# Möbius Homology

Amit Patel  Department of Mathematics, Colorado State University  Primoz
Skraba  School of Mathematical Sciences, Queen Mary University London

###### Abstract

This paper introduces Möbius homology, a homology theory for representations
of finite posets into abelian categories. While the connection between poset
topology and Möbius functions is classical, we establish a direct connection
between poset topology and Möbius inversions. More precisely, the Möbius
homology categorifies the Möbius inversion because its Euler characteristic is
equal to the Möbius inversion of the dimension function of the representation.
We also introduce a homological version of Rota’s Galois Connection Theorem
which relates the Möbius homology over two posets connected by a Galois
connection. Our main application is to persistent homology over general
posets. We show that under one definition, the persistence diagram is an Euler
characteristic over a poset of intervals and hence Möbius homology is a
categorification of the persistence diagram. This provides a new invariant for
persistent homology over general posets. Finally, we use our homological
Rota’s Galois Connection Theorem to prove several results about the
persistence diagram.††This work is partially funded by the Leverhulme Trust
grant VP2-2021-008.

##  1 Introduction

Let $\mathcal{G}$ be an abelian group and $P$ a finite poset. For all
functions $m:P\to\mathcal{G}$, there exists a unique function $\partial
m:P\to\mathcal{G}$ such that for all $b\in P$,

$m(b)=\sum_{a:a\leq b}\partial m(a).$

The function $\partial m$ is called the _Möbius inversion_ of $m$. Often the
function $m$ is the shadow of a richer algebraic structure. That is, there is
often an abelian category $\mathcal{C}$ and a $P$-module, i.e., a functor,
$M:P\to\mathcal{C}$ whose _dimension function_ is $m$. Here, the dimension
function is the assignment to every $b\in P$ the representative
$\big{[}M(b)\big{]}$ of the object $M(b)$ in the Grothendieck group
$\mathcal{G}$ of $\mathcal{C}$. In this case, it is natural to ask for an
invariant of $M$ that decategorifies to the Möbius inversion $\partial m$.

This paper introduces a homology theory for $P$-modules we call _Möbius
homology_. Every module $M$ gives rise to a simplicial cosheaf over the order
complex of $P$. We define the Möbius homology of $M$ at an element $b\in P$,
denoted $H^{\downarrow}_{\ast}(b;M)$, as the homology of the simplicial
cosheaf localized to $b$. Our first result (Theorem 3.13) states that the
Euler characteristic of the Möbius homology is the Möbius inversion. That is,
for all $b\in P$,

$\partial m(b)=\sum_{d\geq 0}(-1)^{d}\big{[}H^{\downarrow}_{d}(b;M)\big{]}.$

Our second result (Theorem 5.3) is a homological version of Rota’s Galois
connection theorem. Rota’s theorem is an important tool for studying and
computing Möbius inversions, and likewise we use our result to prove several
properties of Möbius homology.

#### Previous Work

Poset topology is the study of the order complex associated to a poset. It
originates in Rota’s seminal 1964 paper on Möbius functions [6] and is now a
rich subject motivated by questions from diverse fields [13]. Interest in the
Möbius function stems from its connection to the Euler characteristic of the
order complex [11]. However, a topological approach to the Möbius inversion
was never fully developed. To the best of our knowledge, Backlawski was the
first to study the simplicial sheaf associated to a $P$-module in his
definition of Whitney homology for geometric lattices [1]. In a followup
paper, Backlawski studies Galois connections between these sheaves but falls
short of making a clear connection to Möbius inversions [2].

#### Application: Persistent Homology

Our discovery of Möbius homology is motivated by questions in persistent
homology. Traditionally, persistent homology starts with a family of
topological spaces parameterized by a finite, totally ordered poset $P$. Apply
homology using field coefficients and the result is a $P$-module of vector
spaces $M_{\ast}:P\to{\mathrm{Vec}}$, one for each dimension. The _persistence
diagram_ of each $M_{\ast}$ is a complete combinatorial invariant that encodes
the birth and death of its generators along $P$.

There are many equivalent definitions of the persistence diagram. The
definition proposed by Cohen-Steiner, Edelsbrunner, and Harer uses an
inclusion-exclusion formula [5]. From this viewpoint, the persistence diagram
of $M_{\ast}$ is the Möbius inversion of a ${\mathbb{Z}}$-valued function on
the poset of intervals of $P$ [10]. There are now two well developed
generalizations of this definition to modules over any finite poset. The
persistence diagram of Kim and Mémoli is a Möbius inversion of the rank
invariant of the module [8] whereas the persistence diagram of Gülem,
McCleary, and Patel is the Möbius inversion of the birth-death invariant of
the module [9, 7]. When $P$ is not totally ordered, neither are complete
invariants of the module. We show how Möbius homology categorifies the birth-
death approach and is a stronger invariant than simply the Möbius inversion.
Finally, we demonstrate how the homological version of Rota’s Galois
connection theorem helps in computing persistent Möbius homology.

##  2 Background

This section provides a brief introduction to Möbius inversions and simplicial
cosheaves. We intentionally avoid the language of derived functors in our
treatment of cosheaf homology in the hope of reaching a wider audience.

###  2.1 Möbius Inversion

Let $P$ be a finite poset. For $a\leq c$, the (closed) _interval_
$[a,c]:=\\{b\in P:a\leq b\leq c\\}$. Let ${\mathrm{Int}}\,P$ be the set of all
intervals. The _${\mathbb{Z}}$ -incidence algebra_ on $P$, denoted
${\mathrm{Inc}}(P)$, is the set of all functions
$\alpha:{\mathrm{Int}}\,P\to{\mathbb{Z}}$ with three operations: scaling,
addition, and multiplication. The first two are clear. Multiplication of two
functions $\alpha$ and $\beta$ is

$(\alpha\ast\beta)[a,c]:=\sum_{b:a\leq b\leq c}\alpha[a,b]\cdot\beta[b,c].$

The multiplicative identity is

${\mathbbm{1}}[a,b]=\begin{cases}1&\text{if $a=b$}\\\
0&\text{otherwise}.\end{cases}$

The _zeta function_ is $\zeta[a,b]=1$, for all $a\leq b$. The zeta function is
invertible, and its inverse, denoted $\mu$, is the _Möbius function_.

######  Definition 2.1:

Let $P$ be a finite poset and $\mathcal{G}$ an abelian group. The Möbius
inversion of any function $f:P\to\mathcal{G}$ is the function $\partial
f:P\to\mathcal{G}$ defined as

$\partial f(c)=\sum_{b:b\leq c}f(c)\cdot\mu[b,c].$ (1)

The Möbius inversion is the unique function satisfying the following equation
for all $c\in P$:

$\displaystyle\sum_{b:b\leq c}\partial f(b)$ $\displaystyle=\sum_{b:b\leq
c}\partial f(b)\cdot\zeta[b,c]$ $\displaystyle=\sum_{b:b\leq
c}\left(\sum_{a:a\leq b}f(a)\cdot\mu[a,b]\right)\cdot\zeta[b,c]$
$\displaystyle=\sum_{a:a\leq c}f(a)\left(\sum_{b:a\leq b\leq
c}\mu[a,b]\cdot\zeta[b,c]\right)$ $\displaystyle=\sum_{b:b\leq
c}f(b)\cdot{\mathbbm{1}}[b,c]$ $\displaystyle=f(c).$

That is, $\partial f$ is the derivative of $f$.

###  2.2 Simplicial Cosheaves

Let $K$ be a finite simplicial complex. We write $\tau\geq\sigma$ to mean
$\tau$ is a coface of $\sigma$ and $\tau>_{1}\sigma$ to mean
$\dim\tau=\dim\sigma+1$. We also think of $K$ as a category where
$\tau\to\sigma$ whenever $\tau\geq\sigma$. For the moment, let $\mathcal{C}$
be any abelian category.

