Exit paths, part 1

This post is meant to set up all the necessary ideas to define the category of exit paths.

Preliminaries

 Let $X$ be a topological space and $C$ a category. Recall the following terms:

• $\Delta$: The category whose objects are finite ordered sets $[n]=(1,\dots,n)$ and whose morphisms are non-decreasing maps. It has several full subcategories, including
  • $\Delta_s$, comprising the same objects of $\Delta$ and only injective morphisms, and
  • $\Delta_{\leqslant n}$, comprising only the objects $[0],\dots,[n]$ with the same morphisms.
• equalizer: An object $E$ and a universal map $e:E\to X$, with respect to two maps $f,g:X\to Y$. It is universal in the sense that all maps into $X$ whose compositions with $f,g$ are equal factor through $e$. Equalizers and coequalizers are described by the diagram below, with universality given by existence of the dotted maps.

• fibered product or pullback: The universal object $X\times_Z Y$ with maps to $X$ and $Y$, with respect to maps $X\to Z$ and $Y\to Z$.
• fully faithful: A functor $F$ whose morphism restriction $\Hom(X,Y)\to \Hom(F(X),F(Y))$ is surjective (full) and injective (faithful).
• locally constant sheaf: A sheaf $\mathcal F$ over $X$ for which every $x\in X$ has a neighborhood $U$ such that $\mathcal F|_U$ is a constant sheaf. For example, constructible sheaves are locally constant on every stratum. 
• simplicial object: A contravariant functor from $\Delta$ to any other category. When the target category is $\text{Set}$, it is called a simplicial set. They may also be viewed as a collection $S = \{S_n\}_{\geqslant 0}$ for $S_n=S([n])$ the value of the functor on each $[n]$. Simplicial sets come with two natural maps:
  • face maps $d_i:S_n\to S_{n-1}$ induced by the map $[n-1]\to [n]$ which skips the $i$th piece, and
  • degeneracy maps $s_i:S_n\to S_{n+1}$ induced by the map $[n+1]\to[n]$ which repeats the $i$th piece.
• stratification: A property of a cover $\{U_i\}$ of $X$ for which consecutive differences $U_{i+1}\setminus U_i$ have ``nicer" properties than all of $X$. For example, $E_i\to U_{i+1}\setminus U_i$ is a rank $i$ vector bundle, but there is no vector bundle $E\to X$ that restricts to every $E_i$.

Now we get into new territory.

Definition: The nerve of a category $C$ is the collection $N(C) = \{N(C)_n = Fun([n],C)\}_{n\geqslant 0}$, where $[n]$ is considered as a category with objects $0,\dots,n$ and a single morphism in $\Hom_{[n]}(s,t)$ iff $s\leqslant t$.

Note that the nerve of $C$ is a simplicial set, as it is a functor from $\Delta^{op}\to Fun(\Delta,C)$. Moreover, the pieces $N(C)_0$ are the objects of $C$ and $N(C)_1$ are the morphisms of $C$, so all the information about $C$ is contained in its nerve. There is more in the higher pieces $N(C)_n$, so the nerve (and simplicial sets in general) may be viewed as a generalization of a category.

Kan structures

Let $\text{sSet}$ be the category of simplicial sets. We may consider $\Delta^n = \Hom_\Delta(-,[n])$ as a contravariant functor $\Delta\to \text{Set}$, so it is an object of $\text{sSet}$.

Definition: Fix $n\geqslant 0$ and choose $0\leqslant i\leqslant n$. Then the $i$th $n$-horn of a simplicial set is the functor $\Lambda^n_i\subset \Delta^n$ generated by all the faces $\Delta^n(d_j)$, for $j\neq i$.

