$F\colon\textup{\bf{vect}}^{Q}\to\textup{\bf{vect}}^{P}$ such that

    1. 1.

if $M$ is indecomposable, then so is $F(M)$,

    2. 2.

$F(M)\cong F(N)$ if and only if $M\cong N$,

    3. 3.

$F(cf+dg)=cF(f)+dF(g)$ for all $f,g\colon M\to N$ and $c,d\in k$.

Thus, if $P$ is wild, a full classification of the indecomposable $P$-modules
would yield a classification of the indecomposable $Q$-modules for any finite
poset $Q$. Obtaining such a classification is thought to be a hopelessly
difficult task.

The following “trichotomy” result is remarkable and surprising.

######  Theorem 8.8 (Drozd [83]).

Over an algebraically closed field $k$, any finite poset is of finite, tame,
or wild representation type.

###  8.2 The Representation Theory of $[m]\times[n]$

It turns out that the representation type of a commutative grid is wild even
for small grid sizes. To see this, observe that any module over the 5-star
poset embeds as multiparameter persistence module over $[5]\times[5]$:

${k^{n}}$${k^{n}\oplus k^{n}}$${k^{n}\oplus k^{n}}$${k^{n}\oplus
k^{n}}$${k^{n}\oplus k^{n}}$${0}$${k^{n}}$${k^{n}\oplus k^{n}}$${k^{n}\oplus
k^{n}}$${k^{n}\oplus k^{n}}$${0}$${0}$${k^{n}}$${k^{n}\oplus
k^{n}}$${k^{n}\oplus k^{n}}$${0}$${0}$${0}$${k^{n}}$${k^{n}\oplus
k^{n}}$${0}$${0}$${0}$${0}$${k^{n}}$$\scriptstyle{A}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{B}$$\scriptstyle{B}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{C}$$\scriptstyle{C}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{D}$$\scriptstyle{D}$$\scriptstyle{=}$$\scriptstyle{E}$

In particular, $[m]\times[n]$ must be of wild type for $m,n\geq 5$. The
following theorem gives a complete description of the representation type of
finite grids of varying sizes.

######  Theorem 8.9 ([138, Theorem 5], [137, Theorem 2.5]).

The poset $[m]\times[n]$ is of finite type if

  * •

$m=1$ or $n=1$,

  * •

$(m,n)\in\\{(2,2),(2,3),(2,4),(3,2),(4,2)\\}$.

It is of tame type if

  * •

$(m,n)\in\\{(2,5),(3,3),(5,2)\\}$.

It is of wild type in all other cases.

###  8.3 Homology in Dimension $\geq 1$

The previous discussion shows that the representation theory becomes very
complicated even for small grids. Moreover, Carlsson and Zomorodian have
observed that, when working over a prime field $k$, one can realize any
persistence module over a finite grid by applying $H_{1}(-;k)$ to a cellular
or simplicial sublevel filtration. As an example, Fig. 8.1 illustrates how one
can realize the following indecomposable persistence module, previously
considered in Example 4.6:

${k}$${k}$${0}$${k}$${k^{2}}$${k}$${0}$${k}$${k}$$\scriptstyle{=}$$\scriptstyle{[1,0]^{T}}$$\scriptstyle{=}$$\scriptstyle{[1,1]}$$\scriptstyle{[1,0]}$$\scriptstyle{[0,1]^{T}}$$\scriptstyle{=}$$\scriptstyle{=}$
(8.2)

Figure 8.1: Applying $H_{1}(-;k)$ to the above bifiltration yields the
indecomposable bipersistence module in Eq. 8.2.

It is shown in [40] that infinite families of indecomposables on a finite grid
can arise as the homology modules of function-Rips bifiltrations.

###  8.4 Homology in Dimension 0

Several bifiltrations that appear naturally in multiparameter persistence have
the property that their $0$-th homology modules are surjective in at least one
parameter direction. As an example, for $\gamma\colon P\to\mathbb{R}$ as in
Definition 5.1,

$H_{0}(\mathrm{Rips^{\downarrow}}(\gamma)_{(a,r),(a,s)})\colon
H_{0}(\mathrm{Rips^{\downarrow}}(\gamma)_{a,r})\to
H_{0}(\mathrm{Rips^{\downarrow}}(\gamma)_{a,s})$

is surjective for all $r\leq s$.

######  Remark 8.10.

$H_{0}$ factors as $H_{0}(-;k)\cong{\rm Free}\circ\pi_{0}$, where $\pi_{0}$
denotes path-components, and ${\rm
Free}\colon\mathbf{Set}\to\textup{\bf{Vect}}_{k}$ is the free functor sending
a set $X$ to the free $k$-vector space with basis $X$. The fact that
$H_{0}(\mathrm{Rips^{\downarrow}}(\gamma))_{(a,r),(a,s)}$ is an epimorphism
for $r\leq s$ is a direct consequence of this factorization. See [58, 38] for
a further discussion of persistence modules factoring through the category of
sets.