######  Definition 2.2:

A simplicial cosheaf over $K$ valued in $\mathcal{C}$ is a functor
$\underline{{A}}:K\to\mathcal{C}$.

We now construct the chain complex $C_{\bullet}(K;\underline{{A}})$ associated
to a simplicial cosheaf $\underline{{A}}$. Orient every simplex of $K$. The
$d$-th chain object is

$C_{d}(K;\underline{{A}}):=\bigoplus_{\sigma:\dim\sigma=d}\underline{{A}}(\sigma).$

For $\tau>_{1}\sigma$, restrict the orientation on $\tau$ to an orientation on
$\sigma$. Write $[\tau:\sigma]=1$ if the restriction agrees with the
orientation on $\sigma$, and write $[\tau:\sigma]=-1$ if the restriction
disagrees. The boundary operator $\partial_{d}:C_{d}(K;\underline{{A}})\to
C_{d-1}(K;\underline{{A}})$ is generated summand-wise as follows. For a
$d$-simplex $\tau$, let $\iota_{\tau}:\underline{{A}}(\tau)\to
C_{d}(K;\underline{{A}})$ be the canonical morphism into the direct sum, and
let $\pi_{\tau}:C_{d}(K;\underline{{A}})\to\underline{{A}}(\tau)$ be the
canonical morphism out of the direct sum. Define the restriction of
$\partial_{d}$ to $\tau$ as

$\partial_{d}|_{\tau}:=\sum_{\sigma:\tau>_{1}\sigma}[\tau:\sigma]\cdot\big{(}\iota_{\sigma}\circ\underline{{A}}(\tau\geq\sigma)\circ\pi_{\tau}\big{)}.$

The full boundary operator is the sum
$\partial_{d}:=\sum_{\tau:\dim\tau=d}\partial_{d}|_{\tau}.$ Of course, the
ordinary boundary operator applied twice to any simplex is zero. This fact
combined with commutativity of $\underline{{A}}$, implies
$\partial_{d-1}\circ\partial_{d}=0$. Thus, we have a chain complex

${C_{\bullet}(K;\underline{{A}}):\cdots}$${C_{2}(K;\underline{{A}})}$${C_{1}(K;\underline{{A}})}$${C_{0}(K;\underline{{A}})}$${0.}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

######  Definition 2.3:

The $d$-th cosheaf homology of $\underline{{A}}$, denoted
$H_{d}(K;\underline{{A}})$, is the $d$-th homology object of the chain complex
$C_{\bullet}(K;\underline{{A}})$.

We are also interested in relative simplicial cosheaf homology. Let
$L\subseteq K$ be a subcomplex, and let $\underline{{B}}:L\to\mathcal{C}$ be
the restriction of the cosheaf $\underline{{A}}$ to $L$. The associated
relative chain complex is

${C_{\bullet}(K,L;\underline{{A}}):\cdots}$${\dfrac{C_{2}(K;\underline{{A}})}{C_{2}(L;\underline{{B}})}}$${\dfrac{C_{1}(K;\underline{{A}})}{C_{1}(L;\underline{{B}})}}$${\dfrac{C_{0}(K;\underline{{A}})}{C_{0}(L;\underline{{B}})}}$${0.}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

######  Definition 2.4:

The $d$-th relative cosheaf homology of $\underline{{A}}$ with respect to the
subcomplex $L\subseteq K$, denoted $H_{d}(K,L;\underline{{A}})$, is the $d$-th
homology object of the chain complex $C_{\bullet}(K,L;\underline{{A}})$.

We now explore the Euler characteristic of a simplicial cosheaf valued in an
abelian category $\mathcal{C}$. Assume $\mathcal{C}$ is equivalent to a small
category. Denote by $[A]$ the isomomorphism class of an object $A$ in
$\mathcal{C}$.

######  Definition 2.5:

The Grothendieck group of $\mathcal{C}$, denoted $K(\mathcal{C})$, is the free
abelian group generated by the set of isomorphism classes of objects in
$\mathcal{C}$ with the relation $[B]=[A]+[C]$ for every short exact sequence
$0\to A\to B\to C\to 0$ in $\mathcal{C}$.

We now give three working examples of abelian categories along with their
Grothendieck groups. The calculation of each group is left to the reader.

######  Example 2.6:

If ${\mathrm{Vec}}$ is the category of finite dimensional
${\mathsf{k}}$-vector spaces, for some fixed field ${\mathsf{k}}$, then
$K({\mathrm{Vec}})\cong{\mathbb{Z}}$. For an object $A$, the corresponding
element $[A]\in K({\mathrm{Vec}})$ is the dimension of $A$.

######  Example 2.7:

Let $\mathrm{End}(\mathbb{C})$ be the category of endomorphisms on finite
dimensional complex vector spaces. That is, an object is an endomorphism
$\phi:\mathbb{C}^{m}\to\mathbb{C}^{m}$, and a morphism from an object
$\phi:\mathbb{C}^{m}\to\mathbb{C}^{m}$ to an object
$\psi:\mathbb{C}^{n}\to\mathbb{C}^{n}$ is a linear map
$f:\mathbb{C}^{m}\to\mathbb{C}^{n}$ such that $f\circ\phi=\psi\circ f$. Its
Grothendieck group, $K\big{(}\mathrm{End}(\mathbb{C})\big{)}$, is isomorphic
to $\oplus_{\lambda\in\mathbb{C}}{\mathbb{Z}}$. For an object $A$, the
corresponding element $[A]\in K\big{(}\mathrm{End}(\mathbb{C})\big{)}$ is the
map $\mathbb{C}\to{\mathbb{Z}}$ that assigns to every $\lambda\in\mathbb{C}$
the number of times $\lambda$ occurs as an eigenvalue of $A$.

######  Example 2.8:

Let $\mathrm{FinAb}$ be the category of finite abelian groups. Its
Grothendieck group, $K(\mathrm{FinAb})$, is isomorphic to $\oplus_{p\text{
prime}}{\mathbb{Z}}$. For an object $A$, the corresponding element $[A]\in
K(\mathrm{FinAb})$ is the assignment to each prime the number of times it
occurs in the prime decomposition of $A$. For example, if $A$ is
${\mathbb{Z}}/16{\mathbb{Z}}$, then $[A]$ is the assignment $2\mapsto 4$.

######  Definition 2.9:

The Euler characteristic of a simplicial cosheaf
$\underline{{A}}:K\to\mathcal{C}$, denoted $\chi(K;\underline{{A}})$, is the
following element of $K(\mathcal{C})$:

$\chi(K;\underline{{A}}):=\sum_{d\geq
0}\sum\limits_{\sigma:\dim(\sigma)=d}(-1)^{d}\big{[}\underline{{A}}(\sigma)\big{]}.$
(2)

If we think of $\big{[}H_{i}(K;\underline{{A}})\big{]}\in K(\mathcal{C})$ as
the $i$-th “Betti number” of $\underline{{A}}$, then the Euler characteristic
of $\underline{{A}}$ is the familiar signed sum of Betti numbers as follows.