We purposefully do not describe what "$\subset$" or "generated by" mean for functors, hoping that intuition fills in the gaps. In some sense the horn feels like a partially defined functor (though it is a true simplicial set), well described by diagrams, for instance with $n=2$ and $i=1$ we have

Definition: A simplicial set $S$ is a Kan complex whenever every map $f:\Lambda^n_i\to S$ factors through $\Delta^n$. That is, when there exists a

The map $\iota$ is the inclusion. Moreover, $S$ is an $\infty$-category, or quasi-category, if the extending map $f'$ is unique.

Example: Some basic examples of $\infty$-categories, for $X$ a topological space, are

• $Sing(X)$, made up of pieces $Sing(X)_n = \Hom(\Delta^n,X)$, and
• $LCS(X)$, the category of locally constant sheaves over $X$. Here $LCS(X)_n$ over an object $A$, whose objects are $B\to A$ and morphisms are the appropriate commutative diagrams

Definition: A morphism $p\in \Hom_{\text{sSet}}(S,T)$ is a Kan fibration if for every commutative diagram (of solid arrows)

the dotted arrow exists, making the new diagram commute.

Definition: Let $C,D,A$ be categories with functors $F:C\to D$ and $G:C\to A$.

• The left Kan extension of $F$ along $G$ is a functor $A\xrightarrow L D$ and a universal natural transformation $F\stackrel \lambda \rightsquigarrow L\circ G$.
• The right Kan extension of $F$ along $G$ is a functor $A\xrightarrow R D$ and a universal natural transformation $R\circ G \stackrel \rho\rightsquigarrow F$.

Exit paths

The setting for this section is constructible sheaves over a topological space $X$. We begin with a slightly more technical definition of a stratification.

Definition: Let $(A,\leqslant)$ be a partially ordered set with the upset topology. That is, if $x\in U$ is open and $x\leqslant y$, then $y\in A$. An $A$-stratification of $X$ is a continuous function $f:X\to A$.

We now begin with a Treumann's definition of an exit path, combined with Lurie's stratified setting.

Definition: An exit path in an $A$-stratified space $X$ is a continuous map $\gamma:[0,1]\to X$ for which there exists a pair of chains $a_1\leqslant \cdots \leqslant a_n$ in $A$ and $0=t_0\leqslant \cdots \leqslant t_n=1$ in $[0,1]$ such that $f(\gamma(t))=a_i$ whenever $t\in (t_{i-1},t_i]$.

This really is a path, and so gives good intuition for what is happening. Recall that the geometric realization of the functor $\Delta^n$ is $|\Delta^n| = \{(t_0,\dots,t_n)\in \R^{n+1}\ :\ t_0+\cdots+t_n=1\}$. Oserving that $[0,1]\cong|\Delta^1|$, Lurie's definition of an exit path is more general by instead considering maps from $|\Delta^n|$.

Definition: The category of exit paths in an $A$-stratified space $X$ is the simplicial subset $Sing^A(X)\subset Sing(X)$ consisting of those simplices $\gamma:|\Delta^n|\to X$ for which there exists a chain $a_0\leqslant \cdots \leqslant a_n$ in $A$ such that $f(\gamma(t_0,\dots,t_i,0,\dots,0))=a_i$ for $t_i\neq 0$.

Example: As with all new ideas, it is useful to have an example. Consider the space $X=\Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$ of a closed manifold $M$ (see post "A constructible sheaf over the Ran space" 2017-06-24 for more). With the poset $(A,\leqslant)$ being $(a\leqslant b\leqslant c)$ and stratifying map
$$\begin{array}{r c l} f\ :\ X & \to & A, \\ (P,t) & \mapsto & \begin{cases} a & \text{ if } P\in \Ran^1(M), \\ b & \text{ if } P\in \Ran^2(M), t\leqslant d(P_1,P_2), \\ c & \text{ else,} \end{cases} \end{array}$$
we can make a continuous map $\gamma:\Delta^3\to X$ by
$$\begin{array}{r c l} (1,0,0) & \mapsto & (P\in \Ran^1(M),0), \\ (t_0,t_1\neq 0,0) & \mapsto & (P\in \Ran^2(M), d(P_1,P_2)), \\ (t_0,t_1,t_2\neq 0) & \mapsto & (P\in \Ran^2(M), t>d(P_1,P_2)). \end{array}$$
Then $f(\gamma(t_0\neq 0,0,0))=a$, and $f(\gamma(t_0,t_1\neq 0,0))=b$, and $f(\gamma(t_0,t_1,t_2\neq 0))=c$, as desired. The embedding of such a simplex $\gamma$ is described by the diagram below.