A natural question is to what extent such additional structure reduces the
algebraic complexity. For an instance of a result in this direction, one can
show that that if $M$ is an $[m]\times[n]$-module with surjections in one
direction and injections in the other, then $M$ is interval-decomposable; see
Theorem 10.4. On the other hand, the two-parameter families of persistence
modules from Example 8.5 all embed as $[5]\times[5]$-modules with all
morphisms surjective. This suggests that the reduction in complexity can only
be rather limited. The following theorem confirms that.

######  Theorem 8.11 ([8]).

The category of $[m]\times[n]$-modules whose horizontal maps are surjective is
equivalent to the category of $[m]\times[n-1]$-modules, up to a finite,
explicit list of indecomposables.

An analogue of Theorem 8.11 for the case where all morphisms are surjective is
also given in [8]. Furthermore, Theorem 8.11 and Theorem 8.9 together
immediately imply the following:

######  Corollary 8.12.

For $[m]\times[n]$-modules whose horizontal maps are surjective, the cases of
finite type are:

  * •

$m=1$ or $n\leq 2$ (all modules are interval-decomposable)

  * •

$(m,n)\in\\{(2,3),(2,4),(2,5),(3,3),(4,3)\\}$,

and the tame cases are: $(m,n)\in\\{(2,6),(3,4),(5,3)\\}$.

###  8.5 Computing Decompositions

While explicitly classifying the indecomposable multiparameter persistence
modules is hopeless, one can still compute the indecomposable summands of any
given persistence module. General purpose polynomial-time algorithms for
decomposing modules over finite-dimensional algebras exist in the literature
and implementations are readily available; see e.g,. the MeatAxe algorithm
[115], which computes a decomposition of a $P$-module in time
$\tilde{O}\left(\left(\sum_{p\in P}\mathop{\bf dim}M_{p}\right)^{6}\right)$,
and is implemented in GAP [104]. However, such algorithms are impractical in
the context of TDA.

Dey and Xin [81] have recently provided a decomposition algorithm tailored to
multiparameter persistence. Their algorithm takes as input a presentation
matrix of a $d$-parameter persistence module $M$. Under the assumption that no
two rows or columns have the same label, the algorithm outputs a decomposition
of $M$ into indecomposables. When this assumption does not hold, the algorithm
still provides a decomposition of $M$ into summands, but the summands may not
be indecomposabale. For an input presentation matrix with a total of $N$ rows
and columns, the algorithm runs in time $O(N^{(d-1)(2\omega+1)})$, where
$\omega<2.373$ is the exponent for matrix multiplication. While this work
represents significant progress on the efficient decomposition of MPH modules,
new ideas are needed for how to use such decompositions in the context of data
analysis.

##  9 Signed Barcodes

Recall that the rank invariant $\operatorname{Rk}M$ of a persistence module
$M$ is the function $(s,t)\mapsto\operatorname{Rk}(M_{s}\to M_{t})$, where
$s\leq t$. As noted in Section 4.2, the rank invariant is a complete invariant
of a 1-parameter persistence module. Thus, the barcode and rank invariant of a
1-parameter persistence module determine each other. Though the rank invariant
is incomplete outside of the 1-parameter setting, one may hope that the
correspondence between barcodes and rank invariants can be extended to the
multiparameter setting. As we will explain, this is indeed possible if one
considers _signed barcodes_.

In this section we will consider three signed barcode constructions appearing
in the recent TDA literature, which we will call the _rectangle_ , _interval_
, and _hook_ barcodes. The rectangle barcode [36] and hook barcodes [36, 37]
are both equivalent to the rank invariant; the former can be constructed via
Möbius inversion, while the latter is constructed via relative homological
algebra. The hook barcode has the advantage of being stable in a stronger
sense than the rectangle barcode. The interval barcode (also known as the
_generalized persistence diagram_) [36, 127, 2, 127] is an analogue of the
rectangle barcode which is equivalent to the generalized rank invariant
(Definition 4.7).

###  9.1 Rectangle Barcodes via Decomposition of the Rank Invariant

In Example 4.6, we saw a module $M$ whose rank invariant is not equal to the
rank invariant of any interval-decomposable module. On the other hand, the
rank invariant of $M$ can be expressed as the _difference_ between the rank
invariants of two interval-decomposable modules:

(9.1)

Such a decomposition need not be unique:

(9.2)

It turns out, however, that the decomposition is unique if we restrict to
rectangles.

######  Definition 9.1.

A _rectangle_ in a product of totally ordered sets $T_{1}\times\cdots\times
T_{n}$ is a poset of the form $I_{1}\times\cdots\times I_{n}$, where each
$I_{j}$ is an interval in $T_{j}$.

######  Theorem 9.2 ([36]).

Let $P=T_{1}\times\cdots\times T_{n}$, where each $T_{i}\subseteq\mathbb{R}$,
and let $M$ be a finitely presented $P$-module. There exists a _minimal_ pair
$(\mathcal{R},\mathcal{S})$ of finite multisets of rectangles in $P$
satisfying

$\operatorname{Rk}M=\operatorname{Rk}\left(\bigoplus_{I\in\mathcal{R}}k_{I}\right)-\operatorname{Rk}\left(\bigoplus_{J\in\mathcal{S}}k_{J}\right);$
(9.3)

here minimal means that if $(\mathcal{R}^{\prime},\mathcal{S}^{\prime})$ is
any other such pair satisfying Eq. 9.3, then
$\mathcal{R}\subseteq\mathcal{R}^{\prime}$ and
$\mathcal{S}\subseteq\mathcal{S}^{\prime}$.

