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.