######  Proposition 2.10:

$\chi(K;\underline{{A}})=\sum_{d\geq
0}(-1)^{d}\big{[}H_{d}(K;\underline{{A}})\big{]}.$

###### Proof.

Recall the chain complex $C_{\bullet}(K;\underline{{A}})$. Let
$Z_{d}(K;\underline{{A}}):={\mathrm{ker}}\,\partial_{d}$ and
$B_{d}(K;\underline{{A}}):={\mathrm{im}}\,\partial_{d+1}$. Consider the
following short exact sequences:

${0}$${Z_{d}(K;\underline{{A}})}$${C_{d}(K;\underline{{A}})}$${B_{d-1}(K;\underline{{A}})}$${0}$${0}$${B_{d}(K;\underline{{A}})}$${Z_{d}(K;\underline{{A}})}$${H_{d}(K;\underline{{A}})}$${0.}$$\scriptstyle{\partial_{d}}$

The two sequences imply
$\big{[}C_{d}(K;\underline{{A}})\big{]}=\big{[}Z_{d}(K;\underline{{A}})\big{]}+\big{[}B_{d-1}(K;\underline{{A}})\big{]}$
and
$\big{[}H_{d}(K;\underline{{A}})\big{]}=\big{[}Z_{d}(K;\underline{{A}})\big{]}-\big{[}B_{d}(K;\underline{{A}})\big{]}$
in $K(\mathcal{C})$. Thus

$\displaystyle\chi(K;\underline{{A}})$ $\displaystyle=\sum_{d\geq
0}\sum_{\sigma:\dim\sigma=d}(-1)^{d}\big{[}\underline{{A}}(\sigma)\big{]}$
$\displaystyle=\sum_{d\geq 0}(-1)^{d}\big{[}C_{d}(K;\underline{{A}})\big{]}$
$\displaystyle=\sum_{d\geq
0}(-1)^{d}\big{[}Z_{d}(K;\underline{{A}})\big{]}+\sum_{d\geq
0}(-1)^{d}\big{[}B_{d-1}(K;\underline{{A}})\big{]}$ $\displaystyle=\sum_{d\geq
0}(-1)^{d}\left(\big{[}Z_{d}(K;\underline{{A}})\big{]}-\big{[}B_{d}(K;\underline{{A}})\big{]}\right)$
$\displaystyle=\sum_{d\geq 0}(-1)^{d}\big{[}H_{d}(K;\underline{{A}})\big{]}.$

∎

######  Definition 2.11:

The relative Euler characteristic of $\underline{{A}}$ with respect to the
subcomplex $L\subseteq K$, denoted $\chi(K,L;\underline{{A}})$, is

$\chi(K,L;\underline{{A}}):=\sum_{d\geq 0}\sum_{\sigma\in K\setminus
L:\dim(\sigma)=d}(-1)^{d}\big{[}\underline{{A}}(\sigma)\big{]}.$

Similarly, the relative Euler characteristic is the signed sum of relative
Betti numbers.

######  Proposition 2.12:

$\chi(K,L;\underline{{A}})=\sum_{d\geq
0}(-1)^{d}\big{[}H_{d}(K,L;\underline{{A}})\big{]}.$

##  3 Möbius Inversions and Euler Characteristics

We define two invariants associated to a $P$-module valued in an abelian
category. The first is the Möbius inversion of the module’s _dimension
function_. The second is _Möbius homology_. Our main theorem (Theorem 3.13)
shows that the Möbius inversion is the Euler characteristic of the Möbius
homology.

######  Definition 3.1:

Let $P$ be a finite poset and $\mathcal{C}$ any small abelian category. A
$P$-module is a functor $M:P\to\mathcal{C}$.

###  3.1 Dimension Function

The first invariant considers only the objects of $M$ while forgetting all its
morphisms.

######  Definition 3.2:

The dimension function of $M$ is the function $m:P\to K(\mathcal{C})$ that
assigns to every $a\in P$ the element $\big{[}M(a)\big{]}\in K(\mathcal{C})$.

We now look at two similar examples of the dimension function $m$ and its
Möbius inversion $\partial m$ in two different categories.

######  Example 3.3:

Consider the poset $P$ and the two $P$-modules $M$ and $N$ in Figure 1. The
dimension function $m:P\to{\mathbb{Z}}$ for both modules are the same. The
Möbius inversion $\partial m$ is $\partial m(b)=1$, $\partial m(a)=0$, and
$\partial m(c)=1$.

${M}$${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${N}$${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${P}$${b}$${a}$${c}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{\leq}$$\scriptstyle{\geq}$ Figure 1: Poset $P$
along with two $P$-modules $M$ and $N$ valued in ${\mathrm{Vec}}$

######  Example 3.4:

Consider the poset $P$ and the two $P$-modules $M$ and $N$ in Figure 2. Recall
an element of $K\big{(}\mathrm{End}(\mathbb{C})\big{)}$ is the assignment to
every complex number a multiplicity. The dimension function $m:P\to
K(\mathcal{C})$ for both modules are the same. That is,
$m(b)=\\{\lambda\mapsto 1\\}$, $m(a)=\\{\lambda\mapsto 2\\}$, and
$m(c)=\\{\lambda\mapsto 2\\}$. The Möbius inversion $\partial m$ is $\partial
m(b)=\\{\lambda\mapsto 1\\}$, $\partial m(a)=0$, and $\partial
m(c)=\\{\lambda\mapsto 1\\}$.

${M}$${\mathbb{C}}$${\mathbb{C}^{2}}$${\mathbb{C}}$${N}$${\mathbb{C}}$${\mathbb{C}^{2}}$${\mathbb{C}}$${P}$${b}$${a}$${c}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda&0\\\
0&\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda&0\\\
0&\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}\lambda\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{\leq}$$\scriptstyle{\geq}$ Figure 2: Poset $P$
along with two $P$-modules $M$ and $N$ valued in $\mathrm{End}(\mathbb{C})$

###  3.2 Order cosheaf

For our second invariant, we start by building a simplicial complex. A chain
of _length_ $n$ in $P$ is a sequence $a_{0}<\cdots<a_{n}$ of $n+1$ distinct
elements. A _subchain_ of a chain $a_{0}<\cdots<a_{n}$ is a chain obtained by
deleting any number of its elements.

######  Definition 3.5:

The order complex of $P$, denoted $\Delta P$, is the simplicial complex whose
(open) $i$-simplices are chains of length $i$. A simplex $a_{0}<\cdots<a_{j}$
is a coface of a simplex $b_{0}<\cdots<b_{i}$ if the later is a subchain of
the former.

Given a simplex $\sigma\in\Delta P$, denote by $\min\sigma$ and $\max\sigma$
the minimal and maximal element, respectively, of its corresponding chain.
Note that for every $\tau\geq\sigma$ in $\Delta P$, $\min\tau\leq\min\sigma$
and $\max\sigma\leq\max\tau$.

######  Definition 3.6:

The order cosheaf of $M$ is the simplicial cosheaf $\underline{{M}}:\Delta
P\to\mathcal{C}$ defined as follows. For $\sigma\in\Delta P$, let
$\underline{{M}}(\sigma):=M(\min\sigma).$ For $\tau\geq\sigma$, the morphism
$\underline{{M}}(\tau\geq\sigma):\underline{{M}}(\tau)\to\underline{{M}}(\sigma)$
is $M(\min\tau\leq\min\sigma)$.

We now consider a local order cosheaf homology.

######  Definition 3.7:

The lower complex of $b\in P$ is the subcomplex $\Delta P_{\leq
b}:=\\{\sigma\in\Delta P:\max\sigma\leq b\\}$ of $\Delta P$. The strict lower
complex of $b\in P$ is the subcomplex $\Delta P_{<b}:=\big{\\{}\sigma\in\Delta
P:\max\sigma<b\big{\\}}$ of $\Delta P$.