We shall refer to the pair $(\mathcal{R},\mathcal{S})$ as the _rectangle
barcode_ of $\operatorname{Rk}M$. (It is called a _minimal rank decomposition_
in [36].) Note that if $d=1$, then by Theorem 3.4 and the uniqueness of the
signed barcode, we must have that $\mathcal{S}=\emptyset$.

The rectangle barcode can be constructed by taking a Möbius inversion of the
restriction of the rank invariant to a finite grid [36].

######  Remark 9.3.

The idea of using Möbius inversion to associate a signed barcode to the rank
invariant was introduced by Patel [155], who worked in the setting of
1-parameter persistence modules taking values in certain symmetric monoidal
categories. Subsequently, Möbius inversions have been applied to persistence
modules in several settings [139, 2, 127, 18]. See [36, Section 1] for a
detailed discussion of the connection between the various approaches.

The 1-parameter slices of rectangle barcodes are stable, in the sense of the
following theorem:

######  Theorem 9.4 ([36]).

Given signed barcodes $(\mathcal{R},\mathcal{S})$ and
$(\mathcal{R}^{\prime},\mathcal{S}^{\prime})$ of finitely presented
$\mathbb{R}^{d}$-persistence modules $M$ and $M^{\prime}$, we have:

$d_{\mathrm{match}}\left(\bigoplus_{I\in\mathcal{R}}k_{I}\oplus\bigoplus_{J^{\prime}\in\mathcal{S}^{\prime}}k_{J^{\prime}},\bigoplus_{I^{\prime}\in\mathcal{R}^{\prime}}k_{I^{\prime}}\oplus\bigoplus_{J\in\mathcal{S}}k_{J}\right)\leq
d_{\mathrm{match}}(M,M^{\prime}).$

One may ask whether Theorem 9.4 still holds if one replaces
$d_{\mathrm{match}}$ with the generalized bottleneck distance $d_{B}$
(Definition 6.4). The answer is no, even up to a constant factor [37,
Proposition 3.1]. In contrast, the hook barcode, which we will introduce in
Section 9.3, is Lipschitz stable with respect to $d_{B}$.

##### Visualization

Visualizing the rectangle barcode of a bipersistence module by directly
plotting the rectangles can be messy for examples of realistic size, since
many rectangles may overlap. To obtain a cleaner visualization, [36] proposes
to instead represent each rectangle via a line segment (“bar”) connecting its
infimum to its supremum, as illustrated in Fig. 9.1 (a). Examples involving
larger persistence modules can be found in [36].

One can read the rank invariant off of such a visualization:
$\operatorname{Rk}(M_{s}\to M_{t})$ is the number of bars in $\mathcal{R}$
intersecting both the sets $s^{-}=\\{u\in P\colon u\leq s\\}$ and
$t^{+}=\\{u\in P\colon u\geq t\\},$ minus the number of bars in $\mathcal{S}$
intersecting both sets; see Fig. 9.1 (b).

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$ (a)

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$(0,1)^{-}$$(1,1)^{+}$
(b)

Figure 9.1: (a) Visualization of the signed barcode of Eq. 9.2. Rectangles
with positive and negative sign are represented using blue and red bars,
respectively. (b) The rank of the map $M_{(0,1)}\to M_{(1,1)}$ (represented by
the green arrow) is 1; This is the number of blue bars intersecting both gray
regions minus the number of red bars intersecting both regions.

##### Computation

The rectangle barcode of a p.f.d. persistence module $M$ indexed over a finite
grid $\mathcal{G}=\prod_{i=1}^{d}[1,n_{i}]$ can be computed by means of a
simple inclusion-exclusion formula. Let $\alpha_{[s,t]}$ denote the
multiplicity of the rectangle

$[s,t]=\\{u\in\mathbb{R}^{d}\mid s\leq u\leq t\\},$

in the rectangle barcode $(\mathcal{R},\mathcal{S})$. That is, if
$\alpha_{[s,t]}>0$ then $[s,t]$ appears with multiplicity $\alpha_{[s,t]}$ in
$\mathcal{R}$, and if $\alpha_{[s,t]}<0$ then $[s,t]$ appears with
multiplicity $-\alpha_{[s,t]}$ in $\mathcal{S}$. Then we have:

$\alpha_{[s,t]}=\sum_{\begin{smallmatrix}s^{\prime}\leq s\\\
\|s^{\prime}-s\|_{\infty}\leq
1\end{smallmatrix}}\sum_{\begin{smallmatrix}t^{\prime}\geq t\\\
\|t^{\prime}-t\|_{\infty}\leq
1\end{smallmatrix}}(-1)^{\|s^{\prime}-s\|_{1}+\|t^{\prime}-t\|_{1}}\,\operatorname{Rk}M(s^{\prime},t^{\prime}).$
(9.4)

######  Remark 9.5.

The case $d=1$ gives the well-known inclusion-exclusion formula relating the
persistence diagram of a one-parameter persistence module to its rank
invariant [67]. The case $d=2$ specializes to the inclusion-exclusion formula
for computing the multiplicities of summands in rectangle-decomposable
2-parameter persistence modules [35].

