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)

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\[i^1_*:\Shv(A_1) \to \Shv(A_2),
\hspace{1cm}
j^2_*:\Shv(A_2\setminus A_1) \to \Shv(A_2).\]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 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\[\mathcal F(U) = \bigsqcup_{k=1}^N S_k \delta_{U,A_K\setminus A_{k-1}},
\hspace{1cm}
\mathcal F_{(P,t)} = \bigsqcup_{k=1}^N S_k \delta_{(P,t),\text{cl}(,A_K\setminus A_{k-1})},\]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")

Sheaves, derived and perverse

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map. We let $\Shv(X)$ be the category of sheaves on $X$, $D(\Shv(X))$ the derived category of sheaves on $X$, and $D_b(\Shv(X))$ the bounded variant. Recall that $D(\mathcal A)$ for an abelian category $\mathcal A$ is constructed first by taking $C(\mathcal A)$, the category of cochains of elements of $\mathcal A$, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let $\mathcal F\in \Shv(X)$. Recall:

• a section of $\mathcal F$ is an element of $\mathcal F(U)$ for some $U\subseteq X$,
• a germ of $\mathcal F$ at $x\in X$ is an equivalence class in $\{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x$,
• $s\sim_x t$ iff every neighborhood $W$ of $x$ in $U\cap V$ has $s|_W = t|_W$, for $s\in \mathcal F(U)$, $t\in \mathcal F(V)$,
• the support of the section $s\in\mathcal F(U)$ is $\supp(s) = \{x\in U\ :\ s \nsim_x 0\}$,
• the support of the sheaf $\mathcal F$ is $\supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}$.

Definition: The map $f$ induces functors between categories of sheaves, called
\[\begin{array}{r r c l}
\text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt]
\text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\
& (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt]
\text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right).
\end{array}\]

Above we used that $f:X\to Y$ is proper if $f^{-1}(K)\subseteq X$ is compact, for every $K\subseteq Y$ compact. Next, recall that a functor $\varphi:\mathcal A\to \mathcal B$ induces a functor $R\varphi:D(\mathcal A)\to D(\mathcal B)$, called the (first) derived functor of $\varphi$, given by $R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet)$.

Remark: Each of the maps $f_*,f^*,f_!$ have their derived analogues $Rf_*, Rf^*,Rf_!$, respectively. For reasons unclear, $Rf_!$ has a right adjoint, denoted $Rf^!:D(\Shv(Y))\to D(\Shv(X))$. This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let $A^\bullet \in D(\Shv(X))$. Then:

• the $i$th cohomology sheaf of $A^\bullet$ is $H^i(A^\bullet) = \ker(d^i)/\im(d^i)$,
• $A^\bullet$ is a constructible complex if $H^i(A^\bullet)$ is a constructible sheaf for all $i$,
• $A^\bullet$ is a perverse sheaf if $A^\bullet\in D_b(\Shv(X))$ is constructible and $\dim(\supp(H^{-i}(P))) \leqslant i$ for all $i\in \Z$ and for $P=A^\bullet$ and $P=(A^\bullet)^\vee = (A^\vee)^\bullet$ the dual complex of sheaves.

We finish off with an example.

Example: Let $X = \R$ be a stratified space, with $X_0=0$ the origin and $X_1 = \R\setminus 0$. Let $\mathcal F\in \Shv(X)$ be an $\R$-valued sheaf given by $\mathcal F(U) = \inf_{x\in U} |x|$, and define a chain complex $A^\bullet$ in the following way:
\[0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.\]
Note that for any $U\subseteq \R$, we have $H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset$ if $0\not\in U$, and $0$ otherwise. Hence $\supp(H^{-1}(A^\bullet)) = \R\setminus 0$, whose dimension is 1. Next, $H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0$, and so $\dim(\supp(H^0(A^\bullet))) = 0$. Note that $A^\bullet$ is self-dual and constructible, as the cohomology sheaves are locally constant. Hence $A^\bullet$ is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")

Ordering simplicial complexes with unlabeled vertices

The goal of this post is to describe a partial order on the collection of simplical complexes with $\leqslant n$ unlabeled vertices that is nice in the context of the space $X=\Ran^{\leqslant n}(M)\times \R_{>0}$.