######  Definition 3.8:

The Möbius chain complex module of $M:P\to\mathcal{C}$, denoted
$C^{\downarrow}_{\bullet}M:P\to\mathrm{Ch}(\mathcal{C})$, is the assignment to
every $a\in P$ the relative chain complex $C_{\bullet}(\Delta P_{\leq
a},\Delta P_{<a};\underline{{M}})$ and to every $a\leq b$ the morphism

$C_{\bullet}\big{(}\Delta P_{\leq a},\Delta
P_{<a};\underline{{M}}\big{)}\rightarrow C_{\bullet}\big{(}\Delta P_{\leq
b},\Delta P_{<b};\underline{{M}}\big{)}$

induced by the inclusion of the pair of complexes. Apply the homology functor
and the result is the Möbius homology module of $M$, denoted
$H^{\downarrow}_{\ast}M:P\to\mathcal{C}$.

The internal morphisms of the Möbius homology module $H^{\downarrow}_{\ast}M$
are all zero. This is because for every $a<b$, the homology
$H^{\downarrow}_{\ast}M(a)$ is supported over the set of simplices $\Delta
P_{\leq a}\setminus\Delta P_{<a}$ which is a subset of $\Delta P_{<b}$.

######  Example 3.9:

Consider the poset $P$ and the two $P$-modules $M,N:P\to{\mathrm{Vec}}$ in
Figure 1. The table below lists the Möbius homology of $M$ and $N$ at each
element. Notice the Möbius inversion is the Euler characteristic of the Möbius
homology.

| $b$ | $a$ | $c$
---|---|---|---
$H^{\downarrow}_{0}M$ | ${\mathsf{k}}$ | ${\mathsf{k}}$ | ${\mathsf{k}}$
$H^{\downarrow}_{1}M$ | $0$ | ${\mathsf{k}}$ | $0$
$H^{\downarrow}_{0}N$ | ${\mathsf{k}}$ | $0$ | ${\mathsf{k}}$
$H^{\downarrow}_{1}N$ | $0$ | $0$ | $0$

######  Example 3.10:

Consider the poset $P$ and the two $P$-modules
$M,N:P\to\mathrm{End}(\mathbb{C})$ in Figure 2. The table below lists the
Möbius homology of $M$ and $N$ at each element. Note that the Möbius inversion
is the Euler characteristic of the Möbius homology.

| $b$ | $a$ | $c$
---|---|---|---
$H^{\downarrow}_{0}M$ | $\begin{pmatrix}\lambda\end{pmatrix}$ | $\begin{pmatrix}\lambda\end{pmatrix}$ | $\begin{pmatrix}\lambda\end{pmatrix}$
$H^{\downarrow}_{1}M$ | $0$ | $\begin{pmatrix}\lambda\end{pmatrix}$ | $0$
$H^{\downarrow}_{0}N$ | $\begin{pmatrix}\lambda\end{pmatrix}$ | $0$ | $\begin{pmatrix}\lambda\end{pmatrix}$
$H^{\downarrow}_{1}N$ | $0$ | $0$ | $0$

######  Example 3.11:

Consider the cusp map $f:\mathbb{R}^{2}\to\mathbb{R}^{2}$ in Figure 3(a). The
critical values of $f$ partition the codomain into five cells: one 0-cell, two
$1$-cells, and two $2$-cells. Let $P$ be the face poset of this partition and
consider the module $M:P\to{\mathrm{Vec}}$ in Figure 3(b) that assigns to
every cell its fiberwise zero-dimensional homology to every face relation the
natural map between the fibers. The table below lists the Möbius homology of
$M$ at each cell. Notice all non-trivial entries occur at the codimension of
the cell. Möbius homology of $M$ identifies the normal component of $f$ to
each cell.

| $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---
$H^{\downarrow}_{0}M$ | ${\mathsf{k}}^{3}$ | ${\mathsf{k}}$ | $0$ | $0$ | $0$
$H^{\downarrow}_{1}M$ | $0$ | $0$ | ${\mathsf{k}}^{2}$ | ${\mathsf{k}}^{2}$ | $0$
$H^{\downarrow}_{2}M$ | $0$ | $0$ | $0$ | $0$ | ${\mathsf{k}}$

(a) Cusp map

$k^{3}$$k$$k$$k^{2}$$k^{2}$$\begin{bmatrix}1&1&0\\\ 0&0&1\end{bmatrix}$
$\begin{bmatrix}1&0&0\\\ 0&1&1\end{bmatrix}$ $\begin{bmatrix}0\\\
1\end{bmatrix}$ $\begin{bmatrix}1\\\ 0\end{bmatrix}$
$\begin{bmatrix}1\end{bmatrix}$ $\begin{bmatrix}1&1\end{bmatrix}$
$\begin{bmatrix}1&1\end{bmatrix}$ (b) $H_{0}$ module

Figure 3: Module over face poset for the cusp map

######  Example 3.12:

Consider the modules $M$ and $N$ of subgroups of ${\mathbb{Z}}/4{\mathbb{Z}}$
and ${\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$,
respectively, in Figures 4(a) and 4(b). Both modules are valued in
$\mathrm{FinAb}$. The table below lists the Möbius homology of $M$ and $N$ at
the top element.

| $\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\oplus\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}$ | $\frac{{\mathbb{Z}}}{4{\mathbb{Z}}}\vspace{2pt}$
---|---|---
$H^{\downarrow}_{0}$ | $0$ | $\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\vspace{2pt}$
$H^{\downarrow}_{1}$ | $\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}$ | $0\vspace{2pt}$
$H^{\downarrow}_{2}$ | $0$ | $0$

${\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\oplus\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}}$${\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}}$${\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}}$${\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}}$${0}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$ (a) $M:P\to\mathrm{FinAb}$

${\frac{{\mathbb{Z}}}{4{\mathbb{Z}}}}$${\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}}$${0}$$\scriptstyle{\footnotesize{\begin{pmatrix}2\end{pmatrix}}}$
(b) $N:Q\to\mathrm{FinAb}$

Figure 4: Modules of subgroups

###  3.3 Main Theorem

The following theorem relates Möbius inversions to Möbius homology.

######  Theorem 3.13:

Let $m:P\to K(\mathcal{C})$ be the dimension function of a $P$-module
$M:P\to\mathcal{C}$. Then for all $b\in P$,

$\partial m(b)=\sum_{d\geq 0}(-1)^{d}\big{[}H^{\downarrow}_{\ast}M(b)\big{]}.$

The proof is based on the following observation.

######  Lemma 3.14 (Philip Hall’s Theorem, Prop 3.8.5 [12]):

For $a\leq b$ in $P$, let $n_{i}$ be the number of chains of length $i$
starting at $a$ and ending at $b$. Note that $n_{1}=1$ whenever $a<b$. Then,

$\mu[a,b]=\sum_{i>0}(-1)^{i}\cdot n_{i}.$

###### Proof of Theorem 3.13.

Let $\sigma(a,b)$ be the simplex $a\leq b$ in $\Delta P$. It is a $0$-simplex
if $a=b$ and a $1$-simplex otherwise. Define its (open) star as

${\mathrm{st}}\,\sigma(a,b):=\big{\\{}\tau\in\Delta
P:\min\tau=a,\max\tau=b\big{\\}}.$

Let $n_{i}$ be the number of chains of length $i>0$ starting at $a$ and ending
at $b$. In other words, $n_{i}$ is the number of $i$-simplices in
${\mathrm{st}}\,\sigma(a,b)$. For convenience,

$\tilde{\chi}\big{(}{\mathrm{st}}\,\sigma(a,b)\big{)}:=\sum_{i>0}(-1)^{i}\cdot
n_{i}.$

When $a\neq a^{\prime}$, the intersection
${\mathrm{st}}\,\sigma(a,b)\cap{\mathrm{st}}\,\sigma(a^{\prime},b)$ is empty.
Further, every simplex $\tau\in\Delta P_{\leq b}\setminus\Delta P_{<b}$
belongs to a unique star, namely ${\mathrm{st}}\,\sigma(\min\tau,b)$.
Therefore,