A simple inspection of the formula in Eq. 9.4 reveals that computation is
bounded in time $O\left(2^{2d}\,\prod_{i=1}^{d}n_{i}^{2}\right)$, assuming
constant-time access to the ranks $\operatorname{Rk}M(s^{\prime},t^{\prime})$
and constant-time arithmetic operations. When the module $M$ comes from a
simplicial filtration over the grid $\mathcal{G}$ with $n=\max_{i}n_{i}$
simplices in total, the rank invariant itself can be pre-computed and stored,
e.g., by naively computing the ranks $\operatorname{Rk}M(s,t)$ for each pair
$s\leq t\in\mathcal{G}$ independently, which takes $O(n^{2d+\omega})$ time in
total, where $\omega<2.373$ is the exponent for matrix multiplication [145].
Adding in the computation time for the signed barcode yields a bound in
$O(n^{2d+\omega}+(2n)^{2d})$. In the special case where $d=2$, assuming the
filtration is $1$-critical (i.e. each simplex has a unique minimal index of
appearance in the filtration), there is an $O(n^{4})$-time algorithm to
compute the rank invariant [146, 35], and computing its rectangle barcode also
takes $O(n^{4})$ time.

###  9.2 Interval Barcodes via Decomposition of the Generalized Rank Invariant

While a smaller signed barcode using intervals of other shapes than rectangles
is sometimes possible, it comes at the expense of uniqueness, as illustrated
by Eq. 9.1 and Eq. 9.2. It is however possible to retain uniqueness if one
instead decomposes the generalized rank invariant (Definition 4.7).

Let $P$ be an arbitrary poset. We say a collection $\mathcal{I}$ of intervals
in $P$ is _locally finite_ if for any $I,K\in\mathcal{I}$ the set
$\\{J\in\mathcal{I}:I\subseteq J\subseteq K\\}$ is finite.

######  Theorem 9.6 ([36]).

Let $\mathcal{I}$ be a locally finite collection of intervals in $P$, and let
$M$ be a p.f.d. $P$-module. Then there exists a pair
$(\mathcal{R},\mathcal{S})$, where $\mathcal{R}$ and $\mathcal{S}$ are finite
multi-sets of intervals in $\mathcal{I}$, such that

$\operatorname{Rk}_{\mathcal{I}}M=\operatorname{Rk}_{\mathcal{I}}\left(\bigoplus_{I\in\mathcal{R}}k_{I}\right)-\operatorname{Rk}_{\mathcal{I}}\left(\bigoplus_{J\in\mathcal{S}}k_{J}\right),$
(9.5)

and if $(\mathcal{R}^{\prime},\mathcal{S}^{\prime})$ is any other such pair
satisfying Eq. 9.5, then $\mathcal{R}\subseteq\mathcal{R}^{\prime}$ and
$\mathcal{S}\subseteq\mathcal{S}^{\prime}$.

######  Remark 9.7.

When $\mathcal{I}$ consists of all intervals in an essentially finite poset
$P$, the _generalized persistence diagram_ of [127], constructed via Möbius
inversion of the generalized rank invariant, is essentially the same as the
decomposition $(\mathcal{R},\mathcal{S})$ of Theorem 9.6.

###  9.3 Hook Barcodes via Relative Homological Algebra

Several recent works use _relative homological algebra_ to study generalized
persistence. In brief, the idea is to identify an _exact structure_ for a
given invariant, i.e., a distinguished class of exact sequences which respects
the structure of the invariant. There is a notion of _relative minimal
projective resolution_ associated to any exact structure, which is used to
define a signed barcode. This approach to generalized persistence was
initiated for the rank invariant in [36], and in parallel for other invariants
in [28]; see also [153, 3, 57, 37] for other recent work, and [42] for an
excellent introduction to exact categories.

In the special case of the rank invariant, this approach yields the _hook
barcode_ , a signed barcode which is bottleneck-stable with respect to the
interleaving distance [37]. We now explain this case in more detail.

######  Definition 9.8.

  * (i)

A short exact sequence

$0\to M\to M^{\prime}\to M^{\prime\prime}\to 0$

of $\mathbb{R}^{n}$-persistence modules is _rank exact_ if
$\operatorname{Rk}M^{\prime}=\operatorname{Rk}M+\operatorname{Rk}M^{\prime\prime}$,

  * (ii)

A long exact sequence

$\cdots\xrightarrow{f_{i+1}}X_{i}\xrightarrow{f_{i}}X_{i-1}\xrightarrow{f_{i-1}}\cdots$

is _rank exact_ if $\operatorname{im}f_{i}\to
X_{i-1}\to\operatorname{im}f_{i-1}$ is rank exact for all $i$,

  * (iii)

An $\mathbb{R}^{n}$-persistence module $M$ is _rank projective_ if ${\rm
Hom}(M,-)$ takes each rank exact sequence to an exact sequence of vector
spaces.

