A naive constructible sheaf

In this post we describe a constructible sheaf over $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ valued in simplicial complexes, for a compact, smooth, connected manifold $M$. We note however that it does not capture all the information about the underlying space. Thanks to Joe Berner for helpful ideas.

Recall the category $SC$ of simplicial complexes and simplicial maps, as well as the full subcategories $SC_n$ of simplicial complexes with $n$ vertices (the vertices are unordered). Let $A = \bigcup_{k=1}^n SC_n$ with the ordering $\leqslant_A$ as in a previous post ("Ordering simplicial complexes with unlabeled vertices," 2017-12-03), and $f:X\to A$ the stratifying map. Let $\{A_k\}_{k=1}^N$ be a cover of $X$ by nested open sets of the type $f^{-1}(U_S) = f^{-1}(\{T\in A\ :\ S\leqslant_A T\})$, whose existence is guaranteed as $A$ is finite. Note that $f(A_1)$ is a singleton containg the complete simplex on $n$ vertices.

Remark: For every simplicial complex $S\in A$, there is a locally constant sheaf over $f^{-1}(S)\subseteq X$. Given the cover $\{A_k\}$ of $X$, denote this sheaf by $\mathcal F_k \in \Shv(A_k\setminus A_{k-1})$ and its value by $S_k\in SC$.

Let $i^1:A_1\hookrightarrow A_2$ and $j^2:A_2\setminus A_1 \hookrightarrow A_2$ be the natural inclusion maps . Note that $A_1$ is open and $A_2\setminus A_1$ is closed in $A_2$. The maps $i^1,j^2$ induce direct image functors on the sheaf categories\[i^1_*:\Shv(A_1) \to \Shv(A_2),
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j^2_*:\Shv(A_2\setminus A_1) \to \Shv(A_2).\]The induced sheaves in $\Shv(A_2)$ are extended by 0 on the complement of the domain from where they come. Note that since $A_2\setminus A_1\subseteq A_2$ is closed, $j^2_*$ is the same as $j^2_!$, the direct image with compact support. We then have the direct sum sheaf $i^1_*\mathcal F_1 \oplus j_*^2\mathcal F_2 \in \Shv(A_2)$, which we interpret as the disjoint union in $SC$. Then\[\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)(U) = \begin{cases}
S_1 & \text{ if }U\subseteq A_1, \\
S_2 & \text{ if }U\subseteq A_2\setminus A_1, \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}
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\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)_{(P,t)} = \begin{cases}
S_1 & \text{ if } (P,t)\in A_1, \\
S_2 & \text{ if }(P,t)\in \text{int}(A_2\setminus A_1), \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}\]for $U\subseteq A_2$ open and $(P,t)\in A_2$. Generalizing this process, we get a sheaf on $X$. The diagram

may be helpful to keep in mind. We use the fact that direct sums commute with colimits (used in the definition of the direct image sheaf) to simplify notation. We then get sheaves\[\begin{array}{r c l}
\mathcal F^1 & \in & \Shv(A_1), \\
i_*^1\mathcal F^1 \oplus j_*^2 \mathcal F^2 & \in & \Shv(A_2), \\
i_*^2i_*^1\mathcal F^1 \oplus i_*^2j_*^2 \mathcal F^2 \oplus j_*^3 \mathcal F^3 & \in & \Shv(A_3), \\
i_*^3i_*^2i_*^1\mathcal F^1 \oplus i_*^3i_*^2j_*^2 \mathcal F^2 \oplus i_*^3j_*^3 \mathcal F^3 \oplus j_*^4 \mathcal F^4 & \in & \Shv(A_4),
\end{array}\]and finally\[i_*^{N-1\cdots 1}\mathcal F^1 \oplus \left(\bigoplus_{k=2}^{N-1} i_*^{N-1\cdots k}j_*^k \mathcal F^k \right) \oplus j_*^N \mathcal F^N \in \Shv(A_N=X),\]where $i_*^{N-1\cdots k}$ is the composition $i_*^{N-1} \circ i_*^{N-2} \circ \cdots \circ i_*^k$ of direct image functors. Call this last sheaf simply $\mathcal F \in \Shv(X)$. Each $i_*^k$ extends the sheaf by 0 on an ever larger domain, so every summand in $\mathcal F$ is non-zero on exactly one stratum as defined by $f:X\to A$. We now have a functor $\mathcal F:Op(X) \to SC$ defined by\[\mathcal F(U) = \bigsqcup_{k=1}^N S_k \delta_{U,A_K\setminus A_{k-1}},
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\mathcal F_{(P,t)} = \bigsqcup_{k=1}^N S_k \delta_{(P,t),\text{cl}(,A_K\setminus A_{k-1})},\]where $\delta_{U,V}$ is the Kronecker delta that evaluates to the identity if $U\cap V \neq \emptyset$ and zero otherwise.

Remark: The sheaf $\mathcal F$ is $A$-constructible, as $\mathcal F|_{f^{-1}(S)}$ is a constant sheaf evaluating to the simplicial complex $S\in A$. However, if we want the cohomology groups to capture how the simplicial complexes change between strata, then we must use a different approach - all groups die when leaving a stratum because of the extension by zero construction.

References: nLab (article "Simplicial complexes")