Artin gluing a sheaf 1: a small example

The goal of this post is to describe a sheaf on a particular stratified space using locally constant sheaves defined on the strata. Thanks to Joe Berner for helpful discussions.

Recall the direct image and inverse image sheaves from a previous post ("Sheaves, derived and perverse," 2017-12-05). Let $M$ be a smooth, compact, connected manifold, and $X = \Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$. Let $SC$ be the category of abstract simplicial complexes and simplicial maps. All sheaves will be functors $\text{Op}(-)^{op}\to SC$. The space $X$ looks like the diagram below.

Let $Y = A\cup B$. Note that $A\subseteq Y$ is open, $B\subseteq Y$ is closed, $Y\subseteq X$ is open, and $C\subseteq X$ is closed. There is a natural stratified map $f:X\to \{1,2,3\}$, with $\{1,2,3\}$ given the natural ordering. The map $f$ is described by $f^{-1}(3) = A$, $f^{-1}(2) = B$, and $f^{-1}(1) = C$. Define the inclusion maps \begin{align*}
i\ &:\ A \hookrightarrow Y, & k\ &:\ Y\hookrightarrow X,\\
j\ &:\ B \hookrightarrow Y, & \ell\ &:\ C\hookrightarrow X.
\end{align*} Define the following constant sheaves on $A,B,C$, respectively:

If $U = \emptyset$, all three give back the simplicial complex on a single vertex. We will now attempt to define a sheaf on all of $X$ by gluing sheaves on the strata. Choose some subsets of $X$ as below on which to test the sheaves.

Step 1: Extend $\mathcal F$ and $\mathcal G$ to a sheaf on $Y$.

The direct image of $\mathcal F$ via $i$, as a sheaf on $Y$, is

for any $U\subseteq Y$. The inverse image of $i_*\mathcal F$ via $j$, as a sheaf on $B$, is

for any $U\subseteq B$. Note $j^*i_*\mathcal F(B')$ is the 0-simplex and $j^*i_*\mathcal F(B'')$ is the 1-simplex. The inverse image sheaf is actually defined as the sheafification of the presheaf obtained by taking the colimit, but the sheaf axioms are easily seen to be satisfied here, as the support is on a closed subset.

Following the MathOverflow question, we need to define a map $\mathcal G \to j^*i_*\mathcal F$ of sheaves on $B$. Since the support of $j^*i_*\mathcal F$ is only $\text{cl}(A)\cap B$, it suffices to define the map here, and we can do it on stalks. There is a natural simplicial map

which we use as the sheaf map. It seems we should now have a sheaf on all of $Y$ now, but the result is not immediate. Following the proof of Theorem 3.10 in Chapter 2 of Milne, we need to take the fiber product, or pullback, of $i_*\mathcal F$ and $j_*\mathcal G$ over $j_*j^*i_*\mathcal F$, call it $\mathcal K$. Consider the pullback diagram on sets like $B'''$:

Hence it makes sense that $\mathcal K(B''')$ is two 0-simplicies. We now have a sheaf $\mathcal K$ on $Y$ given by

Step 2: Extend $\mathcal K$ and $\mathcal H$ to a sheaf on $X$.

The direct image of $\mathcal K$ via $k$, as  a sheaf on $X$, is

for any $U\subseteq X$. The inverse image of $k_*\mathcal K$ via $\ell$, as a sheaf on $C$, is

for any $U\subseteq C$. We need to again define a map $\mathcal H\to \ell^*k_*\mathcal K$ of sheaves on $C$. On stalks we naturally have maps

due to the fact that both complexes are symmetric, so sending to one or the other vertex is the same. Let $\mathcal L$ be the sheaf we should now have defined over all of $X$, by taking the fiber product of $\ell_*\mathcal H$ and $k_*\mathcal K$ over $\ell_*\ell^*k_*\mathcal K$. Let us consider its pullback diagrams for the sets $L',M',N'$.

It seems that we should set $\mathcal L(L') = \mathcal L(M') = \mathcal L(N')$ to be the 0-simplex. We now have a sheaf $\mathcal L$ on $X$ given by

The next goal is to extend this approach to $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$. An immediate difficulty seems to be finding canonical simplicial maps like $\varphi$ and $\psi$, but hopefully a choice of increasing nested open cover of the startifying set of $X$ will solve this problem.

References: MathOverflow (Question 54037), Milne (Etale cohomology, Chapter 2.3)