######  Definition 9.9.

A _hook_ is an interval in $\mathbb{R}^{n}$ of the form
$\\{p\in\mathbb{R}^{n}:r\leq p\not\geq s\\}$ for
$r<s\in\mathbb{R}^{n}\cup\\{\infty\\}$.

One can show that if $M\colon\mathbb{R}^{n}\to\textup{\bf{Vect}}$ is finitely
presented and rank projective, then $M$ is _hook-decomposable_ , i.e., $M$ is
interval-decomposable and each interval in ${\mathcal{B}}(M)$ is a hook.

######  Theorem 9.10 ([36, 37]).

A finitely presented $\mathbb{R}^{n}$-persistence module $M$ admits a minimal
rank projective resolution

$0\to X_{k}\to X_{k-1}\to\cdots\to X_{1}\to X_{0}\to M,$

with each $X_{i}$ finitely presented (and hook-decomposable) and $k\leq 2n-2$.

It follows from the theory of exact categories that a minimal rank projective
resolution is unique up to isomorphism. Given a finitely presented
$\mathbb{R}^{n}$-persistence module $M$, let
$\beta_{i}(M)={\mathcal{B}}(X_{i}),$ where $X_{i}$ is as in Theorem 9.10.

######  Definition 9.11.

We define the _hook barcode_ of $M$ to be the pair

$\left(\beta_{2\mathbb{N}}(M),\beta_{2\mathbb{N}+1}(M)\right):=\left(\bigcup_{i\text{
even}}\beta_{i}(M),\bigcup_{i\text{ odd}}\beta_{i}(M)\right).$

######  Remarks 9.12.

  * (i)

By virtue of the exact structure, one has that

$\operatorname{Rk}M=\sum_{i=0}^{k}(-1)^{i}\operatorname{Rk}X_{i}=\sum_{i\text{
even}}\operatorname{Rk}X_{i}-\sum_{i\text{ odd}}\operatorname{Rk}X_{i}.$

This implies that we can interpret the hook barcode of $M$ as a decomposition
of $\operatorname{Rk}M$, as for the rectangle barcode.

  * (ii)

In contrast to the rectangle barcode, the two components of the hook barcode
may have intervals in common. Removing common hooks from the decomposition in
(i) yields a minimal decomposition of the rank function in terms of hook
modules. However, the main stability property of the hook barcode (Theorem
9.13 below) does not hold for this minimal decomposition [37, Remark 3.2].

Like the rectangle barcode, the hook barcode can be visualized by a collection
of line segments, as the following rank projective resolution illustrates.

The hook barcode is stable in the following sense:

######  Theorem 9.13 ([37]).

If $M$ and $N$ are finitely presented $\mathbb{R}^{n}$-persistence modules,
then

$d_{B}\left(\beta_{2\mathbb{N}}(M)\cup\beta_{2\mathbb{N}+1}(N),\beta_{2\mathbb{N}}(N)\cup\beta_{2\mathbb{N}+1}(M)\right)\leq(2n-1)^{2}d_{I}(M,N).$

The proof is an application of the algebraic stability of hook-decomposable
modules (Theorem 6.5 (iv)).

######  Remark 9.14.

Computation of minimal resolutions and _relative Betti numbers_ in exact
categories is an area of active research. Recently, Chachólski et al. [57]
provided a framework for computing relative Betti numbers of exact structures
on upper semi-lattices.

##  10 Local Conditions for Interval Decomposability

In this section, we introduce local conditions on a bipersistence module $M$
which force $M$ to be interval-decomposable, with each interval in
${\mathcal{B}}(M)$ of a particularly simple form. We will see that such
conditions arise naturally in the study of the _interlevel persistence_ of
$\mathbb{R}$-valued functions.

###  10.1 Decomposition into Blocks and Rectangles

Let $S$ and $T$ be totally ordered sets.

######  Definition 10.1.

Given $P\subseteq S\times T$ and a persistence module $M\colon
P\to\mathbf{Vect}$,

  * (i)

$M$ is _middle-exact_ [32, 66, 49, 48] if

$M_{a}\xrightarrow{M_{a,b}\oplus M_{a,c}}M_{b}\oplus
M_{c}\xrightarrow{(M_{b,d}-M_{c,d})}M_{d}$ (10.1)

is exact (i.e., the image of the first map equals the kernel of the second
map) for all $a,b,c,d\in P$ of the form

$a=(x,y),\quad b=(x,y^{\prime}),\quad c=(x^{\prime},y),\quad
d=(x^{\prime},y^{\prime}).$

  * (ii)

$M$ is _weakly-exact_ [34, 35] if for all $a,b,c,d$ as above,

$\displaystyle\mathrm{Im}(M_{a}\to M_{d})$ $\displaystyle=\mathrm{Im}(M_{b}\to
M_{d})\cap\mathrm{Im}(M_{c}\to M_{d}),$
$\displaystyle\operatorname{Ker}(M_{a}\to M_{d})$
$\displaystyle=\operatorname{Ker}(M_{a}\to M_{b})+\ker(M_{a}\to M_{c}).$

One can check that if $M$ is middle-exact then it is weakly-exact.

Recall from Example 8.2 that a persistence module over a commutative square is
interval-decomposable. A simple case-by-case inspection of the intervals gives
the following characterization of middle-exact and weakly-exact persistence
modules.

