Visualizing paths in configuration space

The goal of this post is to visualize how point configurations induce persistent homology, and how paths between point samples induce changes in the simplicial complexes producing the homology. We use the Čech simplicial complex construction of a finite subset of $\R^N$.

Definition: For $M$ a Riemannian mandifold, $\Conf_n(M):= \{P\subseteq M : |P|=n\}$ is the configuration space of $n$ points on $M$.

The space $\Conf_n(M)$ is itself a topological space, with topology induced by the Hausdorff distance of subsets. Let $\SC$ be the set of abstract simplicial complexes $(V,S)$, where $V$ is a set and $S\subseteq P(V)$ closed under subsets. Let $\uSC$ be the set of unlabeled abstract simplicial complexes, with the natural projection map $\SC\to \uSC$.

Definition: The Čech map is the function $\check C\colon \Conf_n(M)\times \R_{\geqslant 0}\to \SC$ given by $V(\check C(P,r))=P$ and $P'\in S(\check C(P,r))$ whenever $\bigcap_{p\in P'} B(p,r) \neq \emptyset$, for every $P'\subseteq P$. The unlabeled Čech map is the composition of $\check C$ with the projection to $\uSC$.

We will consider the case $M=\R^2$ and $n=4$. To describe an implementation of the Čech map, we only need to consider double and triple intersections. Finding if $B(P_1,r)\cap B(P_2,r)$ is empty or not is easy, but to determine if $B(P_1,r)\cap B(P_2,r)\cap B(P_3,r)$ is empty or not requires more care. Below is an implementation in Mathematica.

(* CechPt : Finds the coordinate where balls of the same radii around a,b,c will first intersect *)
(* Input : 3 coordinates {x, y}. Output : 1 coordinate {x, y} *)
CechPt[a_,b_,c_] := Module[{
    cenx = Det[{{Norm[a]^2, a[[2]], 1}, {Norm[b]^2, b[[2]], 1}, {Norm[c]^2, c[[2]], 1}}],
    ceny = Det[{{a[[1]], Norm[a]^2, 1}, {b[[1]], Norm[b]^2, 1}, {c[[1]], Norm[c]^2, 1}}],
    scal = 2*Det[{{a[[1]], a[[2]], 1}, {b[[1]], b[[2]], 1}, {c[[1]], c[[2]], 1}}]},
   cen = {cenx/scal, ceny/scal};
   If[Max[ArcCos[(b-a).(c-a)/(Norm[b-a]*Norm[c-a])],
           ArcCos[(a-b).(c-b)/(Norm[a-b]*Norm[c-b])],
           ArcCos[(a-c).(b-c)/(Norm[a-c]*Norm[b-c])]] < Pi/2, cen,
     If[Norm[cen-(a+b)/2] < Norm[cen-(a+c)/2],
       If[Norm[cen-(a+b)/2] < Norm[cen-(b+c)/2], (a+b)/2, (b+c)/2],
       If[Norm[cen-(a+c)/2] < Norm[cen-(b+c)/2], (a+c)/2, (b+c)/2]]]];

Here cen is the circumcenter of the input points, which corresponds to our desired point only if it lies within the convex hull of the points. Now $B(P_1,r)\cap B(P_2,r)\cap B(P_3,r)$ is non-empty if and only if the distance from each of $P_1$, $P_2$, $P_3$ to CechPt[$P_1$, $P_2$, $P_3$] is less than or equal to $r$.

Let $\gamma\colon I\to \Conf_4(\R^2)$ be a path, and $\gamma(0)= \{P_1,P_2,P_3,P_4\}$. At each $t\in I$ and for every pair and triple $P'\subseteq\gamma(t)$, we can find the smallest $r$ such that $\bigcap_{p\in P'}B(P,r)\neq \emptyset$. This gives 6 curves for the pairs $P'$, and 4 curves for the triples $P'$, which we can plot all together in Mathematica.

PList[t_] := {P1[t],P2[t],P3[t],P4[t]};
(* Graphs of pairwise distances *)
DistGraph1 = Plot[Table[Norm[pair[[1]]-pair[[2]]]/2, {pair,Subsets[PList[t],{2}]}], 
               {t, 0, 1}, PlotRange -> {{0,1},{0,1.5}}, PlotStyle -> {Gray}, AspectRatio -> 1];
(* Graphs of minimum distance from every triple to its CechPt*)
DistGraph2 = Plot[Table[Max[
                  Table[Norm[triple[[k]]-CechPt@@triple],{k,1,3}]], {triple,Subsets[PList[t],{3}]}],
               {t, 0, 1}, PlotRange -> {{0,1},{0,1.5}}, PlotStyle -> {Orange}, AspectRatio -> 1];

The code is given so that it may be easily generalized to more than 4 points. Next, use the Manipulate command to add interactivity to the graphs.

Manipulate[{
  Show[DistGraph2, DistGraph1],
  Show[
    ParametricPlot[PList[t],{t,0,X[[1]]},PlotRange -> {{-2,2},{-2,2}},PlotStyle -> {Black}],
    Graphics[Join[
      {Opacity[.2],Red}, Table[Disk[point,X[[2]]],{point,PList[X[[1]]]}],
      {Opacity[1],Red}, Table[Circle[point,X[[2]]],{point,PList[X[[1]]]}],
      {Red,Disk[P1[X[[1]]],.05]},
      {Blue,Disk[P2[X[[1]]],.05]},
      {Darker[Green],Disk[P3[X[[1]]],.05]},
      {Yellow,Disk[P4[X[[1]]],.05]}]]],
  Graphics[Join[
    {Black, Thick},
    Flatten[Table[{{Opacity[0], Opacity[.3]}[[Boole[X[[2]] >= Norm[pair[[1]][[1]][X[[1]]] 
               - pair[[2]][[1]][X[[1]]]]/2] + 1]], Line[#[[2]]&/@pair]},
               {pair,Subsets[{{P1,{0,0}},{P2,{2,0}},{P3,{0,2}},{P4,{2,2}}},{2}]}]],
    Flatten[Table[{{Opacity[0], Opacity[.3]}[[Boole[X[[2]] >= Max[Table[Norm[triple[[k]][[1]][X[[1]]]
               - CechPt@@(#[[1]][X[[1]]]&/@triple)], {k,1,3}]]] + 1]], Polygon[#[[2]]&/@triple]},
               {triple,Subsets[{{P1,{0,0}},{P2,{2,0}},{P3,{0,2}},{P4,{2,2}}},{3}]}]],
    {Opacity[1], Red, Disk[{0,0},.07], Blue, Disk[{2,0},.07], Darker[Green], Disk[{0,2},.07],
               Yellow, Disk[{2,2},.07]}]]
  }, {{X, {.1, .1}}, Locator}]

This produces the interactive visualization below, allowing the user to drag the crosshairs on the graph on the left (graphs of when double and triple intersections are reached). The paths of the individual points $P_1,P_2,P_3,P_4$ are in the middle and the image of the unlabeled Čech map is on the right.

The graphs on the left stratify the strip $I\times \R_{\geqslant 0}$, so that the unlabeled Čech map is constant on each stratum. Computing the Betti numbers of each simplicial complex gives the CROCKER plot (see TZH) of the stratified space. We use the Čech instead of the Rips complex, so perhaps this should be called the CROCKEČ plot. The stratified space, 0-dimensional, and 1-dimensional plots are given below.

Here the Betti numbers were computed by inspection, since the complexes are so small. An extension would be to make this computation automatic once the input path $\gamma$ is given.

The Mathematica code for this post is available online.

References: Topaz, Ziegelmeier, Halverson (Topological Data Analysis of Biological Aggregation Models)

Enriched and straightened categories

Definition: A category $\mathcal C$ is monoidal if it is accompanied by

• a functor $\otimes \colon \mathcal C\times \mathcal C\to \mathcal C$,
• an object $\mathbf{1}\in \Obj(\mathcal C)$, and
• isomorphisms
• $\alpha_{X,Y,Z}\in \Hom_{\mathcal C}((X\otimes Y)\otimes Z,X\otimes(Y\otimes Z))$,
• $\lambda_X\in \Hom_{\mathcal C}(\mathbf{1}\otimes X,X)$, and
• $\rho_X\in \Hom_{\mathcal C}(X\otimes \mathbf{1},X)$,

for all $X,Y,Z,W\in \Obj(\mathcal C)$, such that $\otimes$ is unital and $\alpha$ is associative over $\otimes$. That is, the diagrams below commute.

Definition: Let $\mathcal C$ be monoidal as above. A category $\mathcal D$ is enriched over $\mathcal C$ if it is accompanied by

• an object $\mathcal D(P,Q)\in \Obj(\mathcal C)$ for every $P,Q\in \Obj(\mathcal D)$, and
• morphisms
• $\gamma_{P,Q,R}\in \Hom_{\mathcal C}(\mathcal D(Q,R)\otimes \mathcal D(P,Q),\mathcal D(P,R))$, and
• $i_P\in \Hom_{\mathcal C}(\mathbf{1},\mathcal D(P,P))$,

for all $P,Q,R,S\in \Obj(\mathcal D)$, such that $\gamma$ is unital and associative over $\otimes$. The category $\mathcal D$ is weakly enriched over $\mathcal C$ if $\gamma$ is unital and associative over $\otimes$ up to homotopy. That is, the diagrams below commute for $\mathcal D$ enriched, and commute up to homotopy for $\mathcal D$ weakly enriched.

Definition: A topological space $X$ is compactly generated if its basis of topology of closed sets is given by continuous images of compact Hausdorff spaces $K$ whose preimages are closed in $K$. A topological space is weakly Hausdorff if the continous image of every compact Hausdorff space is closed in $X$.

We write $\mathcal{CG}$ for the category of compactly generated and weakly Hausdorff spaces. This is a monoidal category with the usual product of topological spaces.

Example: Here are some examples of enriched categories.

• A topological category is a category enriched over $\mathcal {CG}$.
• A bicategory, or weak 2-category, is a category weakly enriched over $\mathcal Cat$, the category of small categories.