First note that there is a natural order on (abstract) simplicial complexes, given by set inclusion. Interpreting elements of $X$ as simplicial complexes induces a more restrictive order, as new vertices must "split off" from existing ones rather than just be introduced anywhere. Also note that the category usually denoted by $SC$ of simplicial complexes and simplicial maps contains objects with unordered vertices. Here we assume an order on them and consider the action of the symmetric groups to remove the order.

Definition: Let $SC_k$, for some positive integer $k$, be the collection of simplicial complexes with $k$ uniquely labeled vertices. This collection is a poset, with $S\leqslant T$ iff $\sigma\in T$ for every $\sigma\in S$.

The symmetric group on $k$ elements acts on $SC_k$ by permuting the vertices, and taking the image under this action we get $SC_k/S_k$, the collection of simplicial complexes with $k$ unlabeled vertices. This set also has a partial order, with $S\leqslant T$ in $SC_k/S_k$ iff $S'\leqslant T'$ in $SC_k$, for some $S'\in q_k^{-1}(S)$ and $T'\in q_k^{-1}(T)$, where $q_k:SC_k \twoheadrightarrow SC_k/S_k$ is the quotient map.

Definition: For all $i=1,\dots,k$, let $s_{k,i}$ be the $i$th splitting map, which splits the $i$th vertex in two. That is, if the vertices of $S\in SC_k$ are labeled $v_1,\dots,v_k$, then $s_{k,i}$ is defined by
\[\begin{array}{r c l}
s_{k,i}\ :\ SC_k & \to & SC_{k+1}, \\
S & \mapsto & \left\langle S'\cup \{v_i,v_{i+1}\} \cup \displaystyle \bigcup_{\{v_i,w\}\in S'} \{v_{i+1},w\} \right\rangle ,
\end{array}\]where $S'$ is $S$ with $v_j$ relabeled as $v_{j+1}$ for all $j>i$, and $\langle T\rangle$ is the simplicial complex generated by $T$.

By "generated by $T$" we mean generated in the Vietoris-Rips sense, that is, if $\{v_a,v_b\}\in T$ for all $a,b$ in some indexing set $I$, then $\{v_c\ :\ c\in I\}\in \langle T\rangle$. The $i$th splitting map is essentially the $i$th face map used for simplicial sets.

Let $A = \bigcup_{k=1}^n SC_k/S_k$. The splitting maps induce a partial order on $A$, with $S\leqslant T$, for $S\in SC_k/S_k$ and $T\in SC_{k+1}/S_{k+1}$, iff $s_{k,i}(S')\leqslant T'$ in $SC_k$, for some $S'\in q_k^{-1}(S)$, $T'\in q_{k+1}^{-1}(T)$, and $i\in \{1,\dots,k\}$. This generalizes via composition of the splitting maps to any pair $S,T\in A$, and is visually decribed by the diagram below.

Now, let $M$ be a smooth, compact, connected manifold embedded in $\R^N$, and $X=\Ran^{\leqslant n}(M)\times \R_{>0}$. Let $f:X\to A$ be given by $(P,t)\mapsto VR(P,t)$, the Vietoris-Rips complex around the points of $P$ with radius $t$.

Proposition: The map $f:X\to A$ is continuous.

Proof: Let $S\in A$ and $U_S \subseteq A$ be the open set based at $S$. Take any $(P,t)\in f^{-1}(U_S)\subseteq X$, for which we will show that there is an open ball $B\owns (P,t)$ completely within $f^{-1}(U_S)$.

Case 1: $t\neq d(P_i,P_j)$ for all pairs $P_i,P_j\in P$. Then set
\[\epsilon = \min\left\{t, \min_{i<j} |t-d(P_i,P_j)|, \min_{i<j} d(P_i,P_j) \right\}.\]Set $B = B^{\Ran^{\leqslant n}(M)}_{\epsilon/4}(P) \times B^{\R_{>0}}_{\epsilon/4}(t)$, which is an open neighborhood of $(P,t)$ in $X$. It is immediate that $f(P',t')$, for any other $(P',t')\in B$, has all the simplices of $f(P,t)$, as $\epsilon \leqslant |t-d(P_i,P_j)|$ for all $i<j$. If $P_i$ has split in two in $P'$, then for every simplex containing $P_i$ in $f(P,t)$ there are two simplices in $f(P't')$, with either of the points into which $P_i$ split. That is, there may be new simplices in $f(P',t')$, but $f(P',t')$ will be in the image of the splitting maps. Equivalently, $f(P,t)\leqslant f(P',t')$ in $A$, so $B\subseteq f^{-1}(U_S)$.

