ are as below.It is immediate that every 3-horn can only be filled in one unique way (as there is only one element of $N(A)_3$), as well as that every 2-horn can be filled in one unique way (as every sequence of two composable morphisms appears as a horn of exactly one element of $N(A)_2$).

In Appendix A.9 of Higher Algebra, Lurie says that there is an equivalence of categories \[(A\text{-constructible sheaves on }X) \cong \left[(A\text{-exit paths on }X),\mathcal S\right],\] given that $X$ is conically stratified, and for $\mathcal S$ the $\infty$-category of spaces (equivalently $N(Kan)$, the nerve of all the simplicial sets that are Kan complexes). So, instead of trying to define a particular constructible sheaf on $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, (as in previous posts "Stratifying correctly," 2017-09-17 and "A constructible sheaf over the Ran space," 2017-06-24) we will try to make a functor that takes an exit path of $X$ and gives back a space.

Fix $n\in \Z_{>0}$ and set $X = \Ran^{\leqslant n}\times \R_{\geqslant 0}$. Let $SC$ be the category of simplicial complexes and simplicial maps, with $SC_n$ the full subcategory of simplicial complexes with at most $n$ vertices. There is a map

\[\begin{array}{r c l}

g\ :\ X & \to & SC_n \\

(P,t) & \mapsto & VR(P,t),

\end{array}\]

allowing us to say

\[X = \bigcup_{S\in SC_n}g^{-1}(S).\]

Here we consider that two elements $P_i,P_j\in P$ give an edge of $VR(P,t)$ whenever $t>d(P_i,P_j)$ (this is chosen instead of $t\geqslant d(P_i,P_j)$ so that the boundaries of the strata ``facing downward," with respect to the poset ordering, are open). Now we define a stratifying poset $A$ for $X$.

**Definition:**Let $A = \{a_S\ :\ S\in SC_n\}$ and define a relation $\leqslant$ on $A$ by\[ \left(a_S\leqslant a_T\right)\ \ \Longleftarrow\ \ \left(

\begin{array}{c}

\exists\ \sigma\in \Sing(X)_1\ \text{such that}\\

g(\sigma(0))=S,\ g(\sigma(t>0))=T.

\end{array}

\right)\]

Let $(A,\leqslant)$ be the poset generated by relations of the type given above.

We claim that $f:X\to A$ given by $f(P,t)=a_{g(P,t)}$ is a stratifying map, that is, continuous in the upset topology on $A$. To see this, take the open set $U_S = \{a_T\in A\ :\ a_S\leqslant a_T\}$ in the basis of the upset topology of $A$, for any $S\in SC_n$, and consider $x\in f^{-1}(U_S)$. If for all $\epsilon>0$ we have $B_X(x,\epsilon)\cap f^{-1}(U_S)^C\neq \emptyset$, then there exists $T_\epsilon\in SC_n$ with $B_X(x,\epsilon)\cap f^{-1}(a_{T_\epsilon})\neq\emptyset$, for $S\not\leqslant T_\epsilon$ (as $T_\epsilon\not\in U_S$). This means there exists $\sigma\in \Sing(X)_1$ with $\sigma(0)=x$ and $\sigma(t>0)\in f^{-1}(a_{T_\epsilon})$, which in turn implies $S\leqslant T_\epsilon$, a contradiction. Hence $f$ is continuous, so $f:X\to A$ is a stratification.

As all morphisms in $\Sing(X)$ are compsitions of the face maps $s_i$ and degenracy maps $d_i$, so are all morphisms in $\Sing^A(X)$. There is a natural functor $F:\Sing^A(X)\to N(A)$ defined in the following way:

\[\begin{array}{r r c l}

%%

%% L1

%%

\text{objects} & \left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\

a_0\leqslant \cdots \leqslant a_k\subseteq A \\

f(\sigma(t_0,\dots,t_i\neq 0,0,\dots,0)) = a_i

\end{array}

\right) & \mapsto & \left( a_0\to\cdots\to a_k\in N(A)_k\right) \\[20pt]

%%

%% L2

%%

\text{face maps} & \left(

\begin{array}{c}

\left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A

\end{array}

\right)\\[10pt]

\downarrow \\[10pt]

\left(

\begin{array}{c}

\tau:|\Delta^{k+1}|\to X \\ a_0\leqslant \cdots \leqslant a_i\leqslant a_i\leqslant \cdots a_k\subseteq A

\end{array}

\right)

\end{array}

\right) & \mapsto &

\left(\begin{array}{c}

\left(a_0\to\cdots \to a_k\right)\\[10pt]

\downarrow\\[10pt]

\left(a_0\to\cdots \to a_i\xrightarrow{\text{id}}a_i\to\cdots \to a_k\right)

\end{array}\right)\\[40pt]

%%

%% L3

%%

\text{degeneracy maps} & \left(

\begin{array}{c}

\left(

\begin{array}{c}

\sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A

\end{array}

\right)\\[10pt]

\downarrow \\[10pt]

\left(

\begin{array}{c}

\tau:|\Delta^{k-1}|\to X \\ a_0\leqslant \cdots \leqslant a_{i-1}\leqslant a_{i+1}\leqslant \cdots a_k\subseteq A

\end{array}

\right)

\end{array}

\right) & \mapsto &

\left(\begin{array}{c}

\left(a_0\to\cdots \to a_k\right)\\[10pt]

\downarrow\\[10pt]

\left(a_0\to\cdots \to a_{i-1}\xrightarrow{\circ}a_{i+1}\to\cdots \to a_k\right)

\end{array}\right)

\end{array}\]

As all maps in $\Sing^A(X)$ are generated by compositions of face and degeneracy maps, this completely defines $F$. Naturality of $F$ follows precisely because of this.

A poset (which can be viewed as a directed simple graph) may be naturally viewed as a 1-dimensional simplicial set, moreover an $\infty$-category (by virtue of being a \emph{simple} graph, with no multi-edges or loops). Hence there is a natural map, the inclusion, that takes $N(A)$ into $N(\mathcal Kan) = \mathcal S$. Finally, Construction A.9.2 of Lurie describes a map that takes a functor from $A$-exit paths into spaces and gives back an $A$-constructible sheaf over $X$, which Theorem A.9.3 shows to be an equivalence, given the following conditions:

- $X$ is paracompact,
- $X$ is locally of singular shape,
- the $A$-stratification of $X$ is conical, and
- $A$ satisfies the ascending chain condition.

*References:*Stacks Project, Lurie (Higher algebra, Appendix A), Ayala, Francis and Tanaka (Local structures on stratified spaces, Sections 2 and 3)