Definition: Let $\mathcal C,\mathcal D$ be bicategories. An assignment $F\colon \mathcal C\to \mathcal D$ is a pseudofunctor when it has

• an object $F(X)\in \Obj(\mathcal D)$,
• a functor $F(X,Y)\colon \mathcal C(X,Y)\to \mathcal D(F(X),F(Y))$, and
• invertible 2-morphisms
• $F(\id_X)\colon \id_X \Rightarrow F(X,X)(\id_X)$, and
• $F(X,Y,Z)(f,g) \colon F(Y,Z)(g)\circ F(X,Y)(f)\Rightarrow F(X,Z)(g\circ f)$,

for all $X,Y,Z\in \Obj(C)$, such that $F(X,Y)$ is unital and associative over composition. The assignment $F$ is a lax functor when the last two morphisms are not necessarily invertible.

Definition: Let $\mathcal C,\mathcal D$ be categories and $F\colon\mathcal C\to \mathcal D$ a functor. A morphism $f\in \Hom_{\mathcal C}(A,B)$ is $F$-cartesian if

commutes for some unique $g\in \Hom_{\mathcal C}(A,Y)$ (all the vertical arrows are $F$).

This definition can be rephrased in the language of simplicial sets: the morphism $f$ is $F$-cartesian if whenever $Ff=d_1\Delta^2$ for some $\Delta^2\in \mathcal D_2$, then every $\Lambda^2\in \mathcal C$ with $\Lambda^2_1 = f$ and $F\Lambda^2_0 = d_0\Delta^2$ can be filled in by $g$ with $Fg=d_2\Delta^2$.

Definition: Let $f\colon \mathcal C\to \mathcal D$ be a functor.

• The category $\mathcal C$ is $F$-fibered over $\mathcal D$ if for every morphism $h\in \Hom_{\mathcal D}(U,V)$ and every $B\in \Obj(\mathcal C)$ with $F(B)=V$, there is some $F$-cartesian $f\in \Hom_{\mathcal C}(-,B)$ with $Ff=h$.
• A cleavage of an $F$-fibered category $\mathcal C$ is a class of cartesian morphisms $K$ in $\mathcal C$ such that for every morphism $h\in \Hom_{\mathcal D}(U,V)$ and every $B\in \Obj(\mathcal C)$ with $F(B)=V$, there is a unique $F$-cartesian $f\in K$ with $Ff=h$.
• A cleavage of $\mathcal C$ is a splitting if it contains all the the identity morphisms and is closed under composition.

If $\mathcal C$ is $F$-fibered over $\mathcal D$ and $\mathcal C'$ is $F'$-fibered over $\mathcal D$, then a functor $\mathcal F\colon \mathcal C\to \mathcal C'$ is a \emph{morphism of fibered categories} if $F = F'\circ \mathcal F$ and $\mathcal Ff$ is $F'$-cartesian whenever $f$ is $F$-cartesian.

Theorem: Let $\mathcal C$ be $F$-fibered over $\mathcal D$.

• Every cleavage of $\mathcal C$ defines a pseudofunctor $\mathcal D\to \mathcal Cat$.
• Every pseudofunctor $\mathcal D\to \mathcal Cat$ defines an $F'$-fibered category $\mathcal C'$ with a cleavage over $\mathcal D$.

The above result follows from sections 3.1.2 and 3.1.3 of Vistoli. Theorem 2.2.1.2 of Lurie generalizes this and provides an equivalence between the category of fibered simplicial sets over $S\in \sSet$ and the category of functors $\text{sCat}\to \sSet$. The forward direction is called straightening and he backward direction is called unstraightening.

References: nLab (articles "Monoidal category," "enriched category," and "pseudofunctor."), Strickland (The category of CGWH spaces), Vistoli (Notes on Grothendieck topologies, Chapter 3), Noohi (A quick introduction), Lurie (Higher Topos Theory, Section 2.2)

Integral transforms

Let $X,Y$ be topological spaces.

Definition: A set $U\subseteq X$ is constructible if it is a finite union of locally closed sets. A function $f\colon X\to Y$ is constructible if $f^{-1}(y)\subseteq X$ is constructible for all $y\in Y$.

Write $CF(X)$ for the set of constructible functions $f\colon X\to \Z$. Recall if $U\subseteq X$ is constructible, it is triangulable.

Definition: Let $X\subseteq \R^N$ be constructible and $\{X_r\}_{r\in \R}$ a filtration of $X$ by constructible sets $X_r$. The $k$th persistence diagram of $X$ is the set $PD(X_r,k)= \{(a,b)\subseteq (\R\cup \{\pm\infty\})^2 : a<b \}$, where each element represents the longest sequence of identity morphisms in the decomposition of the image of the $k$th persistent homology functor $PH(X_r,k)\colon (\R,\leqslant )\to Vect$ to each component.

Write $D$ for the set of all persistence diagrams.

Definition: Let $X,Y\subseteq \R^N$ be constructible, $S\subseteq X\times Y$ also constructible with $\pi_1,\pi_2$ the natural projections, and $\sigma$ a simplex in a triangulation of $X$. The Euler integral of elements of $CF(X)$ is the assignment \[ \begin{array}{r c l}
\displaystyle \int_X\ \cdot\ d\chi\colon CF(X) & \to & \Z, \\
\mathbf{1}_\sigma & \mapsto & (-1)^{\dim(\sigma)}.
\end{array} \] The Radon transform of elements of $CF(X)$ is the assignment \[ \begin{array}{r c l}
\mathcal R_S \colon CF(X) & \to & CF(Y), \\
(x\mapsto h(x)) & \mapsto & \left(y\mapsto \displaystyle \int_{\pi_2^{-1}(y)} \pi_1^*h\ d\chi\right).
\end{array} \] The persistent homology transform of $X$ is the assignment \[ \begin{array}{r c l}
PHT_X \colon S^{N-1} & \to & D^N, \\
v & \mapsto & \left\{PD(\{x\in X:x\cdot v\leqslant r\},0),\dots,PD(\{x\in X : x\cdot v\leqslant r\},N-1)\right\}
\end{array} \]

The Euler integral is also called the Euler transform, or the Euler charateristic transform. The Radon transform has a weighted version, where every simplex in $S$ is assigned a weight.

References: Schapira (Tomography of constructible functions), Baryhsnikov, Ghrist, Lipsky (Inversion of Euler integral transforms), Turner, Mukherjee, Boyer (Persistent homology transform).

Induced orders on sets

The goal of this post is to understand when a map from a poset to an unordered set induces a partial order, and how that applies to the specific case of the set of simplicial complexes. Thanks to Yanlong Hao for spotting some mistakes in my seminar talk on the same topic yesterday.

Definition 1: Let $(A,\leqslant_A)$ be a poset and $f\colon A\to B$ a map of sets. The relation $\leqslant_B$ on $B$, with $a\leqslant_Aa'$ implying $f(a)\leqslant_Bf(a')$, is the relation induced by $f$ on $B$. The map $f$ is monotonic if whenever $b\leqslant_b b'$,

1. if $a\in f^{-1}(b)$, $a'\in f^{-1}(b)$ are comparable, then $a\leqslant_A a'$, and
2. if $a'\in f^{-1}(b')$, then there exists $a\in f^{-1}(b)$ such that $a\leqslant_A a'$.

Since $f$ may not be surjective, there may be $b\in B$ with $f^{-1}(b) = \emptyset$. For such $b$ we only have $b\leqslant_B b$ and $b$ is not comparable to any other element of $B$.

Lemma 2: If $f\colon A\to B$ is monotonic, then the induced relation $\leqslant_B$ is a partial order on $B$.

Proof: For reflexivity, take any $a\in A$, which has $a\leqslant_A a$ by reflexivity of $\leqslant_A$. Then $f(a)\leqslant_Bf(a)$, so every $b\in \im(f)$ satisfies reflexivity. Every $b\not\in\im(f)$ also satisfies reflexivity by the comment above.

For anti-symmetry, suppose that $b\leqslant_Bb'$ and $b'\leqslant_Bb$. Since $b\leqslant_B b'$, there is some $a\in f^{-1}(b)$ and $a'\in f^{-1}(b')$ such that $a\leqslant_A a'$. Similarly, there is some $c'\in f^{-1}(b')$ and $c\in f^{-1}(b)$ such that $c'\leqslant_A c$. Since $c\in f^{-1}(b)$ and $c'\in f^{-1}(b')$ are comparable, and the first assumed relation is $b\leqslant_Bb'$, by property 1 of Definition 1, we must have $c\leqslant_A c'$. By anti-symmetry of $A$, we now have that $c=c'$, so it follows that $b=f(c)=f(c')=b'$.

For transitivity, suppose that $b\leqslant_Bb'$ and $b'\leqslant_Bb''$. Take $a''\in f^{-1}(b'')$, for which property 2 of Definition 1 guarantees that there exists $a'\in f^{-1}(b')$ such that $a'\leqslant_Aa''$. Similarly, the first assumed relation and the same property guarantees there exists $a\in f^{-1}(b)$ such that $a\leqslant_Aa'$. By transitivity of $A$, we have $a\leqslant_A a''$. By the definition of $\leqslant_B$, we have $b=f(a) \leqslant_B f(a'') = b''$. $\square$

Let $M$ be a piecewise linear, compact, connected, embedded manifold in $\R^N$, and $SC$ the category of simplicial complexes. Let $A= \{1<2a>2b<3\}$. The product $A^N$ has the product order. Fix $n\in \Z_{>0}$ and let $T$ be the set of all distinct 2-,3-,...,$n$-tuples in $\{1,\dots,n\}$, or $T := \bigcup_{k=2}^n\left(\{1,\dots,n\}^k\setminus \Delta\right)/_{S_k}$. This set has size $\sum_{k=2}^n \binom nk = 2^n-n-1$. Assume every $v\in T$ is ordered in the canonical way. Then $v$ induces a natural projection $\pi_v\colon M^n \to M^{v}$, as well as another map \[ \begin{array}{r c l}
\pi_v'\colon M^n \times \R_{>0} & \to & A, \\
(P,t) & \mapsto & \begin{cases}
1 & \forall\ i,j, \pi_v(P)_i = \pi_v(P)_j, \\
2a & \exists\ i,j \text{ s.t. }\pi_v(P)_i\neq\pi_v(P)_j \text{ and } \textstyle\bigcap_{i=1}^{|v|} B(\pi_v(P)_i,t) \neq \emptyset, \\
2b & \exists\ i,j \text{ s.t. }\pi_v(P)_i\neq\pi_v(P)_j \text{ and } \textstyle\bigcap_{i=1}^{|v|} B(\pi_v(P)_i,t)=*, \\
3 & \exists\ i,j \text{ s.t. }\pi_v(P)_i\neq\pi_v(P)_j \text{ and } \textstyle\bigcap_{i=1}^{|v|} B(\pi_v(P)_i,t) = \emptyset.
\end{cases}
\end{array} \] Here all the balls $B$ are closed, and $M^n$ has the Hausdorff topology.

