Artin gluing a sheaf 3: the Ran space

The goal of this post is to extend earlier ideas, of a sheaf defined on $\Conf_n(M)\times \R_{\geqslant 0}$, to a family of sheaves defined on $\bigcup_{k=1}^n \Conf_n(M)\times \R_{\geqslant 0} = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$.

Recall our main map $f:\Conf_n(M)\times \R_{\geqslant 0} \tov{VR(-)} SC_n \tov{\Hom(\Delta^\bullet,-)} \sSet$. Following Definition 1 and Proposition 2 in a previous post ("Artin gluing a sheaf 2: simplicial sets and configuration spaces," 2018-01-31), define a sheaf $\mathcal F_k$ on $X_k$ by \[ \mathcal F_k(U) = \begin{cases}
S_{k,\max\{1\leqslant \ell\leqslant N_k\ :\ U\cap B_{\ell}\neq \emptyset\}} & \text{ if $U$ is good,}\\
S_\emptyset & \text{ else if }U\neq\emptyset,
\end{cases} \hspace{2cm} (1) \] for all $k=1,\dots,n$. We have assumed a total order on all simplicial complexes on $k$ vertices, induced by a cover $U_k,\dots,U_{k,N_k}$ of nested opens of $X_k$. This induces a total order $S_{k,1},\dots,S_{k,N_k}$ on the image of $\Ran^k(M)\times \R_{\geqslant 0}$ in $\sSet$, and by the product order, a total order on all of $\sSet' := f(\Ran^{\leqslant n}(M)\times \R_{\geqslant 0})$.

A small example

Let $n=3$, so $X = \Ran^{\leqslant 3}(M)\times \R_{\geqslant 0}$. We already have $\mathcal F_1,\mathcal F_2,\mathcal F_3$ on $X_1,X_2,X_3$, respectively, and we will extend them from the top down to sheaves over all of $X$, as in the diagram below.

The map $i$ will be the inclusion of an open set into a larger one, and $j$ the inclusion of a closed set into a larger one. Recall that the pullback of two sheaves is defined equivalently by a map of sheaves on the boundary of the open nd closed sets. With that in mind, for $U\subseteq X_2\cup X_3$ good, the pullback square

defines $\mathcal F_{d_0}$, where the $d_0$ indicates the face map that skips the $0$th spot. The sheaf $\mathcal F_{d_1}$ is defined similarly, but by the face map $d_1$, and $\mathcal F_{d_2}$ by the face map $d_2$. For each of these three sheaves on $X_3\cup X_2$, we have two other sheaves, based on where the single point maps to. However, we note that for $U\subseteq X$ good and $U\cap X_1\neq\emptyset$, \[ \left((i_*\mathcal F_{d_0}\times j_*\mathcal F_1)(U) \text{ defined by } d_0\right)
\ \ =\ \
\left((i_*\mathcal F_{d_1}\times j_*\mathcal F_1)(U) \text{ defined by } d_0\right), \] where $\times$ denotes the pullback over the appropriate sheaf, and similarly for the other sheaves on good sets intersecting $X_1$. We now have 6 unique shaves on all of $X$.

Generalizing

Now let $n$ be any positive integer, and $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$. We reverse the indexation of the $\mathcal F_k$ and $X_k$ above to make notation less cumbersome (so now $\mathcal F_k$ is $\mathcal F_{n-k+1}$ from (1), over $X_k = \Ran^{n-k+1}(M)\times \R_{\geqslant 0}$). Define pullback sheaves $\mathcal F_{d_{\ell_1}}$ for $\ell_1=0,\dots,n$ on $X_2\cup X_2$ by the diagram

At the $k$th step, for $1<k<n$, we have sheaves $\mathcal F_{d_{\ell_1}\cdots d_{\ell_{k-1}}}$ over $\bigcup_{m=1}^k X_m$, defined by sequences of face maps $d_{\ell_{k-1}}$ when going from $X_k$ to $X_{k-1}$ and so on, where $\ell_m\in \{0,\dots,n-m+1\}$. Define pullback sheaves $\mathcal F_{d_{\ell_1}\cdots d_{\ell_{k-1}}d_{\ell_k}}$, for $\ell_k = 0,\dots,n-k+1$ on $\bigcup_{m=1}^{k+1} X_k$ by the diagram

At the end of this inductive process, we have $n!$ distinct sheaves $\mathcal F_{d_{\ell_1}\cdots d_{\ell_{n-1}}}$ on all of $X$. Note there is a sheaf map $\mathcal F_{d_{\ell_1}\cdots d_{\ell_i}\cdots d_{\ell_{n-1}}} \to \mathcal F_{d_{\ell_1}\cdots d_{\ell_i'}\cdots d_{\ell_{n-1}}}$, given on $U$ good by \[ \mathcal F_{d_{\ell_1}\cdots d_{\ell_i}\cdots d_{\ell_{n-1}}}(U) = S \mapsto \begin{cases}
S & \text{ if } |S_0| \leqslant n-i, \\
(\ell_i\ \ell_i')(S) & \text{ else,}
\end{cases} \] where $(\ell_i\ \ell_i')\in \mathfrak S_n$ (the symmetric group on the numbers $0,\dots,n-1$) is the transposition swaps the $\ell_i$ and $\ell_i'$ indices of $S_0$, the 0-cells of $S$, inducing a map of simplicial sets. If the two sheaves differ in only two indices $\ell_i\neq \ell_i'$ and $\ell_j\neq \ell_j'$, with $i<j$, then we get $S\mapsto (\ell_j\ \ell_j')_{d_{\ell_{i-1}} \cdots d_{\ell_j}}(\ell_i\ \ell_i')(S)$. Here $(\ell_j\ \ell_j')_{d_{\ell_{i-1}} \cdots d_{\ell_j}}$ is the element of $\mathfrak S_{n-i}$ found by taking $(\ell_j\ \ell_j')$ from $\mathfrak S_{n-j}$ to $\mathfrak S_{n-i}$ by the sequence of group inclusion maps induced by the face maps $d_{\ell_j},\dots,d_{\ell_{i-1}}$.

Remark: This construction is not the most satisfying, for several reasons:

• we do not have a single sheaf, rather a family of sheaves, and
• the use of "good" sets leaves something to be desired, as we should be able to consider larger sets.

Both will hopefully be remedied in a later post.