Case 2: $t= d(P_i,P_j)$ for some pairs $P_i,P_j\in P$. Then set
\[\epsilon = \min\left\{t, \min_{i<j \atop t\neq d(P_i,P_j)} |t-d(P_i,P_j)|,\ \min_{i<j} d(P_i,P_j) \right\},\]and define $B$ as above. We are using the definition of Vietoris-Rips complex for which we add an edge between $P_i$ and $P_j$ whenever $t>d(P_i,P_j)$. Now take any $(P',t')\in B$ such that its image and the image of $(P,t)$ under $f$ are both in $SC_k/S_k$. Then any points $P_i,P_j \in P$ with $d(P_i,P_j)=t$ that have moved around to get to $P'$, an edge will possibly be added, but never removed, in the image of $f$ (when comparing with the image of $(P,t)$). This means that we have $f(P,t)\leqslant f(P',t')$ in $SC_k/S_k$, so certainly $f(P,t)\leqslant f(P',t')$ in $A$. The same argument as in the first case holds if points of $P$ split. Hence $B\subseteq f^{-1}(U_S)$ in this case as well.  $\square$

This proposition shows in particular that $X$ is poset-stratified by $A$

Perspectives on the Ran space

This post combines the finite subset approach with the mapping space approach of the Ran space, in the context of stratifications. The goal is to understand the colimit construction of the Ran space, as that leads to more powerful results.

Topology

Let $X,Y$ be topological spaces.

Definition: The mapping space of $X$ with respect to $Y$ is the topological space $X^Y = \{f:Y\to X$ continuous$\}$. The topology on $X^Y$ is the compact-open topology which has as basis finite intersections of sets
\[\{f\in X^Y\ :\ f(K)\subseteq U\}, \hspace{3cm} (1)\]
for all $K\subseteq Y$ compact and all $U\subseteq X$ open.

Now fix a positive integer $n$.

Definition: The Ran space of $X$ is the space $\Ran^{\leqslant n}(X) = \{P\subseteq X\ :\ 0<|P|\leqslant n\}$. The topology on $\Ran^{\leqslant n}(X)$ is the coarsest which contains
\[\left\{P\in \Ran^{\leqslant n}(X)\ :\ P\subseteq \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i \right\} \hspace{3cm} (2)\]
as open sets, for all nonempty finite collection of parwise disjoint open sets $\{U_i\}_{i=1}^k$ in $X$.

From now on, we let $I$ be a set of size $n$ and $M$ be a compact, smooth, connected $m$-manifold. There is a natural map
\[\begin{array}{r c l}
\varphi\ :\ M^I & \to & \Ran^{\leqslant n}(M), \\
(f:I\to M) & \mapsto & f(I).
\end{array}\]
This map is surjective, and for $n>1$, is not injective.

Proposition 1: The map $\varphi$ is continuous and an open map.

Proof: For continuity, take an open set $U\subseteq \Ran^{\leqslant n}(M)$ as in (2) and consider $\varphi^{-1}(U)$. We use the fact that $\{*\}\subset I$ is a compact (in fact open and closed) subset of $I$ and that all the $U_i$ are open, as is their union. Observe that
\begin{align*}\varphi^{-1}(U) & = \left\{f\in M^I \ :\ f(I)\subset \bigcup_{i=1}^k U_i,\ f(I)\cap U_i\neq \emptyset\ \forall\ i \right\} \\
& = \left\{f\in M^I\ :\ f(I)\subset \bigcup_{i=1}^k U_i\right\} \cap \bigcap_{i=1}^k \left\{f\in M^I\ :\ f(*\in I) \in U_i\right\},\end{align*}
which is a finite intersection of sets of the type (1), and so $\varphi^{-1}(U)$ is open in $M^I$.