Lemma 3: The map $\pi_v$ is continuous on $M^v\times \R_{>0}$.

Proof: Every $(Q,s)\in (\pi_v')^{-1}(3)$ has an open ball of radius $\max_{i,j}\{d(\pi_v(Q)_i,\pi_v(Q)_j)\}/2-s$ around it that is still contained within $(\pi_v')^{-1}(3)$. Similarly, every $(Q,s)\in (\pi_v')^{-1}(2a)$ has an open ball of radius \[ \min\left\{\frac12\text{diam}\left(\bigcap_{i=1}^{|v|}B(\pi_v(Q)_i,s)\right),\max_{i,j}\{d(\pi_v(Q)_i,\pi_v(Q)_j)\}\right\} \hspace{2cm} (1) \] around it that is still contained within $(\pi_v')^{-1}(2a)$. The first expression in the $\min$ makes sure the intersection is non-empty, and the second expression makes sure all elements of $Q$ are not the same.

The set $(\pi_v')^{-1}(1<2a)$ is open by the same argument as for $2a\in A$, enlarging the open ball by removing the second expression in the $\min$ of expression (1). Finally, the set $(\pi_v')^{-1}(2a>2b<3)$ is open by the same argument, now enlarging the ball used for $2a\in A$ by removing the first expression in the $\min$ of expression (1). $\square$

Let $q\colon M^n\to \Ran^{\leqslant n}(M)$ be the natural quotient map, and $\check C\colon \Ran^{\leqslant n}(M)\times \R_{>0}\to SC$ be the Čech simplical complex map. For the next propositions, we will use two maps $f$ and $g$ defined as \[ \begin{array}{r c l}
f\colon M^n\times \R_{>0} & \to & A^{2^n-n-1}, \\
(P,t) & \mapsto & \prod_{v\in T} \pi_v'(P,t),
\end{array}
\hspace{2cm}
\begin{array}{r c l}
g\colon \im(f) & \to & SC, \\
f(P,t) & \mapsto & \check C(q(P),t).\end{array} \] The map $g$ is well-defined because $a\in A^{2^n-n-1}$ with non-empty preimage in $M^n\times \R_{>0}$ specifies whether or not every $k$-tuple of points has a simplex spanning it, for all $k=2,\dots,n$. This defines a unique simplicial complex, so choosing any $(P,t)\in f^{-1}(a)$ will give the same Čech complex, up to renaming of vertices.

Proposition: The map $f\colon M^n\times \R_{>0} \to A^{2^n-n-1}$ is continuous.

Proof: Let $a\in A^{2^n-n-1}$ and suppose that $f^{-1}(a)\neq \emptyset$. Let $a_i\in A$ be in the $i$th factor of $a$, and $r_i$ the radius of the open ball decreed by Lemma 3 to still be within $(\pi_v')^{-1}(a_i)$, where $v$ is the $i$th tuple in the chosen order on $T$. Then every $(P,t)\in f^{-1}(a)$ has an open ball of radius $\min_i\{r_i\}$ around it that is still contained within $f^{-1}(a)$, so $f$ is continuous. $\square$

Proposition: The map $g$ is monotonic.

 Note that any relation $S\leqslant_{SC}S'$ may be split up as a chain of relations $S=T_1\leqslant_{SC} \cdots \leqslant_{SC} T_\ell=S'$, where the only differences between $T_i$ and $T_{i+1}$ are either (i) $T_i$ has a $k$-simplex $\sigma$ that $T_{i+1}$ does not have, or (ii) where $T_i$ has a single 0-simplex where a $k$-simplex $\sigma$ and all its faces used to be in $T_{i+1}$. Hence it suffices to show that properties 1 and 2 of Definition 1 are satisfied in cases (i) and (ii).

Proof: Case (i): Suppose that $S\leqslant_{SC}S'$, and take $a\in g^{-1}(S)$, $a'\in g^{-1}(S')$ with $a\leqslant_A a'$. If there is $b\in g^{-1}(S)$ and $b'\in g^{-1}(S')$ such that $b'\leqslant_A b$, then $g(b)$ has the $k$-simplex $\sigma$ that $g(b')$ does not have, but since $b'$ is ordered lower than $b$, it must be that this $k$-simplex has collapsed to a point. Then we would be in case (ii), a contradiction, so property 1 holds in this case.

Now let $i_1,\dots,i_{\sigma}$ be the indices of $a'$ and $a$ representing the $(k+1)$-fold intersection that describes $\sigma$, so $a'_j = 3$ and $a_j = 2b$ for all $j=i_1,\dots,i_\sigma$. Take any $b'\in g^{-1}(S')$, which also has some indices $\ell_1,\dots,\ell_\sigma$ representing this same $(k+1)$-fold intersection, so $b'_j=3$ at all $j=\ell_1,\dots,\ell_\sigma$. Let $b\in A^{2^n-n-1}$ be the element with all the same factors as $b'$, except at indices $\ell_1,\dots,\ell_\sigma$, which have been changed to $2b$. This element $b$ is still in $\im(f)$ as removing only this $k$-simplex still leaves the well-defined simplex $S'$ we assumed at the beginning. Hence $g(b)=S'$ and property 2 holds. \\

Case (ii): Suppose that $S\leqslant_{SC}S'$, and take $a\in g^{-1}(S)$, $a'\in g^{-1}(S')$ with $a\leqslant_A a'$. If there is $b\in g^{-1}(S)$ and $b'\in g^{-1}(S')$ such that $b'\leqslant_A b$, then $g(b')$ has the $k$-simplex $\sigma$ and all its faces that $g(b)$ does not have, but since $b'$ is ordered lower than $b$, it must be that we have introduced $\sigma$ and all its faces. Then we would be in case (i), or a chain of case (i) situations, a contradiction, so property \ref{1mon} holds in this case.

Now let $i_1,\dots,i_{\sigma}$ be the indices of $a'$ and $a$ representing the $(k+1)$-fold intersection that describes $\sigma$, and all the implied $(f+1)$-fold intersections that describe the $f$-faces of $\sigma$, $f>0$. That is, $a'_j = 2a$ and $a_j = 1$ for all $j=i_1,\dots,i_\sigma$. Take any $b'\in g^{-1}(S')$, which also has some indices $\ell_1,\dots,\ell_\sigma$ representing this same $(k+1)$-fold (and lower) intersection, so $b'_j=3$ at all $j=\ell_1,\dots,\ell_\sigma$. Let $b\in A^{2^n-n-1}$ be the element with all the same factors as $b'$, except at indices $\ell_1,\dots,\ell_\sigma$, which have been changed to $1$. This element $b$ is still in $\im(f)$ as collapsing this $k$-simplex and all its faces to a single 0-simplex still leaves the well-defined simplex $S'$ we assumed at the beginning. Hence $g(b)=S'$ and property 2 holds. $\square$

Since $g$ is monotonic, by Lemma 2 the relation $\leqslant_{SC}$ is a partial order on $SC$.

A functor from entry paths to the nerve of simplicial complexes

Fix $n\in \Z_{>0}$ and let $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ for $M$ a compact, connected PL manifold embedded in $\R^N$. Take $\widetilde h\colon X\to (B,\leqslant)$ the conical stratifying map from a previous post (``Conical stratifications via semialgebraic sets," 2018-04-16) compatible with the natural stratification $h\colon X\to SC$. The goal of this post is to construct a functor $F\colon \Sing_B(X) \to N(SC)$ from the $\infty$-category of entry paths that encodes the structure of $X$.

Recall that a simplicial set is a functor, an element of $\text{Fun}(\Delta^{op},\Set)$. A simplicial set $S$ is defined by its collection of $n$-simplices $S_n$, its face maps $s_i:S_{n-1}\to S_n$, and degeneracy maps $d_i:S_{n+1}\to S_k$, for all $i=0,\dots,n$. For the first simplicial set of interest in this post, we have
\begin{align*}
\Sing_B(X)_n & = \Hom_{\Top}^B(|\Delta^n|,X), \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \begin{array}{c}
\left(|\Delta^{n-1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{collapses $i$th with $(i+1)$th vertex, then maps as source}
\end{array}\right)\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto  \left(\begin{array}{c}
\left(|\Delta^{n+1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{maps as $i$th face of source map}
\end{array}\right)
\end{align*}
We write $\Hom^B_{\Top}$ for the subset of $\Hom_{\Top}$ that respects the stratification $B$ in the context of entry paths. For the second simplicial set, the nerve, we have
\begin{align*}
N(SC)_n & = \{(S_0\tov{f_1} \cdots \tov{f_n} S_n)\ :\ S_i\in SC,\ f_i\ \text{are simplicial maps}\}, \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \left(S_0\tov{f_1}\cdots \tov{f_{n-1}} S_{n-1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_i} S_i \tov{\id} S_i\tov{f_{i+1}} \cdots \tov{f_{n-1}} S_{n-1}\right)\right),\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{r l}
i=0: & \left(S_0\cdots S_{n+1} \right) \mapsto \left(S_1\tov{f_2}\cdots \tov{f_{n+1}} S_{n+1} \right) \\
0<i<n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_{i-1}} S_{i-1} \tov{f_{i+1}\circ f_i} S_{i+1} \tov{f_{i+2}} \cdots \tov{f_{n+1}} S_{n+1}\right) \\
i=n: & \left(S_0 \cdots  S_{n+1} \right) \mapsto \left(S_0\tov{f_1}\cdots \tov{f_n} S_n \right)
\end{array} \right). 
\end{align*}
 Define $F$ on $k$-simplices as \[ F\left(\gamma\colon |\Delta^k|\to \Ran^{\leqslant n}(M)\times \R_{>0}\right) = \left(\widetilde h(\gamma(1,0,\dots,0)) \tov{\left(\widetilde h\circ \gamma \circ s_k\circ \cdots \circ s_2\right)\left( |\Delta^1|\right)} \cdots \tov{\left(\widetilde h\circ \gamma \circ s_{k-2}\circ \cdots \circ s_0\right)\left(|\Delta^1|\right)} \widetilde h(\gamma(0,\dots,0,1))\right). \] A morphism in $\Sing_B(X)$ is a composition of face maps $s_i$ and degeneracy maps $d_i$, so $F$ must satisfy the commutative diagrams

for all $s_i$, $d_i$. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.

