In this post we describe a constructible sheaf over X=Ran⩽ 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 categoriesi^1_*:\Shv(A_1) \to \Shv(A_2), \hspace{1cm} 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} \hspace{1cm} \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 finallyi_*^{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}}, \hspace{1cm} \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")
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 categoriesi^1_*:\Shv(A_1) \to \Shv(A_2), \hspace{1cm} 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} \hspace{1cm} \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 finallyi_*^{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}}, \hspace{1cm} \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")