For openness, take an open set $V$ as in (1), so $V = \bigcap_{i=1}^k \{f\in M^I\ :\ f(K)\subseteq U_i\}$ for different subsets $K\subseteq I$. By Lemma 1, we may assume that the $U_i$ are pairwise disjoint. For each $U_i$, let $\{U_{i,j}\}_{j=1}^\infty$ be a sequence of increasing open sets in $U_i$ such that $U_{i,j}\subseteq U_{i,j+1}$ and $U_{i,j}\xrightarrow{\ j\to\infty\ } U_i$. Then
\[\varphi (V) = \underbrace{\left\{P\in M\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}}_{f\in M^I\text{\ with image completely in the\ }U_i} \cup \underbrace{\bigcap_{i=1}^k \bigcup_{j=1}^\infty \left\{ P\in M\ :\ P\subset U_{i,j}\cup \left(\overline{U_{i,j}}\right)^c,\ P\cap U_{i,j}\neq \emptyset,\ P \subset \left(\overline{U_{i,j}}\right)^c \neq \emptyset\right\}}_{f\in M^I\text{\ with image partially in the\ }U_i}.\]
Note that $U_{i,j}$ and $\left(\overline{U_{i,j}}\right)^c$, the complement of the closure of $U_{i,j}$ are both open and disjoint in $M$. Since infinite unions and finite intersections of elements in the topology are also open, we have that $\varphi(V)$ is open in $\Ran^{\leqslant n}(M)$. $\square$

The above proposition says that we may talk equivalently about the compact-open topology on $M^I$ and the Ran space topology on $\Ran^{\leqslant n}(M)$. Viewing the Ran space as a function space allows for more general terminology to be applied.

Lemma 1: Let $U_i\subseteq M$ be open, for $i=1,\dots,k$. Then $\bigcap_{i=1}^k\{f\in M^I\ :\ f(K)\subseteq U_i\}$ may be written as a union of intersections $\bigcap_{j=1}^\ell \{f\in M^I\ :\ f(K)\subseteq V_j\}$ with the $V_j$ open, pairwise disjoint, and $\ell\leqslant k$.

Proof: It suffices to prove this in the case $k=2$. Let $U,V\subseteq M$ open and suppose than $U\cap V\neq \emptyset$. Note that $U\setminus V$ and $V\setminus U$ are separated (that is, $(U\setminus V) \cap \overline{V\setminus U} = \emptyset$ and $(V\setminus U)\cap \overline{U\setminus V} = \emptyset$), and since $\R^N$ is a completely normal space (equivalently, satisfies the $T5$ axiom), there exist disjoint open sets $A,B$ with $U\setminus V\subseteq A$ and $U\setminus V\subseteq B$. So for $A' = A\cap (U\cup V)$ and $B' = B\cap (U\cup V)$, we have
\begin{align*} \{f\in M^I\ &:\ f(K)\subseteq U\} \cap \{f\in M^I\ :\ f(K)\subseteq V\} \\
& = \left(\{f\in M^I\ :\ f(K)\subseteq U\setminus V\} \cap \{f\in M^I\ :\ f(K)\subseteq V\setminus U\}\right) \cup \{f\in M^I\ :\ f(K)\subseteq U\cap V \} \\
& = \left(\{f\in M^I\ :\ f(K)\subseteq A'\} \cap \{f\in M^I\ :\ f(K)\subseteq B'\}\right) \cup \{f\in M^I\ :\ f(K)\subseteq U\cap V \}, \end{align*}
for $A',B',U\cap V$ open, and $A'\cap B' = \emptyset$.  $\square$

Note that in the last calculation of the proof, the intersection of sets in the second line is smaller than the intersection of sets in the last line (as $U\setminus V \subsetneq A$ and $V\setminus U\subsetneq B$). However, all the extra ones in the third line appear in the set $\{f\in M^I\ :\ f(K)\subseteq U\cap V\}$.

Stratifications

Now we compare stratifications on $M^I$ and $\Ran^{\leqslant n}(M)$. As before, $I$ is a set of size $n$.

Corollary: An image-constant $A$-stratification on $M^I$ is equivalent to an $A$-stratification on $\Ran^{\leqslant n}(M)$.

This follows from Proposition 1. By image-constant we mean if $\alpha,\beta\in M^I$ have the same image (that is, $\alpha(I)=\beta(I)$), then $\alpha,\beta$ are sent to the same element of $A$.

Proof: If we start with a continuous map $f:M^I\to A$, setting $g(P) = f(I\to M)$ whenever $(I\to M) \in \varphi^{-1}(P)$ is continuous, as $\varphi(f^{-1}(U))$ is open, by continuity of $f$ and openness of $\varphi$. The assignment $g(P) = f(I\to M)$ whenever $(I\to M) \in \varphi^{-1}(P)$ is well defined, as the stratification is image-constant, so any continuous map from $M^I$ must send every element of $\varphi^{-1}(P)$ to the same place.