Both the image of $(1,0,0)$ and the 1-simplex from $(1,0,0)$ to $(0,1,0)$ lie in the singularity set of $\Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$, which is pairs $(P,t)$ where $t=d(P_i,P_j)$ for some $i,j$. The idea that the simplex "exits" a stratum is hopefully made clear by this image.

References: Lurie (Higher algebra, Appendix A), Lurie (What is... an $\infty$-category?), Groth (A short course on $\infty$-categories, Section 1), Joyal (Quasi-categories and Kan complexes), Goerss and Jardine (Simplicial homotopy theory, Chapter 1), Treumann (Exit paths and constructible stacks)

The Ran space and singularity sets

Fix a manifold $M$ along with an embedding of $M$ into $\R^N$ and set $X=\Ran(M)\times \R_{\geqslant 0}$. The goal of this post is to show that every $(P,t)\in X$ has an open neighborhood that contains no points of the type $(Q,d(Q_i,Q_j))$, for some $i\neq j$. The collection of all such elements of $X$ is called the singularity set of $X$, as the Vietoris-Rips complex at $Q$ with such a radius changes at such elements.

Following Lurie, given a collection of open sets $\{U_i\}_{i=1}^k$ in $M$, set
$$\Ran(\{U_i\}_{i=1}^k) = \left\{P\in \Ran(M)\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}.$$
The topology on $\Ran(M)$ is the smallest topology for which every $\Ran(\{U_i\}_{i=1}^k)$ is open, for any $\{U_i\}_{i=1}^k$, for any $k$. The topology on the product $X$ is the product topology.

Remark: Note that the Ran space $\Ran(M)$ by itself can be split up into the pieces $\Ran^k(M)$, with "singularities" viewed as when a point splits into two (or more) points, or two (or more) combine into one. Then every element of $\Ran(M)$ is on the edge of the singularity set, as any neighborhood of a single point on the manifold contains two points on the manifold.

Fix $(P,t)\in X$ not in the singularity set of $X$, with $P=(P_1,\dots,P_k)$, for $1\leqslant k\leqslant n$. Set
$$\mu = \min\left\{t,\min_{1\leqslant i<j\leqslant k}\left\{|t-d(P_i,P_j)|\right\}\right\},$$
with distance $d$ being Euclidean distance in $\R^N$. The quantity $\mu$ should be thought of as the upper bound on how "far" we may move from $(P,t)$ without hitting the singularity set.

Proposition: Let $(P,t)$ be as above and $t,\alpha,\beta>0$ such that $\alpha+\beta=\mu$. Then
$$U=\Ran\left(\{B(P_i,\alpha/2)\}_{i=1}^k\right) \times \left(t-\beta,t+\beta\right)$$
is an open neighborhood of $(P,t)$ in $X$ and does not contain any points of the singularity set of $X$.

If $t=0$, then having $[0,\beta)$ as the second component of $U$, with $\alpha+\beta=\min_{i,j}d(P_i,P_j)$ works as the open neighborhood of $(P,t)$. The balls $B(x,r)$ are $N$-dimensional in $\R^N$. The proof is mostly applications of the triangle inequality.

Proof: By construction we have that $U$ is open in $X$ and that it contains $(P,t)$. For $(Q,s)\in U$ any other element, we have three cases. We will show that the distance between any two $Q_a,Q_b\in Q$ is never $s$. Fix distinct indices $\ell,m\in \{1,\dots,k\}$.