Commutativity of the diagram on the left is immediate, as considering a simplex $|\Delta^{n-1}|$ as the $i$th face of a larger simplex $|\Delta^n|$ is the same as adding a step that is the identity map in the Hamiltonian path of vertices of $|\Delta^{n-1}|$. Similarly, observing that the image of the shortest path $v_{i-1}\to v_i\to v_{i+1}$ in $|\Delta^{n+1}|$, for $v_i = (0,\dots,0,1,0,\dots,0)$ the $i$th standard basis vector, induced by an element $\gamma\colon |\Delta^{n+1}|\to X$ in $\Sing_B(X)_{n+1}$, is homotopic to the image of the shortest path $v_{i-1} \to v_{i+1}$ shows that the diagram on the right commutes. Since $F$ is a natural transformation between the two functors $\Sing_B(X)$ and $N(SC)$, it is a functor on the functors as simplicial sets.

Remark: The particular choice of $X$ did not seem to play a large role in the arguments above. However, the stratifying map $\widetilde h\colon X\to B$ has image sitting inside $SC$, the nerve of which is the target of $F$, and every morphism in $\Sing_B(X)$ can be interpreted as a relation in $B\subseteq SC$ (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor $\Sing_A(X)\to N(A')$ may exist for a stratified space $X\to A\subseteq A'$.

Exit paths and entry paths through $\infty$-categories

Let $X$ be a topological space, $(A,\leqslant)$ a poset, and $f: X\to (A,\leqslant)$ a continuous map.

Definition: An exit path in an $A$-stratified space $X$ is a continuous map $\sigma: |\Delta^n|\to X$ for which there exists a chain $a_0\leqslant \cdots \leqslant a_n$ in $A$ such that $f(\sigma(t_0,\dots,t_i,0,\dots,0))=a_i$ for $t_i\neq 0$. An entry path is a continuous map $\tau: |\Delta^n|\to X$ for which there exists a chain $b_0\leqslant \cdots \leqslant b_n$ in $A$ such that $f(\tau(0,\dots,0,t_i,\dots,t_n))=b_i$ for $t_i\neq 0$.

Up to reordering of vertices of $\Delta^n$ and induced reordering of the realization $|\Delta^n|$, an exit path is the same as an entry path. The next example describes this equivalence.

Example: The standard 2-simplex $|\Delta^2|$ is uniquely an exit path and an entry path with a chain of 3 distinct elements, stratfied in the ways described below.

Recall the following algebraic constructions, through Joyal's quasi-category model:

• A simplicial set is a functor $\Delta^{op}\to \Set$.
• A Kan complex is a simplicial set satisfying the inner horn condition for all $0\leqslant k\leqslant n$. That is, the $k$th $n$-horn lifts (can be filled in) to a map on $\Delta^n$.
• An $\infty$-category is a simplicial set satisfying the inner horn condition for all $0<k<n$.

Moreover, if the lift is unique, then the Kan complex is the nerve of some category. Recall also the category $\Sing(X) = \{$continuous $\sigma: |\Delta^n|\to X\}$, which can be combined with the stratification $f: X\to A$ of $X$

Remark: The subcategory $\Sing^A(X)$ of exit paths and the subcategory $\Sing_A(X)$ of entry paths are full subcategories of $\Sing(X)$, with $(\Sing^A(X))^{op} = \Sing_A(X)$. If the stratification is conical, then these two categories are $\infty$-categories.

Recall the nerve construction of a category. Here we are interested in the nerve of the category $SC$ of simplicial complexes, so $N(SC)_n = \{$sequences of $n$ composable simplicial maps$\}$. Recall the $k$th $n$-horns, which are compatible diagrams of elements of $N(SC)_n$. In general, they are colimits of a diagram in the category $\Delta$. That is, \[ \Lambda^n_k := \colim \left(\bigsqcup_{0\leqslant i<j\leqslant n} \Delta^{n-2} \rightrightarrows \bigsqcup_{0\leqslant i\leqslant n \atop i\neq k} \Delta^{n-1}\right). \] Example: The images of the 3 different types of 2-horns and 4 different types of 3-horns in $SC$ are given below. Note that they are not unique, and depend on the choice of simplices $S_i$ (equivalently, on the choice of functor $\Delta^{op}\to SC$).

For example, the 0th 2-horn $\Lambda^2_0$ can be filled in if there exists a simplicial map $h: S_1\to S_2$ in $SC$ (that is, an element of $N(SC)_1$) such that $h\circ f = g$. Similarly, the 1st 3-horn $\Lambda^3_1$ can be filled in if there exists a functor $F: [0<1<2]\to SC$ for which $F(0<1)=f_{02}$, $F(0<2)=f_{03}$, and $F(1<2)=f_{23}$ (equivalently, a compatible collection of elements of $N(SC)_2$).

Definition: Let $A,B$ be $\infty$-categories. A functor $F: A\to B$ is a morphism of the simplicial sets $A,B$. That is, $F:A\to B$ is a natural transformation for $A,B\in \text{Fun}(\Delta^{op},\Set)$.

A functor of simplicial sets of a particular type can be identified with a functor of 1-categories. Recall the nerve of a 1-category, which turns it into an $\infty$-category. This construction has a left adjoint.

Definition: Let $\mathcal C$ be an $\infty$-category. The homotopy category $h\mathcal C$ of $\mathcal C$ has objects $\mathcal C_0$ and morphisms $\Hom_{h\mathcal C}(X,Y) = \pi_0(\text{Map}_{\mathcal C}(X,Y))$.

By Lurie, $h$ is left-adjoint to $N$. That is, $h : \sSet \rightleftarrows \text{Cat} : N$, or $\text{Map}_{\sSet}(\mathcal C,N(\mathcal D)) \cong \text{Map}_{\text{Cat}}(h\mathcal C, \mathcal D)$, for any $\infty$-category $\mathcal C$ and any 1-category $\mathcal D$. Our next goal is to describe a functor $\Sing_A(X)\to N(SC)$, maybe through this adjunction, where $SC$ is the 1-category of simplicial complexes and simplicial maps.

References: Lurie (Higher topos theory, Sections 1.1.3 and 1.2.3), Lurie (Higher algebra, Appendix A.6), Goerss and Jardine (Simplicial homotopy theory, Section I.3), Joyal (Quasi-categories and Kan complexes)

Conical stratifications via semialgebraic sets

The goal of this post is to describe a conical stratification of $\Ran_{\leqslant n}(M)\times \R_{\geqslant 0}$ that refines the stratification previously seen (in "Exit paths, part 2," 2017-09-28, and "Refining stratifiations," 2018-03-11). Thanks to Shmuel Weinberger for the key observation that the strata under consideration are nothing more than semialgebraic sets, which are triangulable, and so admit a conical stratification via this triangulation.

Remark: Fix $n\in \Z_{>0}$, let $M$ be a smooth, compact, connected, embedded submanifold in $\R^N$, and let $M^n$ have the Hausdorff topology. We will be interested in $M^n\times \R_{>0}$, though this will be viewed as the compact set $M^n\times [0,K]\subseteq \R^{nN+1}$ for some $K$ large enough (for instance, larger than the diameter of $M$) when necessary. The point 0 is added for compactness.

Stratification of the Ran space by semialgebraic sets

We begin by stratifying $M^n\times \R_{>0}$ by a poset $A$, creating strata based on the pairwise distance between points in each $M$ component. Then we take that to a stratification of the quotient $\Ran^{\leqslant n}(M)\times \R_{>0}$ via the action of the symmetric group $S_n$ and overcounting of points.

Definition: Define a partial order $\leqslant$ on the set $A = \big\{$partitions of ($\{1,\dots,n\}^2\setminus \Delta)/S_2$ into 4 parts$\big\}$ of ordered 4-tuples of sets by \[ (Q,R,S,T) \leqslant \left(Q\setminus Q',\ R\cup Q' \cup S',\ S\setminus (S'\cup S''),\ T\cup S''\right), \] for all $Q'\subseteq Q$ and $S',S''\subseteq S$, with $S'\cap S'' = \emptyset$.

The diagram to keep in mind is the one below, with arrows pointing from lower-ordered elements to higher-ordered elements. Once we pass to valuing the 4-tuple in simplicial complexes, moving between $Q$ and $R$ will not change the simplicial complex type (this comes from the definition of the Vietoris--Rips complex).

Lemma 1: The map $f: M^n\times \R_{>0}\to (A,\leqslant)$ defined by \begin{align*} (\{P_1,\dots,P_n\},t)\mapsto \bigg( \{(i,j>i)\ & :\ P_i=P_j\},\ \{(i,j>i)\ :\ d_M(P_i,P_j)<t\},\\
& \{(i,j>i)\ :\ d_M(P_i,P_j)=t\},\ \{(i,j>i)\ :\ d_M(P_i,P_j)>t\}\bigg) \end{align*} is continuous in the upset topology on $(A,\leqslant)$.