######  Proposition 10.2.

  * (i)

$M$ is middle-exact if and only if its restriction to any square

${M_{c}}$${M_{d}}$${M_{a}}$${M_{b}}$

decomposes into interval modules supported on the set

$B:=\\{\\{a\\},\\{d\\},\\{a,b\\},\\{a,c\\},\\{b,d\\},\\{c,d\\},\\{a,b,c,d\\}\\}.$

  * (i)

$M$ is weakly-exact if and only if its restriction to any square as above
decomposes into interval modules supported on the set

$B\cup\\{\\{b\\},\\{c\\}\\}.$

######  Definition 10.3.

An interval $I\subseteq S\times T$ is called a _block_ if $I$ can be written
in one of the following ways:

  1. 1.

$I=J_{S}\times J_{T}$ for downsets $J_{S}$ and $J_{T}$,

  2. 2.

$I=J_{S}\times J_{T}$ for upsets $J_{S}$ and $J_{T}$,

  3. 3.

$I=J_{S}\times T$ for an interval $J_{S}$,

  4. 4.

$I=S\times J_{T}$ for an interval $J_{T}$.

More generally, recall from Definition 9.1 that a rectangle in $S\times T$ is
an interval of the form $I\times J$, where $I$ and $J$ are intervals in $S$
and $T$. We say that $M$ is _block-decomposable_ (respectively, _rectangle-
decomposable_) if $M$ is interval-decomposable and each interval of
${\mathcal{B}}(M)$ is a block (respectively, a rectangle).

The following theorems tell us that a module is block-decomposable or
rectangle-decomposable if and only if its restriction to each square is.

######  Theorem 10.4 ([32]).

A p.f.d. $S\times T$-module is middle-exact if and only if it is block-
decomposable.

Given a totally ordered set $T$, a subset $T^{\prime}\subset T$ is said to be
_coinitial_ if for all $t\in T$ there exists $t^{\prime}\in T^{\prime}$ with
$t^{\prime}\leq t$. For example, $\mathbb{Z}$ is a coinitial subset of
$\mathbb{R}$.

######  Theorem 10.5 ([34]).

Suppose that $S$ and $T$ are totally ordered sets such that every interval in
$S$ or $T$ admits a countable coinitial subset. Then a p.f.d. $S\times
T$-module is weakly-exact if and only if it is rectangle-decomposable.

######  Remark 10.6.

In the case that $S$ and $T$ satisfy the conditions of Theorem 10.5, Theorem
10.4 follows easily from Theorem 10.5.

######  Remark 10.7.

Proofs of Theorem 10.4 for the special cases $S\times T$ finite and $S\times
T=\mathbb{R}^{2}$ first appeared in [14] and [66], respectively; see also [49,
48], which had a strong influence on these results. For the case $S\times T$
finite, an elementary proof of Theorem 10.5 appeared in [35].

As in Section 5.2, let

$\overline{U}:=\\{(a,b)\in\mathbb{R}^{\text{op}}\times\mathbb{R}\colon a\leq
b\\},$

and let

$U:=\\{(a,b)\in\mathbb{R}^{\text{op}}\times\mathbb{R}\colon a<b\\}.$

As a (non-trivial) application of Theorem 10.4, [32] establishes the following
result.

######  Theorem 10.8 ([32]).

Let $P$ be $U$ or $\overline{U}$. Any p.f.d. middle-exact $P$-module $M$ is
interval-decomposable, and each interval of ${\mathcal{B}}(M)$ is of the form
$P\cap I$, where $I$ is a block in
$\mathbb{R}^{\operatorname{op}}\times\mathbb{R}$.

We shall also refer to the intervals $P\cap I$ arising in Theorem 10.8 as
_blocks (in $P$)_.

###  10.2 Interlevel Persistent Homology

One can use Theorem 10.8 to define _interlevel barcodes_ , fundamental
invariants of $\mathbb{R}$-valued functions which refine the standard sublevel
barcodes. Moreover, these invariants admit extensions given using relative
homology, and the invariants in different homological degrees assemble nicely
into a single object, the _Mayer-Vietoris strip_. We will now explain this.

To formulate the definitions, one has to choose between using open intervals
or closed intervals in $\mathbb{R}$. For consistency with our treatment of
sublevel filtrations (Definition 2.2), we will use closed intervals here,
though using open intervals is arguably more convenient. We briefly discuss
the approach via open intervals in Remark 10.15 below.

Given a continuous function $\gamma\colon W\to\mathbb{R}$, the _interlevel
bifiltration of $\gamma$_ is the functor

$\mathcal{S}(\gamma)\colon\overline{U}\to\mathbf{Top},\quad\mathcal{S}(\gamma)(s,t)=\\{x\in
X:s\leq\gamma(x)\leq t\\}.$

It follows from the Mayer–Vietoris sequence that for all $i\geq 0$, the
restriction of ${H}_{i}\mathcal{S}(\gamma)$ to $U$ is middle-exact. Thus, if
${H}_{i}\mathcal{S}(\gamma)$ is p.f.d., then Theorem 10.8 provides a barcode
of the restriction of ${H}_{i}\mathcal{S}(\gamma)$ to $U$.