Conversely, if we start with a continuous map $g:\Ran^{\leqslant n}(M)\to A$, setting $f(I\to M) = g(\varphi(I\to M))$ is continuous, as $\varphi^{-1}(g^{-1}(U))$ is open, by continuity of $g$ and continuity of $\varphi$. This map is image-constant, as $\varphi(\alpha:I\to M) = \alpha(I)$. $\square$

Next we consider a particular stratification of $M^I$, adapted from Example 3.5.17 of Ayala-Francis-Tanaka, simplified with $P=\{*\}$. That is, the example begins with a stratified space $M\to P$ and proceeds to construct another stratification $M^I\to P'$, but we only consider the trivial stratification $M\to \{*\}$.

Definition: Given $M$ and $I$, let the poset $\mathcal P(I)$ of coincidences on $I$ be the set of equivalence relations on $I$, ordered by reverse set inclusion. Let $f_I:M^I\to \mathcal P(I)$ be the natural stratification that takes a map $\alpha: I\to M$ to the equivalence relation on $I$ describing which elements of $I$ coincide in the image of $\alpha$.

Example: An element of $\mathcal P(I)$ is a subset of $I\times I$ always containing $(a,a)$ for every $a\in I$ (reflexivity), and satisfying the symmetry and transitivity conditions. For example, if $|I|=3$ or 4, then $\mathcal P(I)$ is ordered as in the diagrams below, with order increasing from left to right. We simplify things by writing $[x_1,\dots,x_k]$ for the collection $(x_i,x_j)$ of all $i\neq j$ (the equivalence class).

To check that the map $f_I:M^I\to \mathcal P(I)$ is continuous, we first note that an element $U_{[x_1],\dots,[x_k]}$ in the basis of the upwards-directed topology on $\mathcal P(I)$ contains images of $\alpha\in M^I$ whose images have at most the elements of each equivalence class $[x_i]$ coinciding. Hence
\[ f_I^{-1}(U_{[x_1],\dots,[x_k]}) = \bigcup_{U_1,\dots,U_k\subseteq M \atop \text{open, disjoint}}\ \bigcap_{i=1}^k \left\{\alpha\in M^I\ :\ \alpha(K = \{x\in [x_i]\}) \subseteq U_i\right\},\]
which is an open set in the compact-open topolgy on $M^I$.

The Ran space as a colimit

Beilinson-Drinfeld (Section 3.4) and Ayala-Francis-Tanaka (Section 3.7) describe the Ran space as a colimit, the former of a functor into topological spaces, the latter of a functor into stratified spaces. See Mac Lane for a full treatment of colimits. Both BD and AFT use the category $\Fin^{surj,\leqslant n}$ of finite sets and surjections, that is,
\begin{align*}\Obj(\Fin^{surj,\leqslant n}) & = \{I\in \Obj(\Set)\ :\ 0<|I|\leqslant n\}, \\
\Hom_{\Fin^{surj,\leqslant n}}(I,J) & = \begin{cases}
\emptyset, & \text{\ if\ } |I|<|J|, \\
\left\{\text{surjections\ }I\to J\right\}, & \text{\ if\ } |I|\geqslant |J|.
\end{cases}\end{align*}
AFT uses more involved terminology, with "conically smooth" stratified spaces instead of just poset-stratified. They use a category $\Strat$, which for our purposes we may define as
\begin{align*}
\Obj(\Strat) & = \{\text{poset-stratified topological spaces }X\xrightarrow{ f } A\}, \\
\Hom_{\Strat}(X\xrightarrow{ f } A,Y\xrightarrow{ g } B) & = \{(\mu\in \Hom_{\Top}(X,Y), \nu \in \Hom_{\Set}(A,B)\ :\ g\circ \mu = \nu\circ f\}.
\end{align*}

Remark:  There is a natural functor $\mathcal F_M:(\Fin^{surj,\leqslant n})^{op} \to \Top$, given by $I\mapsto M^I$. A surjection $s:I\to J$ induces a map $M^J\to M^I$, with $(f:J\to M)\mapsto (f\circ s :I\to M)$. BD use this to declare that $\Ran^{\leqslant n}(M) = \colim(\mathcal F_M)$.

Remark: There is also a natural functor $\mathcal G_M:(\Fin^{surj,\leqslant n})^{op} \to \Strat$, given by $I\mapsto (M^I\to \mathcal P(I))$. AFT use this to declare that $(\Ran^{\leqslant n}(M)\to \{1,\dots,n\}) = \colim(\mathcal G_M)$.