Proof: Choose $(Q,R,S,T)\in A$ and consider the open set $U = U_{(Q,R,S,T)}$ based at $(Q,R,S,T)$. Take $(P,t)\in f^{-1}(U)$, which we claim has a small neighborhood still contained within $f^{-1}(U)$. If we move a point $P_i$ slightly that was exactly distance $t$ away from $P_j$, then the pair $(i,j)$ was in $S$, but is now in either $R$ or $T$, and both $(Q,R\cup\{(i,j)\},S\setminus \{(i,j)\},T)$ and $(Q,R,S\setminus \{(i,j)\},T\cup \{(i,j)\})$ are ordered higher than $(Q,R,S,T)$, so the perturbed point is still in $f^{-1}(U)$. If $P_i=P_j$ in $P$ and we move them apart slightly, since $t\in \R_{>0}$, the pair $(i,j)$ will move from $Q$ to $R$, and $(Q,R,S,T) \leqslant (Q\setminus \{(i,j)\},R\cup \{(i,j)\},S,T)$, so the perturbed point is still in $f^{-1}(U)$. For all pairs $(i,j)$ in $R$ or $T$, the distances can be changed slightly so that the pair still stays in $R$ or $T$, respectively. Hence $f$ is continuous. $\square$

This shows that $M^n\times \R_{>0}$ is stratified by $(A,\leqslant)$, using Lurie's definition of a (poset) stratification, which just needs a continuous map to a poset. Our goal is to work with the Ran space of $M$, instead of the $n$-fold product of $M$, which are related by the natural projection map $\pi : M^n\to \Ran^{\leqslant n}(M)$, taking $P=\{P_1,\dots,P_n\}$ to the unordered set of distinct elements in $P$. We also would like to stratify $\Ran^{\leqslant n}(M)\times \R_{>0}$ by simplicial complex type, so we need the following map.

Definition: Let $g: (A,\leqslant)\to SC$ be the map into simplicial complexes that takes $(Q,R,S,T)$ to the clique complex of the simple graph $C$ on $n-k$ vertices, for $|Q|=k(k+1)/2$, defined as follows: 

• $V(C) = \{[i]\ :\ i=1,\dots,n,\ [j]= [i] \text{\ iff\ } (i,j)\in Q\}$,
• $E(C) = \{([i],[j])\ :\ (i,j)\in R\cup S\}$.

We require $C$ to be simple, so if $(i,j)\in Q$ and $(i,\ell),(j,\ell)\in R\cup S$, we only add one edge $([i],[\ell])=([j],[\ell])$ to $C$.

The map $g$ induces a partial order $\leqslant$ on $SC$ from the partial order on $A$, with $C\leqslant C'$ in $SC$ whenever there is $(Q,R,S,T)\in g^{-1}(C)$ and $(Q',R',S',T')\in g^{-1}(C')$ such that $(Q,R,S,T) \leqslant (Q',R',S',T')$ in $A$. Note that if $C\in SC$ is not in the image of $g$, then it is not related to any other element of $SC$. By the universal property of the quotient and continuity of $f$ and $g$ (as $A$ and $SC$ are discrete), there is a continuous map $h:\Ran^{\leqslant n}(M)\times \R_{>0}\to (SC,\leqslant)$ such that the diagram

commutes. Hence $\Ran^{\leqslant n}(M)\times \R_{>0}$ is stratified by $(SC,\leqslant)$.

Remark: The map $\pi$ can be thought of as a quotient by the action of the symmetric group $S_n$, followed by the quotient of the equivalence relation \[ \{P^1_1,\dots,P^{\ell_1}_1,P^1_2,\dots,P^{\ell_2}_2,P^1_3,\dots,P^{\ell_k}_k\} \ \ \sim\ \
\{P^1_1,\dots,P^{\ell_1-1}_1,P^1_2,\dots,P^{\ell_2+1}_2,P^1_3,\dots,P^{\ell_k}_k\} \] on $M^n$, for all possible combinations $\ell_1+\cdots + \ell_k =n$ and $1\leqslant k\leqslant n-1$, where $P_m^i=P_m^j$ for all $1\leqslant i<j\leqslant \ell_m$.

Semialgebraic geometry

Next we move into the world of semialgebraic sets and triangulations, following Shiota. Here we come across a more restrictive notion of stratification of a manifold $X$, which requires a partition of $X$ into submanifolds $\{X_i\}$. If Lurie's stratification $f:X\to A$ gives back submanifolds $\{f^{-1}(a)\}_{a\in A}$, then we have Shiota's stratification. Conversely, the poset $(\{X_i\},\leqslant)$, for $X_i \leqslant X_j$ iff $X_i \subseteq \closure(X_j)$ is always a stratification in the sense of Lurie.

Definition 2: A semialgebraic set in $\R^N$ is a set of the form \[ \bigcup_{\text{finite}} \{x\in \R^N\ :\ f_1(x)=0,f_2(x)>0,\dots,f_m(x)>0\},\] for polynomial functions $f_1,\dots,f_m$ on $\R^N$. A semialgebraic stratification of a space $X\subseteq \R^N$ is a partition $\{X_i\}$ of $X$ into submanifolds that are semialgebraic sets.

Next we observe that the strata of $M^n\times \R_{>0}$ are semialgebraic sets, with the preimage theorem and I.2.9.1 of Shiota, which says that the intersection of semialgebraic sets is semialgebraic. Take $(Q,R,S,T)\in A$  and note that \[ f^{-1}(Q,R,S,T) = \left\{(\{P_1,\dots,P_n\},t)\in M^n\times \R_{>0}\ :\ \begin{array}{r l}
d(P_i,P_j) = 0 & \forall (i,j)\in Q,\\
t-d(P_i,P_j) = 0 & \forall (i,j)\in S, \\
t-d(P_i,P_j) > 0 & \forall (i,j)\in R, \\
d(P_i,P_j) - t > 0 & \forall (i,j)\in T.
\end{array}\right\} \] Here $d$ means distance on the manifold, and we assume the metric to be analytic. Alternatively, $d$ could be Euclidean distance between points on the embedding of $M^n\times \R_{>0}$, induced by the assumed embedding of $M$.

For his main Theorem II.4.2, Shiota uses cells, but we opt for simplices instead, and for cell complexes we use simplicial complexes. Every cell and cell complex admits a decomposition into simplicial complexes, even without introducing new 0-cells (by Lemma I.3.12), so we do not lose any generality.

Definition 3: Let $X,Y$ be semialgebraic sets.

• A map $f: X\to Y$ is semialgebraic if the graph of $f$ is semialgebraic.
• A semialgebraic cell triangulation of a semialgebraic set $X$ is a pair $(C,\pi)$, where $C$ is a simplicial complex and $\pi: |C|\to X$ is a semialgebraic homeomorphism for which $\pi|_{\interior(\sigma)}$ is a diffeomorphism onto its image.
• A semialgebraic cell triangulation $(C,\pi)$ is compatible with a family $\{X_i\}$ of semialgebraic sets if $\pi(\interior(\sigma))\subseteq X_i$ or $\pi(\interior(\sigma))\cap X_i = \emptyset$ for all $\sigma\in C$ and all $X_i$.

A semialgebraic cell triangulation $(C,\pi)$ of $X$ induces a stratification $X\to (C_0 \cup \{\pi(\interior(\sigma))\},\leqslant)$, where the order is the one mentioned just before Definition 2. We use the induced stratification and the cell triangulation interchangeably, specifically in Proposition 4.

A compatible conical stratification

Finally we put everything together to get a conical stratification of $\Ran^{\leqslant n}(M)\times \R_{>0}$. Unfortunately we have to restrict ourselves to piecewise linear manifolds, or PL manifolds, which are homeomorphic images of geometric realizations of simplicial complexes, as otherwise we cannot claim $M$ is a semialgebraic set. We can also just let $M=\R^k$, as the point samples we are given could be coming from an unknown space.

Proposition 4: Let $M$ be a PL manifold embedded in $\R^N$. There is a conical stratification $\widetilde h:\Ran^{\leqslant n}(M)\times \R_{>0}\to (B,\leqslant)$ compatible with the stratification $h: \Ran^{\leqslant n}(M)\times \R_{>0}\to (SC,\leqslant)$.

Proof: (Sketch) The main lifting is done by Theorem II.4.2 of Shiota. Since $M$ is PL, it is semialgebraic, and so $M^n\times \R_{>0}\subseteq \R^{nN+1}$ is semialgebraic, by I.2.9.1 of Shiota. Since the quotient $\pi$ of diagram (1) is semialgebraic, the space $\Ran^{\leqslant n}(M)\times \R_{>0}$ is semialgebraic, by Scheiderer. Similarly, $\{f^{-1}(a)\}_{a\in A}$ is a family of semialgebraic sets, where $f$ is the map from Lemma 1.  Theorem II.4.2 gives that $\Ran^{\leqslant n}(M)\times \R_{>0}$ admits a cell triangulation $(K,\tau)$ compatible with $\{h^{-1}(S)\}_{S\in SC}$. By the comment after Definition \ref{semialgdef}, this means we have a stratification $\Ran^{\leqslant n}(M)\times \R_{>0}\to (K_0\cup \{\tau(\interior(\sigma))\}_{\sigma\in K},\leqslant)$. Further, by Proposition A.6.8 of Lurie, we have a conical stratification $|K|\to (B,\leqslant)$. This is all described by the solid arrow diagram below.

The vertical induced map comes as the poset $B$ has the exact same structure as the abstract suimplicial complex $K$. The diagonal induced map comes as the map $|K|\to \Ran^{\leqslant n}(M)\times \R_{>0}$ is a homeomorphism, and so has a continuous inverse. Composing the inverse with the conical sratification of Lurie, we get a conical stratification of $\Ran^{\leqslant n}(M)\times \R_{>0}$. Composing the vertical induced arrow and the maps to $(SC,\leqslant)$ show that there is a conical stratification of $\Ran^{\leqslant n}\times \R_{>0}$ compatible with its simplicial complex stratification from diagram (1). $\square$

Shiota actually requires that the space that admits a triangulation be closed semialgebraic, and having $\R_{>0}$ violates that condition. Replacing this piece with $\R_{\geqslant 0}$, then applying Shiota, and afterwards removing the $t=0$ piece we get the same result.

Remark: Every (sufficiently nice) manifold admits a triangulation, so it may be possible to extend this result to a larger class of manifolds, but it seems more sophisticated technology is needed.