$\displaystyle\partial m(b)$ $\displaystyle=\sum_{a:a\leq b}m(a)\cdot\mu[a,b]$
by Equation (1) $\displaystyle=\sum_{a:a\leq
b}\big{[}M(a)\big{]}\cdot\mu[a,b]$ by Definition 3.2
$\displaystyle=\sum_{a:a\leq
b}\big{[}M(a)\big{]}\cdot\tilde{\chi}\big{(}{\mathrm{st}}\,\sigma(a,b)\big{)}$
by Lemma 3.14 $\displaystyle=\chi\big{(}\Delta P_{\leq b},\Delta
P_{<b};\underline{{M}}\big{)}$ by Definition 2.11 $\displaystyle=\sum_{d\geq
0}(-1)^{d}\big{[}H^{\downarrow}_{\ast}M(b)\big{]}$ $\displaystyle\text{by
Lemma \ref{prop:relative_reuler}}.$

∎

Note that the only simplices involved in the Möbius chain complex
$C^{\downarrow}_{\ast}M(b)$ are $\Delta P_{\leq b}\setminus\Delta P_{<b}$.
Further, for $a\neq b$, the intersection $\Delta P_{\leq a}\setminus\Delta
P_{<a}\cap\Delta P_{\leq b}\setminus\Delta P_{<b}$ is empty. This, combined
with Proposition 2.10 leads to the following corollary of our main theorem.

######  Corollary 3.15:

Let $m:P\to K(\mathcal{C})$ be the dimension function of a $P$-module
$M:P\to\mathcal{C}$. Then,

$\sum_{a\in P}\partial m(a)=\sum_{d\geq 0}(-1)^{d}\big{[}H_{d}(\Delta
P;\underline{{M}})\big{]}.$

We have seen how Möbius homology is a richer invariant than the Möbius
inversion. However, Möbius homology is not a complete invariant of a
$P$-module.

######  Example 3.16:

Choose an element $\mu\in{\mathsf{k}}$ and consider the module $M_{\mu}$ in
Figure 5. For the choice of a second element $\nu\in{\mathsf{k}}$, the two
modules $M_{\mu}$ and $M_{\nu}$ are not isomorphic [4]. However, according to
our calculations, the element-wise Möbius homologies for the two modules are
isomorphic. Thus, Möbius homology is not a complete invariant.

${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${0}$${0}$${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$$\scriptstyle{\footnotesize\begin{pmatrix}1\\\
0\end{pmatrix}}$$\scriptstyle{\footnotesize\begin{pmatrix}1\end{pmatrix}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
\mu\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&1\\\
1&\mu\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0&1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&1\end{pmatrix}}}$
Figure 5: $M_{\mu}$ is a $P$-module in ${\mathrm{Vec}}$ for every
$\mu\in{\mathsf{k}}$.

##  4 Galois Connections

This section reviews a fundamental topological property of a Galois
connection. We show that a Galois connection between two posets induces a
chain homotopy equivalence between their two order complexes. This will be
useful in the next section.

######  Definition 4.1:

A (monotone) Galois connection from a poset $P$ to a poset $Q$, written
$f:P\leftrightarrows Q:g$, are monotone functions $f:P\to Q$ and $g:Q\to P$
such that $f(a)\leq x$ if and only if $a\leq g(x).$ The function $f$ is called
the _left adjoint_ of $g$ and $g$ is called the _right adjoint_ of $f$

######  Example 4.2:

Consider the following Galois connections $f:P\leftrightarrows Q:g$ where $f$
is given by the red dashed arrows and $g$ by the blue dashed arrows:

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}P}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}P}$

######  Lemma 4.3:

Every Galois connection $f:P\leftrightarrows Q:g$ satisfies the following
properties:

  1. (i)

For all $a\in P$, $a\leq g\circ f(a)$.

  2. (ii)

For all $x\in Q$, $f\circ g(x)\leq x$.

  3. (iii)

If $f(a)=x$, then $f\circ g(x)=x$.

  4. (iv)

$f$ is surjective if and only if $g$ is injective.

  5. (v)

$f$ is injective if and only if $g$ is surjective.

###### Proof.

  1. (i)

In the axiom of a Galois connection above, let $x=f(a)$. Then the inequality
$f(a)\leq f(a)$ implies $a\leq g\circ f(a)$.

  2. (ii)

In the axiom of a Galois connection above, let $a=g(x)$. Then the inequality
$g(x)\leq g(x)$ implies $f\circ g(x)\leq x$.

  3. (iii)

The inequality $f(a)\leq x$ implies $a\leq g(x)$. Since $f$ is monotone,
$x=f(a)\leq f\circ g(x)$. Combined with (ii), $f\circ g(x)=x$.

  4. (iv)

Assume $f$ is surjective. Fix distinct elements $x,y\in Q$, and suppose, for
the sake of contradiction, $g(x)=g(y)$. Since $f$ is surjective, $f(a)=x$ and
$f(b)=y$ for some $a,b\in P$. By (iii), $f\circ g(x)=x$ and $f\circ g(y)=y$.
Thus $x=y$ contradicting $x\neq y$.

Assume $f$ is not surjective. Then there is an $x\in Q$ such that
$f^{-1}(x)=\emptyset$. Let $a=g(x)$ and let $y=f(a)$. By (iii), $x\neq y$. The
inequality $f(a)\leq y$ implies $a\leq g(y)$. By (ii), $f\circ g(x)=y\leq x$.
Apply $g$ to get $g(y)\leq g(x)=a$. Thus $g$ is not injective because
$g(x)=g(y)=a$.

  5. (v)

Assume $f$ is injective. If $a\in P$ and $x=f(a)$, then, by (iii), $f\circ
g(x)=x$. Thus $g(x)=a$ making $g$ surjective.

Assume $f$ is not injective. Then there are are distinct elements $a,b\in P$
such that $f(a)=f(b)=x$. By (iii), $g(x)$ is an element of $f^{-1}(x)$, which
contains both $a$ and $b$. Thus $g$ cannot be surjective.

∎

Fix a Galois connection $f:P\leftrightarrows Q:g$. The function $f$ takes a
chain $a_{0}<\cdots<a_{n}$ in $P$ to the chain $f(a_{0})\leq\cdots\leq
f(a_{n})$ in $Q$. Thus, $f$ gives rise to a simplicial map $\Delta f:\Delta
P\to\Delta Q$. Similarly, $g$ gives rise to a simplicial map $\Delta g:\Delta
Q\to\Delta P$. Denote by $C_{\bullet}(\Delta P;{\mathbb{Z}})$ and
$C_{\bullet}(\Delta Q;{\mathbb{Z}})$ the standard simplicial chain complexes
using ${\mathbb{Z}}$-coefficients associated to $\Delta P$ and $\Delta Q$,
respectively. The simplicial maps $\Delta f$ and $\Delta g$ induce the chain
maps

$\displaystyle\Delta f_{\bullet}:C_{\bullet}(\Delta P;{\mathbb{Z}})\to
C_{\bullet}(\Delta Q;{\mathbb{Z}})$ $\displaystyle\Delta
g_{\bullet}:C_{\bullet}(\Delta Q;{\mathbb{Z}})\to C_{\bullet}(\Delta
P;{\mathbb{Z}}).$

######  Proposition 4.4:

The compositions $\Delta g\circ\Delta f$ and $\Delta f\circ\Delta g$ are chain
homotopic to the identity on $C_{\bullet}(\Delta P;{\mathbb{Z}})$ and
$C_{\bullet}(\Delta Q;{\mathbb{Z}})$, respectively. In other words, the two
chain complexes are chain homotopy equivalent and therefore their homologies
are isomorphic.