The construction of AFT is even more general, as they consider the Ran space of an already stratified space. Here we use their result for $M\to \{*\}$ trivially stratified.

References: Ayala, Francis, and Tanaka (Local structures on stratified spaces, Sections 3.5 and 3.7), Beilinson and Drinfeld (Chiral algebras, Section 3.4), Mac Lane (Categories for the working mathematician, Chapter III.3)

Towards a sheaf of simplicial complexes

The goal of this post is to describe a new stratification of $\Ran^n(M)\times \R_{\geqslant 0}$ that builds on the ideas from a previous post (see "The point-counting stratification of the Ran space is conical (really though) ," 2017-11-15) and some newer ones.

Let $SC_n$ be the set of simplicial complexes on $n$ ordered vertices. There is a natural partial order on $SC_n$ given by inclusion of sets, viewing every simplex as a subset of the power set $\mathbf P(\{1,\dots,n\})$. The symmetric group $S_n$ has a natural action on $SC_n$ and $SC_n/S_n$ has an induced partial order as well. Hence we have a map
\[\begin{array}{r c l}
f\ :\ \Ran^n(M)\times \R_{\geqslant 0} & \to & SC_n/S_n, \\
(P,t) & \mapsto & VR(P,t),
\end{array}\]
where $VR(P,t)$ is the Vietoris-Rips complex on $P$ with radius $t$. We include a $k$-cell in $VR(P,t)$ at the vertices $\{P_0,\dots,P_k\}\subset P$ if $d(P_i,P_j)<t$ for all $0\leqslant i<j\leqslant k$. Because we have strict inequality, the map is continuous in the upwards-directed, or Alexandrov topology on $SC_n/S_n$. Indeed, taking the preimage of an open set $U_S$ in $SC_n/S_n$ based at some simplicial complex $S$ (such $U_S$ form the basis of topology on $SC_n/S_n$), there is an open ball of radius $\min_{i<j} d(P_i,P_j)/2$ in the $\Ran^n(M)$ component and $\min_{(P_i,P_j)\subset f(P,t)} |t-d(P_i,P_j)|$ in the $\R_{\geqslant 0}$ component around any $(P,t)\in f^{-1}(U_S)$.

Remark: The above shows that $\Ran^n(M)\times \R_{\geqslant 0}$ is poset-stratified by $SC_n/S_n$, in the sense of Definition A.5.1 of Lurie. However, the strata are all of the same dimension, so there is no chance of this being a conical stratification, in the sense of Definition A.5.5 of Lurie. We hope to fix that with a different stratification.

Definition: Construct a poset $(A,\leqslant_A)$ in the following way:

• $SC_n/S_n \subset A$, with $S\leqslant_A T$ whenever $S\leqslant_{SC_n/S_n} T$ ,
• for every $S\neq T\in SC_n/S_n$, let $a_{ST}\in A$ with $a_{ST}\leqslant_A S$ and $a_{ST}\leqslant_A T$,
• for every $\{S_1,\dots,S_{k>2}\}\subset SC_n/S_n$, let $a_{S_1\cdots S_k}\in A$ with $a_{S_1\cdots S_k} \leqslant_A a_{S_1\cdots \widehat{S_i}\cdots S_k}$ for all $1\leqslant i\leqslant k$.

Define a map into $(A,\leqslant_A)$ in the following way:
\[\begin{array}{r c l}
g\ :\ \Ran^n(M)\times \R_{\geqslant 0} & \to & A, \\
(P,t) & \mapsto & \begin{cases}
S, & \text{ if $(P,t)\in \text{int}(f^{-1}(S))$ for some }S\in SC_n/S_n, \\
a_{S_1\cdots S_k}, & \text{ if }(P,t)\in \text{cl}(f^{-1}(T))\ \iff\ T\in \{S_1,\dots,S_k\}.
\end{cases}
\end{array}\]

We now claim that $g$ is a stratifying map.

Proposition: The map $g$ is continuous.

Proof: Since $\text{int}(f^{-1}(S))\cap \text{int}(f^{-1}(T)) = \emptyset$ for all $S\neq T\in SC_n/S_n$, the open sets $U_S\subseteq A$ based at $S$ all have open preimage $g^{-1}(U_S) \subseteq X$. Now take $(P,t)\in g^{-1}(U_{a_{S_1\cdots S_k}})$, for $k\geqslant 2$. If every open ball around $(P,t)\in X$ intersects $X_{a_{\mathbf T}}$, for some $\mathbf T \subseteq SC_n/S_n$, then $(P,t)$ must be in the closure of $f^{-1}(T)$, for every $T\in \mathbf T$. Hence the only possible such $\mathbf T$ are $\mathbf T\subseteq \{S_1,\dots,S_k\}$, so $g^{-1}(U_{a_{S_1\cdots S_k}})$ is open in $X$. $\square$

The next step would be to show that this stratification is conical, though it is not clear yet if it is.