References: Shiota (Geometry of subanalytic and semialgebraic sets, Chapters I.2, I.3, II.4), Scheiderer (Quotients of semi-algebraic spaces), Lurie (Higher algebra, Appendix A.6)

Refining stratifications

The goal of this post is to describe a natural stratification associated to any stratification, with hopes of it being conical. Let $X$ be a topological space, $(A,\leqslant_A)$ a finite partially ordered set, and $f:X\to A$ a stratifying map. For every $x\in X$, write $A_{>f(x)} = \{a\in A\ :\ a>f(x)\}\subseteq A$, and analogously for $A_{\geqslant f(x)}$. For every $a\in A$, write $X_a = \{x\in X\ :\ f(x)=a\}$.

Definition: For any other stratified space $g\colon Y\to B$, a stratified map $\varphi\colon (X\to A) \to (Y\to B)$ is a pair of maps $\varphi_{XY}\in \Hom_{\Top}(X,Y)$  and $\varphi_{AB}\in \Hom_{\Set}(A,B)$ such that the diagram

commutes. A stratified map $\varphi$ is an open embedding if both $\varphi_{XY}$ and $\varphi_{XY}|_{X_a}\colon X_a\to Y_{\varphi_{AB}(a)}$ are open embeddings.

Recall the cone $C(Y)$ of a space $Y$ is defined as $Y\times [0,1) / Y\times \{0\}$.

Definition: A stratification $f\colon X\to A$ is conical at $x\in X$ if there exist

• a stratified space $f_x \colon Y\to A_{>f(x)}$,
• a topological space $Z$, and
• an open embedding $Z\times C(Y)\hookrightarrow X$ of stratified spaces whose image contains $x$.

The cone $C(Y)$ has a natural stratification $f_x' \colon C(Y)\to A_{\geqslant f(x)}$, as does the product $Z\times C(Y)$. The space $X$ itself is \emph{conically stratified} if it is conically stratfied at every $x\in X$.

The image to have in mind is that $Z$ is a neighborhood of $x$ in its stratum $X_{f(x)}$, and $C(Y)$ is an upwards-directed neighborhood of $f(x)$ in $A$. Now we describe how to refine the stratification of an arbitrary stratified space to make it conical.

Definition: Let $\leqslant_{\mathbf P(A)}$ be the partial order on $\mathbf P(A)$ defined in the following way:

• For every $x,y\in A$, set $x\leqslant_{\mathbf P(A)} y$ whenever $x\leqslant_A y$, and
• for every $C\in \mathbf P(A)$, set $C\leqslant_{\mathbf P(A)} C'$ for all $C'\in \mathbf P(C)$.

Note that $(A,\leqslant_A)$ is open in $(\mathbf P(A),\leqslant_{\mathbf P(A)})$ in the upset topology. Hence for $i:A\hookrightarrow \mathbf P(A)$ the inclusion map, $i\circ f:X\to A\hookrightarrow \mathbf P(A)$ is also a stratifying map for $X$. We now define another $\mathbf P(A)$-stratification for $X$.

Definition: Let $f_{\mathbf P} \colon X\to \mathbf P(A)$ be defined by $f_{\mathbf P}(x)= \displaystyle\min_{\left(\mathbf P(A),\leqslant_{\mathbf P(A)}\right)} \left\{C\ :\ x\in \closure(f^{-1}(C'))\ \forall\ C'\in C\right\}$.

This map is well defined because for each $x\in X$ there are finitely many strata $f^{-1}(a)$ which contain $x$ in their closure. The element $C\in \mathbf P(A)$ containing all such $a$ is the $C$ to which $x$ gets mapped. We now claim this is a stratifying map for $X$.

Proposition: The map $f_{\mathbf P}\colon X\to \mathbf P(A)$ is continuous.

Proof: Let $C\in \mathbf P(A)$. We will show that the preimage via $f_{\mathbf P}$ of the open set $U_C = \mathbf P(C)\subseteq \mathbf P(A)$ is open in $X$ (and such sets $U_C$ are a basis of topology for $\mathbf P(A)$). By definition of the map $f_{\mathbf P}$, we have \[ f_{\mathbf P}^{-1}(U_C) = f^{-1}(U_{\min\{C'\in C\}}) \setminus \left(\bigcup_{(D,E)\in A\times (A\setminus C)} \closure( f^{-1}(D))\cap \closure (f^{-1}(E))\right). \] By continuity of $f$, the set $f^{-1}(U_{\min\{C'\in C\}})$ is open in $X$, and the sets we are subtracting from this open set are all closed. Hence $f_{\mathbf P}^{-1}(U_C)$ is open in $X$. $\square$

Unfortunately, this stratification is difficult to work with. Recall the space $\Ran_{\leqslant n}(M)\times \R_+$ for a very nice (smooth, compact, connected, embedded) manifold $M$, along with the map \[ \begin{array}{r c l}
f\colon \Ran_{\leqslant n}(M)\times \R_{\geqslant 0} & \to & SC, \\
(P,t) & \mapsto & VR(P,t),
\end{array} \] for $VR$ the Vietoris-Rips complex on $P$ with radius $t$. To put a partial order on $SC$, we first say that $S\leqslant T$ in $SC$ whenever there is a path $\gamma:I\to X$ satisfying

• $\widetilde f(\gamma(0))=S$ and $\widetilde f(\gamma(1))=T$,
• $\widetilde f(\gamma(t))=\widetilde f(\gamma(1))$ for all $t>1$.

Let $(SC,\leqslant_p)$ denote the partial order on $SC$ generated by all relations of this type. We would like to prove some results about $f_{\mathbf P}$ induced by this $f$, and by any stratifying $f$ in general, but the results seem difficult to prove. We give a list, in order of (percieved) increasing difficulty.

• The stratification $f_{\mathbf P}\colon \Ran_{\leqslant n}(M)\times \R_+ \to \mathbf P(SC)$ is conical.
• The stratification $f_{\mathbf P}\colon X\to \mathbf P(A)$ is conical for any stratified space $f\colon X\to A$.
• If $f\colon X\to A$ is already conical, the map $j\colon A\to \mathbf P(A)$ given by $j(a)= \{b\in A\ :\ f^{-1}(a)\subseteq \closure(f^{-1}(b))\}$ is an isomorphism onto its image, and $f_{\mathbf P} = j\circ f$.

References: Ayala, Francis, Tanaka (Local structure on stratified spaces)

Functorial persistence

The goal of this post is to overcome some hurdles encountered by Bauer and Lesnick. In their approach, some geometric information is lost in passing from persistence modules to matchings. Namely, if an interval ends, we forget if the  $k$-cycle it represents becomes part of another $k$-cycle or goes to 0. Recall:

• $(\R,\leqslant)$ is the category of real numbers and unique morphisms $s\to t$ whenever $s\leqslant t$,
• $\Vect$ ($\BVect$) is the category of (based) finite dimensional vector spaces, and
• $\Set_*$ is the category of pointed sets.

We begin by recalling all the classical notions in the TDA pipeline.

Defintion: A persistence module is a functor $F:(\R,\leqslant)\to \Vect$. The barcode of a persistence module $F$ is a collection of pairs $(I,k)$, where $I\subseteq \R$ is an interval and $k\in \Z_{>0}$ is a positive integer.

Crawley-Boevey describes how to find the decomposition of a persistence module into interval modules. The $k$ for each $I$ is usually 1, but is 2 (and more) if the same interval appears twice (or more) in the decomposition. A barcode contains the same information as a \emph{persistence diagram}, though the former is drawn as horizontal bars and the latter is presented on a pair of axes.

Definition: A matching $\chi$ of barcodes $\{(I_i,k_i)\}_i$ and $\{(J_j,\ell_j)\}_j$ is a bijection $I'\to J'$, for some $I'\subseteq \{(I_i,k_i)\}_i$ and $J'\subseteq \{(J_j,\ell_j)\}_j$.

We write matchings as $\chi\colon \{(I_i,k_i)\}_i \nrightarrow \{(J_j,\ell_j)\}_j$.

Definition: A filtered persistence module is a functor $F:(\R,\leqslant) \to \BVect$ for which $F(s\leqslant t)(e_i) =f_j$ or 0, for every $e_i$ in the basis of $F(s)$ and $f_j$ in the basis of $F(t)$.

The notion of filtered persistence module is used for a stronger geometric connection. Indeed, for every filtered space $X$ the persistence module along this filtration is also filtered (once interval modules have been found), as then inclusions $X_s\hookrightarrow X_t$ will induce isomorphisms in homology onto their image. That is, a pair of homology classes from the source may combine in the target, but if the classes come from interval modules, a class from the source can not be in two non-homologous classes of the target.

Remark: The above dicussion highlights that choosing a basis in the definition of a persistence module already uses the decomposition of persistence modules into interval modules.

It is immediate that a morphism of persistence modules is a natural transformation. Let $\BPVect$ be the full subcategory of $\BVect$ consisting of elements in the image of some filtered persistence module (the objects are the same, we just have a restriction of allowed morphisms).

Definition:  Let $\mathcal B$ be the functor defined by \[ \begin{array}{r c l}
\mathcal B\colon \BPVect & \to & \Set_*, \\
(V,\{e_1,\dots,e_n\}) & \mapsto & \{0,1,\dots,n\}, \\
\left(\varphi:(V,\{e_i\}) \to (W,\{f_j\})\right) & \mapsto & \left(
i \mapsto \begin{cases}
j & \text{ if } \varphi(e_i) = f_j, \\ 0 & \text{ if } \varphi(e_i)=0 \text{ or } i=0.
\end{cases} \right)
\end{array} \]

The basepoint of every set in the image of $\mathcal B$ is 0.

Definition: Let $F,G$ be persistence modules and $\eta$ a morphism $F\to G$.

• The persistence diagram of $F$ is the functor $\mathcal B\circ F$.
• The matching induced by $\eta$ is the natural transformation $\mathcal B(\eta): \mathcal B\circ F\to \mathcal B\circ G$.

Bauer and Lesnick's definition of "matching" allow for more freedom to mix and match barcode intervals, but this also restricts how much information of a persistence module morphism can be tracked.

Example: The following example has a horizontal filtration with the degree 0 homology barcode on the left and the degree 1 homology barcode on the right. Linear maps of based vector spaces have also been shown to indicate how homology classes are born, die (column of zeros), and combine (row with more than one 1).