###  4.1 Proof of Proposition 4.4

We prove the proposition in two steps below. Lemma 4.5 proves $\Delta
g\circ\Delta f$ is chain homotopic to ${\mathbbm{1}}_{C_{\bullet}(\Delta
P;{\mathbb{Z}})}$, and Lemma 4.6 proves $\Delta f\circ\Delta g$ is chain
homotopic to ${\mathbbm{1}}_{C_{\bullet}(\Delta Q;{\mathbb{Z}})}$.

#### Left Chain Homotopy

Consider an $n$-simplex in $\Delta P$ given as a chain $a_{0}<\cdots<a_{n}$.
Let $b_{i}=g\circ f(a_{i})$. The following relations follow from Lemma 4.3 (i)
combined with the fact that $g\circ f$ is monotone:

${b_{0}}$${b_{1}}$${\cdots}$${b_{n-1}}$${b_{n}}$${a_{0}}$${a_{1}}$${\cdots}$${a_{n-1}}$${a_{n}.}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{<}$$\scriptstyle{\leq}$$\scriptstyle{<}$$\scriptstyle{\leq}$$\scriptstyle{<}$$\scriptstyle{<}$$\scriptstyle{<}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$

The sequence $b_{0}\leq\cdots\leq b_{n}$ may contain repetitions in which case
it is not an $n$-simplex. Consider the linear map $\phi_{n}:C_{n}(\Delta
P;{\mathbb{Z}})\to C_{n+1}(\Delta P;{\mathbb{Z}})$ generated by the following
simplex-wise assignment:

$\phi_{n}(a_{0}<\cdots<a_{n}):=\sum_{i=0}^{n}(-1)^{i}a_{0}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq b_{n}.$ (3)

Identify every term on the right-hand side that is not an $(n+1)$-simplex to
$0$.

######  Lemma 4.5:

The composition $\Delta g\circ\Delta f:C_{\bullet}(\Delta P;{\mathbb{Z}})\to
C_{\bullet}(\Delta P;{\mathbb{Z}})$ is chain homotopic to the identity.

###### Proof.

For any $n$-simplex $a_{0}<\cdots<a_{n}$ in $\Delta P$ and $b_{i}=g\circ
f(a_{i})$ as above, we show

$(b_{0}\leq\cdots\leq
b_{n})-(a_{0}<\cdots<a_{n})=\partial\circ\phi(a_{0}<\cdots<a_{n})+\phi\circ\partial(a_{0}<\cdots<a_{n}).$

Expand the first term to

$\displaystyle\partial\circ\phi(a_{0}<\cdots<a_{n})=$
$\displaystyle\;\partial\left(\sum_{i=0}^{n}(-1)^{i}a_{0}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq b_{n}\right)$ $\displaystyle=$
$\displaystyle\;\sum_{i=0}^{n}(-1)^{i}\Bigg{(}\sum_{j=0}^{i}(-1)^{j}a_{0}<\cdots<\widehat{a}_{j}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq b_{n}$
$\displaystyle+\sum_{j=i}^{n}(-1)^{j+1}a_{0}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq\widehat{b}_{j}\leq\cdots\leq b_{n}\Bigg{)}.$

Expand the second term to

$\displaystyle\phi\circ\partial(a_{0}<\cdots<a_{n})=$
$\displaystyle\;\phi\Bigg{(}\sum_{j=0}^{n}(-1)^{j}a_{0}<\cdots<\widehat{a}_{j}<\cdots<a_{n}\Bigg{)}$
$\displaystyle=$
$\displaystyle\;\sum_{j=0}^{n}(-1)^{j}\Bigg{(}\sum_{i=0}^{j-1}(-1)^{i}a_{0}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq\widehat{a}_{j}\leq\cdots\leq b_{n}$
$\displaystyle+\sum_{i=j+1}^{n}(-1)^{i-1}a_{0}<\cdots<\widehat{a}_{j}<\cdots<a_{i}\leq
b_{i}\leq\cdots\leq b_{n}\Bigg{)}.$

Note that every term in the second expansion has a corresponding term in the
first expansion but with an opposite sign. This leaves just
$(b_{0}\leq\cdots\leq b_{n})-(a_{0}<\cdots<a_{n})$ as desired. ∎

#### Right Chain Homotopy

Consider an $n$-simplex in $\Delta Q$ given as a chain $x_{0}<\cdots<x_{n}$.
Let $y_{i}=f\circ g(x_{i})$. The following relations follow from Lemma 4.3
(ii) combined with the fact that $f\circ g$ is monotone:

${y_{0}}$${y_{1}}$${\cdots}$${y_{n-1}}$${y_{n}}$${x_{0}}$${x_{1}}$${\cdots}$${x_{n-1}}$${x_{n}.}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\scriptstyle{<}$$\scriptstyle{<}$$\scriptstyle{<}$$\scriptstyle{<}$$\scriptstyle{<}$

The sequence $y_{0}\leq\cdots\leq y_{n}$ may contain repetitions in which case
it is not an $n$-simplex. Consider the linear map $\psi_{n}:C_{n}(\Delta
Q;{\mathbb{Z}})\to C_{n+1}(\Delta Q;{\mathbb{Z}})$ generated by the following
simplex-wise assignment:

$\psi_{n}(x_{0}<\cdots<x_{n}):=\sum_{i=0}^{n}(-1)^{i}y_{0}\leq\cdots\leq
y_{i}\leq x_{i}<\cdots<x_{n}.$ (4)

Identify every term on the right-hand side that is not an $(n+1)$-simplex to
$0$.

######  Lemma 4.6:

The composition $\Delta f\circ\Delta g:C_{\bullet}(\Delta Q;{\mathbb{Z}})\to
C_{\bullet}(\Delta Q;{\mathbb{Z}})$ is chain homotopic to the identity.

###### Proof.

For any $n$-simplex $x_{0}<\cdots<x_{n}$ in $\Delta Q$ and $y_{i}=f\circ
g(x_{i})$ as above, we show

$(x_{0}<\cdots<x_{n})-(y_{0}\leq\cdots\leq
y_{n})=\partial\circ\psi(x_{0}<\cdots<x_{n})+\psi\circ\partial(x_{0}<\cdots<x_{n}).$

Expand the first term to

$\displaystyle\partial\circ\psi(x_{0}<\cdots<x_{n})=$
$\displaystyle\;\partial\left(\sum_{i=0}^{n}(-1)^{i}y_{0}\leq\cdots\leq
y_{i}\leq x_{i}<\cdots<x_{n}\right)$ $\displaystyle=$
$\displaystyle\;\sum_{i=0}^{n}(-1)^{i}\Bigg{(}\sum_{j=0}^{i}(-1)^{j}y_{0}\leq\cdots\leq\widehat{y}_{j}\leq\cdots\leq
y_{i}\leq x_{i}<\cdots<x_{n}$
$\displaystyle+\sum_{j=i}^{n}(-1)^{j+1}y_{0}\leq\cdots\leq y_{i}\leq
x_{i}<\cdots<\widehat{x}_{j}<\cdots<x_{n}\Bigg{)}.$

Expand the second term to

$\displaystyle\psi\circ\partial(x_{0}<\cdots<x_{n})=$
$\displaystyle\;\psi\Bigg{(}\sum_{j=0}^{n}(-1)^{j}x_{0}<\cdots<\widehat{x}_{j}<\cdots<x_{n}\Bigg{)}$
$\displaystyle=$
$\displaystyle\;\sum_{j=0}^{n}(-1)^{j}\Bigg{(}\sum_{i=0}^{j-1}(-1)^{i}y_{0}\leq\cdots\leq
y_{i}\leq x_{i}<\cdots<\widehat{x}_{j}<\cdots<x_{n}$
$\displaystyle+\sum_{i=j+1}^{n}(-1)^{i-1}y_{0}\leq\cdots\leq\widehat{y}_{j}\leq\cdots\leq
y_{i}\leq x_{i}<\cdots<x_{n}\Bigg{)}.$

Note that every term in the second expansion has a corresponding term in the
first expansion but with an opposite sign. This leaves just
$(x_{0}<\cdots<x_{n})-(y_{0}\leq\cdots\leq y_{n})$ as desired. ∎

##  5 Rota’s Galois Connection Theorem

This section establishes a homological version of Rota’s Galois connection
theorem (Theorem 5.3) and explores two immediate consequences. We start with
Rota’s original statement relating the two Möbius functions involved in a
Galois connection.