Furthermore, assuming that $\gamma$ is a Morse function or, more generally,
_of Morse type_ [49, Section 2], ${H}_{i}\mathcal{S}(\gamma)$ is itself
middle-exact, and Theorem 10.8 then provides a barcode
${\mathcal{B}}({H}_{i}\mathcal{S}(\gamma))$ of ${H}_{i}\mathcal{S}(\gamma)$
without having to restrict to $U$.

Taking the intersection of each interval of
${\mathcal{B}}({H}_{i}\mathcal{S}(\gamma))$ with $\partial U=\\{(x,x)\mid
x\in\mathbb{R}\\}$ and projecting onto the first coordinate, we recover the
$i^{\mathrm{th}}$ _levelset barcode_ of $\gamma$, as introduced in [49]; we
denote this as $\mathcal{L}_{i}(\gamma)$. The intervals in this barcode encode
how the homology of the fibers $\gamma^{-1}(r)$ changes as $r$ varies.

######  Example 10.9.

Fig. 10.1 illustrates a function $\gamma\colon W\to\mathbb{R}$ and the
barcodes ${\mathcal{B}}({H}_{i}\mathcal{S}(\gamma))$. By considering how the
intervals in these barcodes intersect $\partial U$, we see that

$\mathcal{L}_{0}(\gamma)=\\{[a_{1},a_{6}],[a_{2},a_{3}),(a_{4},a_{5})\\}$

and $\mathcal{L}_{i}(\gamma)=\emptyset$ for $i>0$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$H_{0}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$\partial
U$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$\partial U$$H_{1}$

Figure 10.1: The interlevel homology of the function $\gamma\colon
W\to\mathbb{R}$ given by $\gamma(x,y)=y$, where $W\subset\mathbb{R}^{2}$ is as
shown in red.

$a_{6}$$a_{5}$$a_{4}$$a_{3}$$a_{2}$$a_{1}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$Q_{1}$$Q_{3}$$Q_{4}$$Q_{2}$$x$$x$$y$$y$$y$$y$$x$$x$

Figure 10.2: Visualization of the poset $Q$ indexing relative interlevel
homology.

##### Relative Interlevel Homology

For each $i\geq 0$, we can extend the above construction to a relative version
as follows: consider the set

$Q=Q_{1}\sqcup Q_{2}\sqcup Q_{3}\sqcup Q_{4}$

of pairs of subsets of $\mathbb{R}$, where

$\displaystyle Q_{1}$ $\displaystyle=\\{([x,y],\emptyset)\colon x\leq y\\}$
$\displaystyle Q_{2}$ $\displaystyle=\\{([x,\infty),[y,\infty))\colon x\leq
y\\}),$ $\displaystyle Q_{3}$
$\displaystyle=\\{((-\infty,y],(-\infty,x])\colon x\leq y\\},$ $\displaystyle
Q_{4}$ $\displaystyle=\\{(\mathbb{R},(-\infty,x]\cup[y,\infty))\colon x\leq
y\\}.$

We regard $Q$ as poset by taking $(A,B)\leq(A^{\prime},B^{\prime})$ if and
only if $A\subseteq A^{\prime}$ and $B\subseteq B^{\prime}$. Let $2\mathbb{R}$
denote the totally ordered set consisting of two disjoint copies of
$\mathbb{R}$, with all elements in the first copy less than all elements in
the second copy. We have a natural identification of $Q$ with an interval in
$(2\mathbb{R}^{\operatorname{op}})\times 2\mathbb{R}$.

Given a continuous function $\gamma\colon W\to\mathbb{R}$, we define a functor
$\mathcal{S}^{\bullet}(\gamma)$ from $Q$ to the category of pairs of
topological spaces by

$\mathcal{S}(\gamma)^{\bullet}(A,B)=(\gamma^{-1}(A),\gamma^{-1}(B)).$

If $\gamma$ is of Morse type, then $H_{i}\mathcal{S}^{\bullet}(\gamma)$ is
interval-decomposable, where each interval is the intersection of a rectangle
in $2\mathbb{R}^{\operatorname{op}}\times 2\mathbb{R}$ with $Q$ [14, 35]. It
is helpful to visualize $Q$ schematically as the decomposition of a tilted
square into four triangles, as shown in Fig. 10.2; we can make formal sense of
this schematic either by reparameterizing $Q$ as in [9], or by assuming that
$\gamma$ is bounded above and below, in which case we may replace $Q$ with a
subposet where $x$ and $y$ are correspondingly bounded, as in [49].

######  Example 10.10.

Fig. 10.3 shows the decompositions of the modules
$H_{0}\mathcal{S}^{\bullet}(\gamma)$ and $H_{1}\mathcal{S}^{\bullet}(\gamma)$
for $\gamma$ the function of Example 10.9 and Fig. 10.2. Observe that the off-
diagonal block in ${H}_{1}\mathcal{S}(\gamma)$ now extends all the way to the
top of $Q$. Moreover, if we reflect the block around the anti-diagonal passing
through the center of $Q$, then we see that the boundary of the block aligns
perfectly with the boundary of the triangle from ${H}_{0}\mathcal{S}(\gamma)$.
Geometrically, this makes sense, as the circle in $\gamma^{-1}[a_{4},a_{5}]$
gives rise to two connected components in all pre-images $\gamma^{-1}[x,y]$
where $[x,y]\subseteq(a_{4},a_{5})$.

