This post contains topics and ideas from ACAT at HIM, April 2017, as presented by Professor Ulrich Bauer (see slide 11 of his presentation, online at
ulrich-bauer.org/persistence-bonn-talk.pdf). The central theme is to assign categories and functors to analyze the process
\[
\text{filtration}\ \longrightarrow\ \text{(co)homology}\ \longrightarrow\ \text{barcode.}
\hspace{3cm}(\text{pipe})
\]
Remark: The categories we will use are below. For filtrations, we have the ordered reals (though any poset $P$ would work) and topological spaces:
\begin{align*}
R\ :\ & \Obj(R) = \R, & \Top\ :\ & \Obj(\Top) = \{\text{topological spaces}\}, \\[5pt]
& \Hom(r,s) = \begin{cases}
\{r \mapsto s\}, & \text{ if } r\leqslant s, \\ \emptyset, & \text{ else,}
\end{cases} && \Hom(X,Y) = \{\text{functions }f:X\to Y\}.
\end{align*}
For (co)homology groups, we have the category of (framed) vector spaces. We write $V^n$ for $V^{\oplus n} = V\oplus V\oplus \cdots \oplus V$, and $e_n$ for a
frame of $V^n$ (see below).
\begin{align*}
\Vect\ :\ & \Obj(\Vect) = \{V^{\oplus n}\ :\ 0\leqslant n< \infty\},\\
& \Hom(V^n,V^m) = \{\text{homomorphisms }f:V^n\to V^m\}, \\[5pt]
\Vect^{fr}\ :\ & \Obj(\Vect^{fr}) = \{V^n\times e^n\ :\ 0\leqslant n<\infty\}, \\
& \Hom(V^n\times e^n,V^m\times e^m) = \{\text{hom. }f:V^n\to V^m,\ g:e^n\to e^m,\ g\in \Mat(n,m)\}.
\end{align*}
Finally for barcodes, we have $\Delta$, the
category of finite ordered sets, and its variants. A
partial injective function, or
matching $f:A\nrightarrow B$ is a bijection $A'\to B'$ for some $A'\subseteq A$, $B'\subseteq B$.
\begin{align*}
\Delta\ :\ & \Obj(\Delta) = \{[n]=(0,1,\dots,n)\ :\ 0\leqslant n<\infty\},\\
& \Hom([n],[m]) = \{ \text{order-preserving functions }f:[n]\to [m]\}, \\[5pt]
\Delta'\ :\
& \Obj(\Delta')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving functions }f:a\to b\}, \\[5pt]
\Delta''\ :\
& \Obj(\Delta'')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving partial injective functions }f:a\nrightarrow b\}.
\end{align*}
Definition: A
frame $e$ of a vector space $V^n$ is equivalently:
- an ordered basis of $V^n$,
- a linear isomorphism $V^n\to V^n$, or
- an element in the fiber of the principal rank $n$ frame bundle over a point.
Frames (of possibly different sizes) are related by full rank elements of $\Mat(n,m)$, which contains all $n\times m$ matrices over a given field.
Definition: Let $(P,\leqslant)$ be a poset. A (
indexed topological)
filtration is a functor $F:P\to \Top$, with
\[
\Hom(F(r),F(s)) = \begin{cases}
\{\iota:F(r) \hookrightarrow F(s)\}, & \text{ if }r\leqslant s, \\ \emptyset, & \text{ else,}
\end{cases}
\]
where $\iota$ is the inclusion map. That is, we require $F(r)\subseteq F(s)$ whenever $r\leqslant s$.
Definition: A
persistence module is the composition of functors $M_i:P \tov{F} \Top \tov{H_i} \Vect$.
Homology will be taken over some field $k$. A
framed persistence module is the same composition as above, but mapping into $\Vect^{fr}$ instead. The framing is chosen to describe how many different vector spaces have already been encountered in the filtration.
Definition: A
barcode is a collection of intervals of $\R$. It may also be viewed as the composition of functors $B_i:P\tov{F}\Top\tov{H_i}\Vect \tov{\dim}\Delta$.
Similarly as above, we may talk about a
framed barcode by instead mapping into $\Vect^{fr}$ and then to $\Delta''$, keeping track of which vector spaces we have already encountered. This allows us to interpret the process $(\text{pipe})$ in two different ways. First we have the unframed approach
\[
\begin{array}{r c c c l}
\Top & \to & \Vect & \to & \Delta, \\
X_t & \mapsto & H_i(X_t;k) & \mapsto & [\dim(H_i(X_t;k))].
\end{array}
\]
The problem here is interpreting the inclusion $X_t\hookrightarrow X_{t'}$ as a map in $\Delta$, for instance, in the case when $H_i(X_t;k)\cong H_i(X_{t'};k)$, but $H_i(X_t\hookrightarrow X_{t'}) \neq \id$. To fix this, we have the framed interpretation of $(\text{pipe})$
\[
\begin{array}{r c c c l}
\Top & \to & \Vect^{fr} & \to & \Delta'', \\
X_t & \mapsto & H_i(X_t;k)\times e & \mapsto & [e].
\end{array}
\]
The first map produces a frame $e$ of size $n$, where $n$ is the total number of different vector spaces encountered over all $t'\leqslant t$, by setting the first $\dim(H_i(X_t;k))$ coordinates to be the appropriate ones, and then the rest. This is done with the second map to $\Delta''$ in mind, as the size of $[e]$ is $\dim(H_i(X_t;k))$, with only the first $\dim(H_i(X_t;k))$ basis vectors taken from $e$. As usual, these maps are best understood by example.
Example: Given the closed curve $X$ in $\R^2$ below, let $\varphi:X\to \R$ be the height map from the line 0, with $X_i=\varphi^{-1}(-\infty,i]$, for $i=r,s,t,u,v$. Let $e_i$ be the standard $i$th basis vector in $\R^N$.
Remark: This seems to make $(\text{pipe})$ functorial, as the maps $X_t\hookrightarrow X_{t'}$ may be naturally viewed as partial injective functions in $\Delta''$, to account for the problem mentioned with the unframed interpretation. However, we have traded locality for functoriality, as the image of $X_t$ in $\Delta''$ can not be calculated without having calculated $X_{t'}$ for all $t'<t$.
References: Bauer (Algebraic perspectives of persistence), Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes)