1.  Case 1: $Q_a,Q_b\in B(P_\ell,\alpha/2)$. The situation looks as in the diagram below.
   
   Observe that $d(Q_a,Q_b)\leqslant d((Q_a,P_\ell)+d(Q_b,P_\ell) <\alpha = \mu-\beta \leqslant t-\beta$. Hence $d(Q_a,Q_b)<s$.
2. Case 2: $Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)>t$. The situation looks as in the diagram below.
   
   Observe that $d(P_\ell,P_m)\leqslant d(P_\ell,Q_b)+d(P_m,Q_b)\leqslant d(P_\ell,Q_a)+d(Q_a,Q_b)+d(P_m,Q_b) < \alpha+d(Q_a,Q_b)$. Since $d(P_\ell,P_m)>t$, the definition of $\mu$ gives us that $\mu \leqslant d(P_\ell,P_m)-t$, so combining this with the previous inequality, we get $d(Q_a,Q_b) > d(P_\ell,P_m)-\alpha\geqslant \mu+t-(\mu-\beta)=t+\beta$. Hence $d(Q_a,Q_b)>s$.
3. Case 3: $Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)<t$. The situation looks as in the diagram below.
   
   Observe that $d(Q_a,Q_b)\leqslant d(P_m,Q_b) + d(P_m,Q_a) \leqslant d(P_\ell,Q_a)+d(P_\ell,P_m)+d(P_m,Q_a)<\alpha+d(P_\ell,P_m)$. Since $d(P_\ell,P_m)<t$, the definition of $\mu$ gives us that $\mu\leqslant t-d(P_\ell,P_m)$, so combining this with the previous inequality, we get $d(Q_a,Q_b)<\mu-\beta+t-\mu = t-\beta$. Hence $d(Q_a,Q_b)<s$. $\square$

As an extension, it would be nice to show that the Vietoris--Rips complex of every element in $U$ is homotopy equivalent. This seems to be intuitively true, but a similar case analysis as above seems daunting.

References: Lurie (Higher Algebra, Section 5.5.1)

New directions in TDA

 Conference topic

This post is informal, meant as a collection of (personally) new things from the workshop "Topological data analysis: Developing abstract foundations" at the Banff International Research Station, July 31 - August 4, 2017. New actual questions:

1. Does there exist a constructible sheaf valued in persistence modules over $\Ran^{\leqslant n}(M)$?
   • On the stalks it should be the persistence module of $P\in \Ran^{\leqslant n}(M)$. What about arbitrary open sets?
   • Is there such a thing as a colimit of persistence modules?
   • Uli Bauer suggested something to do with ordering the elements of the sample and taking small open sets.
2. Can framed vector spaces be used to make the TDA pipeline functorial? Does Ezra Miller's work help?
   • Should be a functor from $(\R,\leqslant)$, the reals as a poset, to $\text{Vect}$ or $\text{Vect}_{fr}$, the category of (framed) vector spaces. Filtration function $f:\R^n\to \R$ is assumed to be given.
   • Framed perspective should not be too difficult, just need to find right definitions.
   • Does this give an equivalence of categories (category of persistence modules and category of matchings)? Is that what we want? Do we want to keep only specific properties?
   • Ezra's work is very dense and unpublished. But it seems to have a very precise functoriality (which is not the main thrust of the work, however).
3. Can the Bubenik-de Silva-Scott interleaving categorification be viewed as a (co)limit? Diagrams are suggestive.
   • Reference is 1707.06288 on the arXiv.
   • Probably not a colimit, because that would be very large, though the arrows suggest a colimit.
   • Have to be careful, because the (co)limit should be in the category of posets, not just interleavings.

New things to learn about:

1. Algebraic geometry / homotopy theory: the etale space of a sheaf, Kan extensions, model categories, symmetric monoidal categories.
2. TDA related: Gromov-Hausdorff distance, the universal distance (Michael Lesnick's thesis and papers), merge trees, Reeb graphs, Mapper (the program).