######  Theorem 5.1 (Rota’s Galois Connection Theorem [6]):

Let $P$ and $Q$ be finite posets and $f:P\leftrightarrows Q:g$ a Galois
connection. Then for any $a\in P$ and $x\in Q$,

$\sum_{b:f(b)=x}\mu_{P}(a,b)=\sum_{y:g(y)=a}\mu_{Q}(x,y).$

The following statement by Gülem and McCleary relates the two Möbius
inversions involved in a Galois connection and is equivalent to Theorem 5.1
when the functions involved are valued in ${\mathbb{Z}}$.

######  Theorem 5.2 (Rota’s Galois Connection Theorem [7]):

Let $m:P\to{\mathbb{Z}}$ and $n:Q\to{\mathbb{Z}}$ be two functions and
$f:P\leftrightarrows Q:g$ a Galois connection. If $n=g\circ m$, then for all
$y\in Q$,

$\partial n(y)=\sum_{b\in f^{-1}(y)}\partial m(b).$

Given a Galois connection $f:P\leftrightarrows Q:g$ and an element $y\in Q$,
consider the following two subcomplexes of $\Delta P$:

$\displaystyle\Delta f^{-1}_{\leq y}:=\big{\\{}\sigma\in\Delta
P:f(\max\sigma)=y\big{\\}}$ $\displaystyle\Delta
f^{-1}_{<y}:=\big{\\{}\sigma\in\Delta P:f(\max\sigma)<y\big{\\}}.$

We now state our theorem and argue how it categorifies Rota’s Galois
connection theorem.

######  Theorem 5.3 (Homological Rota’s Galois Connection Theorem):

For a module $M:P\to\mathcal{C}$ and a Galois connection $f:P\leftrightarrows
Q:g$, let the module $N:=M\circ g:Q\to\mathcal{C}$. For every $y\in Q$, there
is a canonical isomorphism

$H^{\downarrow}_{\ast}N(y)\cong H_{\ast}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}.$ (5)

Theorem 5.3 categorifies Theorem 5.2 as follows. Let $m,n:\to K(\mathcal{C})$
be the dimension functions of the modules $M$ and $N$, respectively. By
Theorem 3.13,

$\partial n(y)=\sum_{d\geq 0}\big{[}H^{\downarrow}_{d}N(y)\big{]}.$

Note that the preimage on the right side of Equation (5) is the following
disjoint union:

$\Delta f^{-1}_{\leq y}\setminus\Delta f^{-1}_{<y}=\bigsqcup_{b\in
f^{-1}(y)}\Delta P_{\leq b}\setminus\Delta P_{<b}.$

This combined with Definition 2.11 and Lemma 2.12 implies

$\sum_{b\in f^{-1}(y)}\partial m(b)=\sum_{d\geq
0}(-1)^{d}\Big{[}H_{\ast}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}\Big{]}.$

######  Remark 5.4:

The right side of Equation 5 can be seen as the limiting term of a Leray-type
spectral sequence. This spectral sequence accounts for the differentials
between the Möbius homology objects $\big{\\{}H^{\downarrow}_{d}M(b)\big{\\}}$
over all $b\in f^{-1}(y)$ and dimensions $d$. Baclawski hints at such a
spectral sequence in [2].

#### Refinements

An immediate consequence of Theorem 5.3 is that Möbius homology is invariant
to refinements of the underlying poset as follows. A _refinement_ of a poset
$P$ is a Galois connection $f:P\leftrightarrows Q:g$ such that $f$ is
injective or, equivalently, $g$ is surjective. In this case, we think of $P$
as a subposet of $Q$.

######  Corollary 5.5:

Given a refinement $f:P\leftrightarrows Q:g$ and a $P$-module $M$, let the
$Q$-module $N:=M\circ g$. For every $b\in P$, $H^{\downarrow}_{\ast}M(b)\cong
H^{\downarrow}_{\ast}N(b)$. For every $y\in Q\setminus P$,
$H^{\downarrow}_{\ast}N(y)=0$ because $\Delta f^{-1}_{\leq y}\setminus\Delta
f^{-1}_{<y}=\emptyset$.

#### Bounding Homological Dimension

When underlying poset of a $P$-module $M$ is a finite distributive lattice,
there is a bound on the highest non-trivial dimension of its Möbius homology.
Recall a _lattice_ is a poset where the meet (greatest lower bound) and join
(least upper bound) is defined for every pair of elements. A lattice is
distributive if for all $a,b,c\in P$, $a\vee(b\wedge c)=(a\vee b)\wedge(a\vee
c)$. An element $c\in P$ _covers_ $a\in P$ if $a<c$ and if $a\leq b<c$, then
$a=b$.

Fix an element $b\in P$ and let $C$ be the set of all elements covered by $b$
along with $b$ itself. Define $\wedge C\subseteq P$ as the sublattice of all
elements expressible as the meet of any number of covering elements. Observe
that $\wedge C$ has a minimum element denoted $0$. By [3, Lemma 2, p.59], the
length of all maximal chains between $0$ and $b$ in $\wedge C$ is $|C|$. The
following corollary uses this sublattice.

######  Corollary 5.6:

Let $P$ be a finite distributive lattice, $b\in P$, and $n$ the number of
elements covered by $b$. Then, for any module $M:P\to\mathcal{C}$,
$H^{\downarrow}_{>n}M(b)=0.$

###### Proof.

There is a Galois connection $f:P_{\leq b}\leftrightarrows\wedge C:g$ as
follows. For $a\in P_{\leq b}$, let $A=\\{b^{\prime}\in C:a\leq
b^{\prime}\\}$. Define $f(a)=\wedge A$ and $g$ as inclusion. Note that
$f^{-1}(1)=\\{b\\}$. Let $N:\wedge C\to\mathcal{C}$ be the module $N=M\circ
g$. Since the length of any chain in $\wedge C$ is at most $n$,
$H^{\downarrow}_{>n}N(1)=0$. The desired result follows from Theorem 5.3. ∎

###  5.1 Proof of Theorem 5.3

The proof constructs a chain homotopy equivalence between
$C_{\bullet}\big{(}\Delta Q_{\leq y},\Delta Q_{<y};\underline{{N}}\big{)}$ and
$C_{\bullet}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}.$ The idea is to adapt a relative version
of Lemmas 4.5 and 4.6 to the cosheaf setting.