##### The Mayer–Vietoris Strip

In the previous example, there is a natural matching between
${\mathcal{B}}(H_{0}\mathcal{S}^{\bullet}(\gamma))$ and
${\mathcal{B}}(H_{1}\mathcal{S}^{\bullet}(\gamma))$. It turns out that one
always has such a matching. To explain, using excision and the boundary map in
the relative Mayer–Vietoris sequence, one obtains maps connecting the diagrams
$H_{i}\mathcal{S}^{\bullet}(\gamma)$ and
$H_{i+1}\mathcal{S}^{\bullet}(\gamma)$ for all $i$. Using these maps and
reflecting every other diagram $H_{i}\mathcal{S}^{\bullet}$ as shown in Fig.
10.3, the diagrams $\\{H_{i}\mathcal{S}^{\bullet}(\gamma)\\}_{i\in\mathbb{N}}$
assemble into one big persistence module called the _Mayer–Vietoris strip_ of
$\gamma$. Let $Q^{\prime}$ denote the indexing poset of this persistence
module. Like $Q$, the poset $Q^{\prime}$ is naturally identified with an
interval in a product of two totally ordered sets, so rectangles in
$Q^{\prime}$ are well defined.

It turns out that the Mayer–Vietoris strip is middle-exact. Using this, one
can prove the following structure theorem.

######  Theorem 10.11 ([9, 14, 34]).

If $\gamma$ is of Morse type, then the Mayer–Vietoris strip of $\gamma$ is
interval-decomposable, where each interval is a maximal rectangle in
$Q^{\prime}$.

######  Example 10.12.

The Mayer–Vietoris strip of the function in Fig. 10.2 (a) is shown in Fig.
10.3.

######  Remark 10.13 (Computation).

The barcode of the Mayer–Vietoris strip of $\gamma$ is fully determined by the
levelset barcodes of $\gamma$ in all homological degrees, and is also fully
determined by the _extended persistence_ barcodes [68] of $\gamma$ in all
homological degrees; this is implicit in [49]. It follows that in practice,
one can obtain the barcode of the Mayer–Vietoris strip by either a
(1-parameter) extended persistence computation or a zigzag persistence
computation. Conversely, the barcode of the Mayer-Vietoris strip determines
both the levelset and extended persistence barcodes.

######  Remark 10.14 (Stability).

The bottleneck distance (Definitions 3.6 and 6.4) generalizes
straightforwardly to barcodes consisting of rectangles in $Q^{\prime}$, and it
can be shown that the barcode of the Mayer–Vietoris strip is stable with
respect to this and the $L^{\infty}$-distance on functions [49, 9]. This is a
direct consequence of the stability of extended persistent homology [68].

######  Remark 10.15 (Interlevel Persistence with Open Intervals).

We now briefly discuss the corresponding interlevel persistence theory using
open intervals. Given a continuous function, $\gamma:W\to\mathbb{R}$, we can
define

$\mathcal{S}(\gamma)^{o}\colon U\to{\rm
Top},\qquad\mathcal{S}(\gamma)^{o}(s,t)=\\{x\in X\mid s<\gamma(x)<t\\}.$

By the Mayer–Vietoris sequence, $\mathcal{S}(\gamma)^{o}$ is middle-exact.
Thus by Theorem 10.8, if $H_{i}\mathcal{S}(\gamma)^{o}$ is p.f.d., then it is
interval-decomposable. Analogously, we can define a variant of the
Mayer–Vietoris strip using open intervals, and we have a structure theorem
[34, 9] which says that if this persistence module is p.f.d., then it is
interval-decomposable where the intervals are maximal rectangles. As in the
case of closed intervals, this decomposition is stable.

######  Remark 10.16 (Related Results).

See Bauer et al. [9] for a careful construction of the Mayer–Vietoris strip
for an arbitrary cohomology theory, and a corresponding structure theorem in
the p.f.d. setting. A structure theorem for middle-exact p.f.d. modules over
the Mayer–Vietoris strip not necessarily coming from functions can be found in
[34]. A related construction and structure theorem in the context of derived
sheaf theory can be found in [15].

$a_{6}$$a_{5}$$a_{4}$$a_{3}$$a_{2}$$a_{1}$$a_{6}$$a_{5}$$a_{4}$$a_{3}$$a_{2}$$a_{1}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$H_{0}$$H_{1}$Mayer–Vietoris
strip$a_{6}$$a_{5}$$a_{4}$$a_{3}$$a_{2}$$a_{1}$$a_{6}$$a_{5}$$a_{4}$$a_{3}$$a_{2}$$a_{1}$

Figure 10.3: Relative interlevel set modules
$H_{i}\mathcal{S}^{\bullet}(\gamma)$ and the corresponding Mayer–Vietoris
strip of the function $\gamma$ from Example 10.9; see Examples 10.10 and 10.12
and the surrounding discussion.

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