Example: Bauer and Lesnick present Example 5.6 to show that functoriality does not work in their setting. We reproduce their example and show that functoriality does work in our setting. Note that vertical ordering of the bars does not matter once they are named.

Apply the functor $\mathcal B$ to the whole diagram to get the matchings induced by $\eta$ and $\xi$, as below.

Next we hope to understand how interleavings fit into this setup.

References: Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes), Crawley-Boevey (Decomposition of pointwise finite-dimensional persistence modules)

Artin gluing a sheaf 4: a single sheaf in two ways

The goal of this post is to give an alternative perspective on making a sheaf over $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, alternative to that of a previous post ("Artin gluing a sheaf 3: the Ran space," 2018-02-05). We will have one unique sheaf on all of $X$, valued either in simplicial complexes or simplicial sets.

Remark: Here we straddle the geometric category $SC$ of simplicial complexes and the algebraic category $\sSet$ of simplicial sets. There is a functor $[\ \cdot\ ]:SC\to \sSet$ for which every $n$-simplex in $S$ gets $(n+1)!$ elements in $[S]$, representing all the ways of ordering the vertices of $S$ (which we would like to view as unordered, to begin with).

Recall from previous posts:

• maps $f:X\to SC$ and $g = [f]:X\to \sSet$,
• the $SC_k$-stratification of $\Ran^k(M)\times \R_{\geqslant 0}$,
• the point-counting stratification of $\Ran^{\leqslant n}(M)$,
• the combined (via the product order) $SC_{\leqslant n}$-stratification of $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$,
• an induced (by the $SC_k$-stratification) cover by nested open sets $B_{k,1},\dots,B_{k,N_k}$ of $\Ran^k(M)\times \R_{\geqslant 0}$,
• a corresponding induced total order $S_{k,1},\dots,S_{k,N_k}$ on $f(\Ran^k(M)\times \R_{\geqslant0})$.

The product order also induces a cover by nested opens of all of $X$ and a total order on $f(X)$ and $g(X)$. We call a path $\gamma:I\to X$ a descending path if $t_1<t_2\in I$ implies $h(\gamma(t_1))\geqslant h(\gamma(t_2))$ in any stratified space $h:X\to A$. Below, $h$ is either $f$ or $g$.

Lemma: A descending path $\gamma:I\to X$ induces a unique morphism $h(\gamma(0))\to h(\gamma(1))$.

Proof: Write $\gamma(0) = \{P_1,\dots,P_n\}$ and $\gamma(1) = \{Q_1,\dots,Q_m\}$, with $m\leqslant n$. Since the path is descending, points can only collide, not split. Hence $\gamma$ induces $n$ paths $\gamma_i:I\to M$ for $i=1,\dots,n$, with $\gamma_i$ the path based at $P_i$. This induces a map $h(\gamma(0))_0\to h(\gamma(1))_0$ on 0-cells (vertices or 0-objects), which completely defines a map $h(\gamma(0))\to h(\gamma(1))$ in the desired category. $\square$

Our sheaves will be defined using colimits. Fortunately, both $SC$ and $\sSet$ have (small) colimits. Finally, we also need an auxiliary function $\sigma:\Op(X)\to SC$ that finds the correct simplicial complex. Define it by \[ \sigma(U)  = \begin{cases}
S_{k,\ell} & \text{ if } U\neq\emptyset, \text{ for } k = \max\{1\leqslant k'\leqslant n\ :\ U\cap \Ran^k(M)\times \R_{\geqslant 0}\neq \emptyset\}, \\ & \hspace{2.23cm} \ell = \max\{1\leqslant \ell'\leqslant N_k\ :\ U \cap B_{k,\ell'}\neq\emptyset\},\\
* & \text{ if }U= \emptyset.
\end{cases} \]

Proposition 1: Let $\mathcal F$ be the function $\Op(X)^{op}\to SC$ on objects given by \[ \mathcal F(U) = \colim\left(\sigma(U)\rightrightarrows S\ :\ \text{every }\sigma(U)\to S \text{ is induced by a descending }\gamma:I\to U\right). \] This is a functor and satisfies the sheaf gluing conditions.

Proof: We have a well-defined function, so we have to describe the restriction maps and show gluing works. Since $V\subseteq U\subseteq X$, every $S$ in the directed system defining $\mathcal F(V)$ is contained in the directed system defining $\mathcal F(U)$. As there are maps $\sigma(V)\to \mathcal F(V)$ and $S\to \mathcal F(V)$, for every $S$ in the directed system of $V$, precomposing with any descending path we get maps $\sigma (U)\to \mathcal F(V)$ and $S\to \mathcal F(V)$, for every $S$ in the directed system of $U$. Then universality of the colimit gives us a unique map $\mathcal F(U)\to \mathcal F(V)$. Note that if there are no paths (decending or otherwise) from $U$ to $V$, then the colimit over an empty diagram still exists, it is just the initial object $\emptyset$ of $SC$.

To check the gluing condition, first note that every open $U\subseteq X$ must nontrivially intersect $\Ran^n(M)\times \R_{\geqslant 0}$, the top stratum (in the point-counting stratification). So for $W = U\cap V$, if we have $\alpha\in \mathcal F(U)$ and $\beta \in \mathcal F(V)$ such that $\alpha|_W = \beta|_W$ is a $k$-simplex, then $\alpha$ and $\beta$ must have been $k$-simplices as well. This is because a simplicial takes a simplex to a simplex, and we cannot collide points while remaining in the top stratum. Hence the pullback of $S\owns \alpha$ and $T\owns \beta$ via some induced maps (by descending paths) from $U$ to $W$ and $V$ to $W$, respectively, will restrict to the identity on the chosen $k$-simplex. Hence the gluing condition holds, and $\mathcal F$ is a sheaf. $\square$

Functoriality of $[\ \cdot\ ]$ allows us to extend the proof to build a sheaf valued in simplicial sets.

Proposition 2: Let $\mathcal G$ be the function $\Op(X)^{op}\to \sSet$ on objects given by \[ \mathcal G(U) = \colim\left([\sigma(U)]\rightrightarrows S\ :\ \text{every }[\sigma(U)]\to S \text{ is induced by a descending }\gamma:I\to U\right). \] This is a functor and satisfies the sheaf gluing conditions.

Remark: The sheaf $\mathcal G$ is non-trivial on more sets. For example, any path contained within one stratum of $X$ induces the identity map on simplicial sets (though not on simplicial complexes). Hence $\mathcal G$ is non-trivial on every open set contained within a single stratum.

References: nLab (article "Simplicial complexes"), n-category Cafe (post "Simplicial Sets vs. Simplicial Complexes," 2017-08-19)

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.

Artin gluing a sheaf 2: simplicial sets and configuration spaces

The goal of this post is to extend the previous stratifying map to simplicial sets, and to generalize the sheaf construction to $X = \Conf_n(M)\times \R_{\geqslant 0}$ for arbitrary integers $n$, where $M$ is a smooth, compact, connected manifold. We work with $\Conf_n(M)$ instead of $\Ran^{\leqslant n}(M)$ because Lemma 1 and Proposition 2 have no chance of extending to $\Ran^{\leqslant n}(M)$ without major modifications (see Remark 3 at the end of this post).

Recall $SC$ is the category of simplicial complexes and simplicial maps, with $SC_n$ the full subcategory of simplicial complexes on $n$ vertices. Our main function is \[ \begin{array}{r c c c l}
f\ :\ X & \tov{f_1} & SC & \tov{f_2} & \sSet, \\
(P,a) & \mapsto & VR(P,a) & \mapsto & \Hom_{\Set}(\Delta^\bullet,VR(P,a)).
\end{array} \] On $\Conf_n(M)$ we have a natural metric, the Hausdorff distance $d_H(P,Q) = \max_{p\in P}\min_{q\in Q}d(p,q)+\max_{q\in Q}\min_{p\in P}d(p,q)$. This induces the 1-product metric on $X$, as \[ d_X((P,a),(Q,b)) = d_H(P,Q) + d(a,b), \] where $d$ without a subscript is Euclidean distance. We could have chosen any other $p$-product metric, but $p=1$ makes computations easier. For a given $(P,t)\in X$, write $P = \{P_1,\dots,P_n\}$ and define its maximal neighborhood to be the ball $B_X(\min\{\delta_1,\delta_2,t\},P)$, where \[ \delta_1 = \min_{i<j}\{d(P_i,P_j)\},
\hspace{1cm}
\delta_2 = \min_{i<j}\{|d(P_i,P_j)-t|\ :\ d(P_i,P_j)\neq t\}. \]

Lemma 1: Any path $\gamma:I\to X$ induces a unique morphism $f(\gamma(0))\to f(\gamma(1))$ of simplicial sets.

Proof: Write $\gamma(0) = \{P_1,\dots,P_n\}$ and $\gamma(1) = \{Q_1,\dots,Q_n\}$. The map $\gamma$ induces $n$ paths $\gamma_i:I\to M$ for $i=1,\dots,n$, with $\gamma_i$ the path based at $P_i$. Let $s:\gamma(0)\to \gamma(1)$ be the map on simplicial complexes defined by $P_i\mapsto \gamma_i(1)$. Since we are in the configuration space, where points cannot collide (as opposed to the Ran space), this is a well-defined map. Then $f_2(s)$ is a morphism of simplicial complexes. $\square$

Note the morphism of simplicial sets induced by any path in a maximal neighborhood of $x\in X$ is the identity morphism. We now move to describing a sheaf over all of $X$.

Definition: Let $X$ be any topological space and $\mathcal C$ a category with pullbacks. Let $A\subseteq X$ open and $B=X\setminus A \subseteq X$ closed, with $i:A\hookrightarrow X$ and $j:B\hookrightarrow X$ the inclusion maps. Let $\mathcal F$ be a $\mathcal C$-valued sheaf on $A$ and $\mathcal G$ a $\mathcal C$-valued sheaf on $B$. Then the \emph{Artin gluing} of $\mathcal F$ and $\mathcal G$ is the $\mathcal C$-valued sheaf $\mathcal H$ on $X$ defined as the pullback, or fiber product, of $i_*\mathcal F$ and $j_*\mathcal G$ over $j_*j^*i_*\mathcal F$ in the diagram below.