References: Lurie (Higher Algebra, Appendix A)

The point-counting stratification of the Ran space is conical (really though)

This post is meant to correct the issues of a previous post ("The point-counting stratification of the Ran space is conical," 2017-11-06). The setting is the same as before.

A key object here will be the Ran space of a disconnected space. If $M$ has $k$ connected components, then $\Ran^{\leq n}(M)$, for $k\leq n$, has $2^k$ connected components (as we must choose whether or not to have at least one point in each component). When $M$ is disconnected with $k\leq n$ components, let $\Ran^{\leq n}(M)'$ be the largest connected component of $\Ran^{\leq n}(M)$, that is, the one with at least one point in every component of $M$.

Proposition 1: The point-counting stratification $f:X\to A$ is conical.

Proof: Fix $P=\{P_1,\dots,P_k\}\in \Ran^k(M)\subseteq \Ran^{\leqslant n}(M)$ and set $2\epsilon = \min_{i<j}d(P_i,P_j)$. Set
\[
Z := \prod_{i=1}^k B^{\R^m}_{\epsilon/3}(0),
\hspace{.5cm}
Y := \Bigg\{(Q^1,\dots,Q^k)\in \underbrace{\Ran^{\leq n}\left(\coprod_{i=1}^k B^{\R^m}_{2\epsilon/3}(0)\right)'}_{W}\ :\ \overbrace{\sum_{i=1}^k \mathbf d(\{0\},Q^i)=\frac{2\epsilon}3}^{\text{the cone condition}}\ ,\ \hspace{-5pt}\underbrace{\sum_{j=1}^{|Q^i|}Q_j^i =0}_{\text{the centroid condition}}\hspace{-5pt}\ \forall\ i\Bigg\}.
\]
The cone condition ensures the right topology at the cone point of $C(Y)$. The centroid condition ensures injectivity of $\varphi_P$ when multiplying by $Z$. Reasoning for the constants $\epsilon/3$ and $2\epsilon/3$ is given in Proposition 2. The balls $B_{\epsilon/3}$ in $Z$ are open and the balls $B_{2\epsilon/3}$ in $Y$ are closed. The collection $Q^i$ is all the points in the $i$th ball $B_{2\epsilon/3}$. The product $Z\times C(Y)$ is given the topology for which a set $U$ is open if $\varphi_P(U)$ is open in the subspace topology on $\im(\varphi_P)\subseteq X$.

If $k=n$, then $Y=\emptyset$ because of the cone and centroid conditions. Then $C(Y)=*$ and $Z\times * \cong Z$ is by construction an open set in $X$, so $X$ is conically stratified at $P$. If $k<n$, then $Y$ has a natural $A_{>k}$ stratification\[(Q^1,\dots,Q^k) \mapsto \sum_{i=1}^k |Q^i| \in \{k+1,\dots,n\},\]
with the image landing in $A_{>k}=\{k+1,\dots,n\}$ because there are at least $k$ points in every element of $W$, and to satisfy the cone condition at least one $Q^i$ must have at least two points. We now claim that
\[\begin{array}{r c l}
\varphi_P\ :\ Z \times C(Y) & \to & X, \\
(R,(Q^1,\dots,Q^k),t\neq 0) & \mapsto & \{P_1+R_1+tQ^1,\dots,P_k+R_k+tQ^k\},\\
(R,*,0) & \mapsto & \{P_1+R_1,\dots,P_k+R_k\}
\end{array}\]
is an open embedding. Here $t\in [0,1)$ is the cone component and $P_i+R_i+Q^i$ is the collection of all $P_i+R_i+Q^i_j,$ for $j=1,\dots,|Q^i|$. Injectivity follows from the cone condition and centroid condition on $Y$: it is clear that for fixed $R\in Z$ the map $\varphi_P$ is injective, as we are taking points at different distances from $P_i+R_i$ (the cone condition). Because of the centroid condition, moving around to different $R$ means the centroid will also move, so we will not get a collection of points we previously had.