Note the definition requires a choice of sheaf map $\varphi:\mathcal G\to j^*i_*\mathcal F$. In the proof below, this sheaf map will be the morphism of simplicial sets from Lemma 1 through the functor $\Hom_\Set(\Delta^\bullet,-) = f_2(-)$.

Recall the ordering of $SC_n$ described by the only definition in a previous post ("Exit paths, part 2," 2017-09-28). Fix a cover $\{A_i\}_{i=1}^{N}$ of $SC_n$ by nested open subsets (so $N=|SC_n|$), with $B_i := f_1^{-1}(A_i)$ and $B_{\leqslant i} := \bigcup_{j=1}^i B_i$. We now have an induced order on and cover of $\im(f)=\sSet'$, as a full subcategory of $\sSet$. Even more, we now have an induced total order on $\sSet' = \{S_1,\dots,S_N\}$, with $S_i$ the unique simplicial set in $A_i\setminus A_{i-1}$. For example, $S_1=\Hom_\Set(\Delta^\bullet,\Delta^n)$ and $S_{N}=\Hom_\Set(\Delta^\bullet,\bigcup_{i=1}^n\Delta^0)$.

For ease of notation, we let $B_0 = \emptyset$ and write $S_\emptyset = \Hom(\Delta^\bullet,\emptyset)$, $S_0 = \Hom(\Delta^\bullet,\Delta^0)$.

Definition 1: Let $\mathcal F_i:\Op(B_i)^{op}\to \sSet$ be the locally constant sheaf given by $\mathcal F_i(U_x) = S_i$, where $U_x$ is a subset of the maximal neighborhood of $x\in B_i$. In general, \[ \mathcal F_i(U) = \begin{cases}
S_i & \text{ if }\begin{array}[t]{l}U\neq \emptyset, \\U\text{ is path connected},\\\text{every loop }\gamma:I\to U\text{ induces }\id:f(\gamma(0))\to f(\gamma(1)),\end{array} \\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases} \] In general, we say $U\subseteq X$ is good if it is non-empty, path connected, and every loop $\gamma:I\to U$ induces the identity morphism on simplicial sets.

Proposition 2: Let $\mathcal F_{\leqslant 1} = \mathcal F_1$, and $\mathcal F_{\leqslant i}$ be the sheaf on $B_{\leqslant i}$ obtained by Artin gluing $\mathcal F_i$ onto $\mathcal F_{\leqslant i-1}$, for all $i=2,\dots,N$. Then $\mathcal F = \mathcal F_{\leqslant N}$ is the $SC_n$-constructible sheaf on $X$ described by \[ \mathcal F(U) = \begin{cases}
S_{\max\{1\leqslant \ell\leqslant N\ :\ U\cap B_{\ell}\neq \emptyset\}} & \text{ if $U$ is good,}\\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases} \hspace{2cm} (1) \]

Proof: We proceed by induction. Begin with the constant sheaf $\mathcal F_1$ on $B_1$ and $\mathcal F_2$ on $B_2$, which we would like to glue together to get a sheaf $\mathcal F_{\leqslant2}$ on $B_{\leqslant 2}$. Since $f_1$ is continuous in the Alexandrov topology on the poset $SC_{\leqslant n}$, $B_1\subseteq B_{\leqslant 2}$ is open and $B_2 \subseteq B_{\leqslant 2}$ is closed. Let $i:B_1\hookrightarrow B_{\leqslant 2}$ and $j:B_2\hookrightarrow B_{\leqslant 2}$ be the inclusion maps. The sheaf $j^*i_*\mathcal F_1$ has support $\closure(B_1)\cap B_2 \neq \emptyset$ with \[ j^*i_*\mathcal F_1(U) = \colim_{V\supseteq j(U)}\left[i_*\mathcal F_1(V)\right] = \colim_{V\supseteq U}\left[\mathcal F_1(V\cap B_1)\right] = \begin{cases}
S_1 & \text{ if }U\cap \closure(B_1)\text{ is good}, \\ S_\emptyset & \text{ else},
\end{cases} \] for any non-empty $U\subseteq B_2$. Let the sheaf map $\varphi:\mathcal F_2\to j^*i_*\mathcal F_1$ be the inclusion simplicial set morphism on good sets (it can be thought of as induced through Lemma 1 by a path starting in $U\cap B_2$ and ending in $V\cap B_1$, for $V$ a small enough set in the colimit above). Note that $S_2 = \Hom_\Set(\Delta^\bullet,\Delta^n\setminus \Delta^1)$, where $\Delta^n\setminus \Delta^1$ is the simplicial complex resulting from removing an edge from the complete simplicial complex on $n$ vertices. Let $\mathcal F_{\leqslant 2}$ be the pullback of $i_*\mathcal F_1$ and $j_*\mathcal F_2$ along $j_*j^*i_*\mathcal F_1$, and $U\subseteq B_{\leqslant 2}$ a good set. If $U\subseteq B_1$, then $\mathcal F_{\leqslant 2}(U) = \mathcal F_1(U)=S_1$, and if  $U\subseteq B_2$, then $\mathcal F_{\leqslant 2}(U) = \mathcal F_2(U) = S_2$. Now suppose that $U\cap B_1 \neq \emptyset$ but also $U\cap B_2\neq\emptyset$, which, since $U$ is good, implies that $U\cap \closure(B_1)\cap B_2\neq\emptyset$. Then we have the pullback square

If $U$ is not good, then the simplicial sets are $S_\emptyset$ or $S_0$, with nothing interesting going on. The pullback over a good set $U$ can be computed levelwise as \[ \mathcal F_{\leqslant 2}(U)_m = \{(\alpha,\beta)\in (S_1)_m\times (S_2)_m\ :\ \alpha=j_*\varphi(\beta)\}. \hspace{2cm} (2)\] Since $j_*\varphi$ is induced by the inclusion $\varphi$, it is the identity on its image. So $\alpha = j_*\varphi(\beta)$ means $\alpha=\beta$, or in other words, $\mathcal F_{\leqslant 2}(U)=S_2$. Hence for arbitrary $U\subseteq B_{\leqslant 2}$, we have \[ \mathcal F_{\leqslant 2}(U) = \begin{cases}
S_{\max\{\ell=1,2\ :\ U\cap B_{\ell}\neq \emptyset\}} & \text{ if $U$ is good,}\\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases}\]

For the inductive step with $k>1$, let $\mathcal F_{\leqslant k}$ be the sheaf on $B_{\leqslant k}$ defined as in Equation (1), but with $k$ instead of $N$. We would like to glue $\mathcal F_{\leqslant k}$ to $\mathcal F_{k+1}$ on $B_{k+1}$ to get a sheaf $\mathcal F_{\leqslant k+1}$ on $B_{\leqslant k+1}$. As before, $B_k \subseteq B_{\leqslant k+1}$ is open and $B_{k+1}\subseteq B_{\leqslant k+1}$ is closed. For $i:B_k\hookrightarrow B_{\leqslant k+1}$ and $j:B_{k+1}\hookrightarrow B_{\leqslant k+1}$ the inclusion maps, the sheaf $j^*i_*\mathcal F_{\leqslant k}$ has support $\closure(B_{\leqslant k})\cap B_{k+1}$, with \[ j^*i_*\mathcal F_{\leqslant k}(U) = \colim_{V\supseteq j(U)}\left[i_*\mathcal F_{\leqslant k}(V)\right] = \colim_{V\supseteq U}\left[\mathcal F_{\leqslant k}(V\cap B_{\leqslant k})\right] = \begin{cases} S_{\max\{1\leqslant \ell\leqslant k\ :\ U\cap \closure(B_\ell)\neq\emptyset\}} & \text{ if }U\cap \closure(B_{\leqslant k})\text{ is good,} \\ S_\emptyset & \text{ else,} \end{cases} \] for any non-empty $U\subseteq B_{k+1}$. Let the sheaf map $\varphi:\mathcal F_{k+1}\to j^*i_*\mathcal F_{\leqslant k}$ be the inclusion simplicial set morphism on good sets (it can be thought of as induced through Lemma 1 by a path starting in $U\cap B_{k+1}$ and ending in $V\cap B_{\leqslant k}$, for $V$ a small enough set in the colimit above). For $U\subseteq B_{\leqslant k+1}$ a good set, if $U\subseteq B_{\leqslant k}$, then $\mathcal F_{\leqslant k+1}(U) = \mathcal F_{\leqslant k}(U)$, and if  $U\subseteq B_{k+1}$, then $\mathcal F_{\leqslant k+1}(U) = \mathcal F_{k+1}(U) = S_{k+1}$. Now suppose that $U\cap B_{\leqslant k} \neq \emptyset$ but also $U\cap B_{k+1}\neq\emptyset$, which, since $U$ is good, implies that $U\cap \closure(B_{\leqslant k})\cap B_{k+1}\neq\emptyset$. Then we have the pullback square

If $U$ is not good, then the simplicial sets are $S_\emptyset$ or $S_0$, with nothing interesting going on. Again, as in Equation (2), the pullback $\mathcal F_{\leqslant k+1}$ on a good set $U$ is \[ \mathcal F_{\leqslant k+1}(U)_m = \{(\alpha,\beta)\in (S_\ell)_m\times (S_{k+1})_m\ :\ \alpha = j_*\varphi(\beta)\}, \] and as before, this implies that $\mathcal F_{\leqslant k+1}(U) = S_{k+1}$. Hence $\mathcal F_{\leqslant k+1}$ is exactly of the form as in Equation (1), with $k+1$ instead of $N$, and by induction we get the desired description for $\mathcal F_{\leqslant N}= \mathcal F$.  $\square$

Remark 3: The statements given in this post do not extend to $\Ran^{\leqslant n}(M)$, at least not as stated. Lemma 1 fails if  somewhere along the path $\gamma$ a point splits in two or more points, as there is no canonical choice which of the "new" points should be the image of the "old" point. This means that the proof of Proposition 2 will also fail, because we relied on a uniquely defined sheaf map $\varphi$ between strata.

Next, we hope to use this approach to describe classic persistent homology results, and maybe link this to the concept of persistence modules.

References: Milne (Etale cohomology, Chapter 2.3)

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)