Further, $Z\times C(Y)$ gets mapped homeomorphically into $X$ because of the topology forced upon it - continuity and openness of the map follow immediately. Finally, since $\im(\varphi_P)$ is open in $X$, $\varphi_P$ is an open embedding, so $X$ is conically stratified at $P$. $\square$

The following proposition gives a better idea of where $\im(\varphi_P)$ actually lands in $X$.

Proposition 2: For $\varphi_P$ as above, we have $B^X_{2\epsilon/3k}(P)\subseteq \im(\varphi_P)\subseteq B^X_{\epsilon}(P)$.

Proof: For the first inclusion, take $S\in B^X_{2\epsilon/3k}(P)$ and for every $1\leq i\leq k$ set
\begin{align*}S^i & := \{s\in S\ :\ d(s,P_i)<d(s,P_j)\ \forall\ j\neq i\}, & (\text{points closest to $P_i$}) \\
T_i & := \frac1{|S^i|}\sum_{j=1}^{|S^i|} S^i_j, & (\text{centroid of the $S^i_j$}) \\
c_i & := \mathbf d(\{T_i\},S^i). & (\text{distance from the $S^i_j$ to their centroid})\end{align*}
If $|S|=k$, then $c_i=0$ for all $i$, and $t=0$, so we are at the cone point, and $S = \varphi_P(T-P,*,0)$. If $|S|>k$, then $0<\sum_i c_i <2\epsilon/3$, as $c_i < 2\epsilon/{3k}$ for all $i$, so there is some $t'\in (0,1)$ such that $\sum_i c_i = t'2\epsilon/3$. Then
\begin{align*}\varphi_P\left(T-P, \left( {\textstyle \frac1{t'}} (S^1-T_1),\dots, {\textstyle \frac1{t'}}(S^k-T_k) \right), t'\right)
& = \left\{ P_1+(T-P)_1 + t'{\textstyle \frac1{t'}}(S^1-T_1), \dots, P_k+(T-P)_k + t'{\textstyle \frac1{t'}}(S^k-T_k)\right\} \\
& = \left\{ P_1+T_1-P_1 + S^1-T_1, \dots, P_k+T_k-P_k + S^k-T_k\right\} \\
& = \left\{S^1, \dots, S^k\right\} \\
& = S.\end{align*}
Note that since $T_i$ is the centroid of the $S^i_j$, and $S^i \subset B^M_{2\epsilon/3k}(P_i)$, and the centroid of a collection of points is in their convex hull, we also have $T_i\in B^M_{2\epsilon/3k}(P_i)$. Since $\frac{2\epsilon}{3k}<\frac{\epsilon}3$ when $k>2$, we have that $P_i-T_i\in B^{\R^m}_{\epsilon/3}(0)$. If $k\leq 2$, then use $k+2$ instead of $k$. Finally, since $\sum_i c_i/t' = 2\epsilon/3$ and $c_i,t\geq 0$, we have that $c_i/t'\leq 2\epsilon/3$, meaning that $\frac1{t'}(S^i-T_i)\subset B^{\R^m}_{2\epsilon/3}(0)$. Hence the argument of $\varphi_P$ given above is in the domain of $\varphi_P$.

For the second inclusion, first fix $i$. For an element in the image of $\varphi_P$, note that $d(P_i,P_i+R_i)\leqslant \epsilon/3$ and $\mathbf d(\{R_i\},tQ^i)<\mathbf d(\{R_i\},Q^i)\leq 2\epsilon/3$. Since $d(P_i,R_i) = \mathbf d(\{P_i\},\{R_i\})$, we have that
\begin{align*}\mathbf d(\{P_i\},P_i+R_i+tQ^i) & \leqslant \mathbf d(\{P_i\},\{P_i+R_i\})+\mathbf d(\{P_i+R_i\},P_i+R_i+tQ^i) & (\text{triangle inequality}) \\
& = \mathbf d(\{0\},\{R_i\}) + \mathbf d(\{0\},tQ^i) & (\text{linearity of $\mathbf d$}) \\
& = \mathbf d(\{0\},\{R_i\}) + t\mathbf d(\{0\},Q^i) & (\text{linearity of $\mathbf d$}) \\
& < \frac{\epsilon}3 +\frac{2\epsilon}3 & (\text{assumption}) \\
& = \epsilon.\end{align*} $\square$

The following diagram describes the last calculation in the proof.