The simplicial map $\Delta f$ induces a chain map

$\Delta\hat{f}_{\bullet}:C_{\bullet}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}\to C_{\bullet}\big{(}\Delta Q_{\leq
y},\Delta Q_{<y};\underline{{N}}\big{)}$

generated simplex-wise as follows. For an $n$-simplex $\sigma\in\Delta
f^{-1}_{\leq y}$, let $\tau:=\Delta f(\sigma)$. By Lemma 4.3 (i),
$\min\sigma\leq g(\min\tau)$ in $P$. If $\tau$ is an $n$-simplex, define the
restriction
$\Delta\hat{f}_{n}|_{\sigma}:\underline{{M}}(\sigma)\to\underline{{N}}(\tau)$
as the morphism $M\big{(}\min\sigma\leq g(\min\tau)\big{)}.$ If $\tau$ is not
an $n$-simplex, then $\Delta\hat{f}_{n}|_{\sigma}$ is the zero morphism. One
easily checks that $\Delta\hat{f}_{\bullet}$ is indeed a chain map.

The simplicial map $\Delta g$ induces a chain map

$\Delta\hat{g}_{\bullet}:C_{\bullet}\big{(}\Delta Q_{\leq y},\Delta
Q_{<y};\underline{{N}}\big{)}\to C_{\bullet}\big{(}\Delta f^{-1}_{\leq
y},\Delta f^{-1}_{<y};\underline{{M}}\big{)}$

induced simplex-wise as follows. For an $n$-simplex $\sigma\in\Delta Q_{\leq
y}$, let $\tau:=\Delta g(\sigma)$. Note $g(\min\sigma)=\min\tau$. If $\tau$ is
an $n$-simplex, define the restriction
$\Delta\hat{g}_{n}|_{\sigma}:\underline{{N}}(\sigma)\to\underline{{M}}(\tau)$
as the identity morphism $M\big{(}g(\min\sigma)=\min\tau\big{)}$. If $\tau$ is
not an $n$-simplex, then $\Delta\hat{g}_{n}|_{\sigma}$ is the zero morphism.
One easily checks that $\Delta\hat{g}_{\bullet}$ is indeed a chain map.

######  Lemma 5.7:

The composition $\Delta\hat{g}\circ\Delta\hat{f}:C_{\bullet}\big{(}\Delta
f^{-1}_{\leq y},\Delta f^{-1}_{<y};\underline{{M}}\big{)}\to
C_{\bullet}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}$ is chain homotopic to the identity.

###### Proof.

The chain homotopy $\phi_{n}$ described by Equation 3 induces a chain homotopy

$\Phi_{n}:C_{n}\big{(}\Delta f^{-1}_{\leq y},\Delta
f^{-1}_{<y};\underline{{M}}\big{)}\to C_{n+1}\big{(}\Delta f^{-1}_{\leq
y},\Delta f^{-1}_{<y};\underline{{M}}\big{)}$

defined simplex-wise as follows. For notational convenience, substitute
$\sigma$ and $\tau_{i}$ for the simplices on the left and right sides of
Equation 3:

$\phi_{n}(\sigma):=\sum_{i=0}^{n}(-1)^{i}\tau_{i}.$

Note that $\min\sigma=\min\tau_{i}$, for all $i$. Thus $\Phi_{n}$ restricted
to the summand $\underline{{M}}(\sigma)$ is the direct sum of signed identity
morphisms:

$\Phi_{n}|_{\sigma}:=\bigoplus_{i=0}^{n}(-1)^{i}M(\min\sigma=\min\tau_{i}).$

By Lemma 4.5, $\Phi_{n}$ is indeed a chain homotopy. ∎

######  Lemma 5.8:

The composition $\Delta\hat{f}\circ\Delta\hat{g}:C_{\bullet}\big{(}\Delta
Q_{\leq y},\Delta Q_{<y};\underline{{N}}\big{)}\to C_{\bullet}\big{(}\Delta
Q_{\leq y},\Delta Q_{<y};\underline{{N}}\big{)}$ is chain homotopic to the
identity.

###### Proof.

The chain homotopy $\psi_{n}$ described by Equation 4 induces a chain homotopy

$\Psi_{n}:C_{n}\big{(}\Delta Q_{\leq y},\Delta
Q_{<y};\underline{{N}}\big{)}\to C_{n+1}\big{(}\Delta Q_{\leq y},\Delta
Q_{<y};\underline{{N}}\big{)}$

defined simplex-wise as follows. For notational convenience, substitute
$\sigma$ and $\tau_{i}$ for the simplices on the left and right sides of
Equation 4:

$\psi_{n}(\sigma):=\sum_{i=0}^{n}(-1)^{i}\tau_{i}.$

By Lemma 4.3 (i), $g(\min\sigma)=g(\min\tau_{i})$ implying
$\underline{{N}}(\sigma)=\underline{{N}}(\tau_{i})$. Thus $\Psi_{n}$
restricted to the summand $\underline{{N}}(\sigma)$ is the direct sum of
signed identity morphisms

$\Psi_{n}|_{\sigma}:=\bigoplus_{i=0}^{n}(-1)^{i}M\big{(}g(\min\sigma)=g(\min\tau_{i})\big{)}.$

By Lemma 4.6, $\Psi_{n}$ is indeed a chain homotopy. ∎

We now complete the proof. If $\sigma\in\Delta f^{-1}_{<y}$, then of course
$f(\max\sigma)<y$. By Lemma 4.3 (iii), $\Delta g\circ\Delta f(\sigma)\in\Delta
f^{-1}_{<y}$. Thus $\Delta\hat{g}\circ\Delta\hat{f}$ passes to a chain
endomorphism:

$\frac{C_{\bullet}\big{(}\Delta f^{-1}_{\leq
y};\underline{{M}}\big{)}}{C_{\bullet}\big{(}\Delta
f^{-1}_{<y};\underline{{M}}\big{)}}\to\frac{C_{\bullet}\big{(}\Delta
f^{-1}_{\leq y};\underline{{M}}\big{)}}{C_{\bullet}\big{(}\Delta
f^{-1}_{<y};\underline{{M}}\big{)}}.$

Now consider the chain homotopy $\phi_{n}$ in Equation 3. By the same
reasoning, if $a_{0}<\cdots<a_{n}$ is in $\Delta f^{-1}_{<y}$, then every
simplex on the right side of the equation is also in $\Delta f^{-1}_{<y}$.
Thus the chain homotopy $\Phi_{n}$ passes to a relative chain homotopy. Now
for the other side. If $\tau\in\Delta Q_{<y}$, then $\max\tau<y$. By Lemma 4.3
(ii), $\Delta f\circ\Delta g(\tau)\in\Delta Q_{<y}$. Thus
$\Delta\hat{f}\circ\Delta\hat{g}$ passes to a chain endomorphism on

$\frac{C_{\bullet}(\Delta Q_{\leq y};\underline{{N}})}{C_{\bullet}(\Delta
Q_{<y};\underline{{N}})}\to\frac{C_{\bullet}(\Delta Q_{\leq
y};\underline{{N}})}{C_{\bullet}(\Delta Q_{<y};\underline{{N}})}.$

Now consider the chain homotopy $\psi_{n}$ in Equation 4. By the same
reasoning, if $x_{0}<\cdots<x_{n}$ is in $\Delta Q_{<y}$, then every simplex
on the right side of the equation is also in $\Delta Q_{<y}$. Thus the chain
homotopy $\Psi_{n}$ passes to a relative chain homotopy.

##  6 Persistent Homology

We are now ready to study the _persistence_ of a $P$-module using Möbius
homology. The idea of persistence is to capture information that persists over
subsets of $P$. The simplest subsets to consider are closed intervals of $P$.
Our main construction is the _birth-death module_ associated to any given
$P$-module. When $P$ is totally ordered, its Möbius homology categorifies the
traditional persistence diagram.

For $a\in P$ and an object $X\in\mathrm{ob}\,\mathcal{C}$, let $X^{\uparrow
a}:P\to\mathcal{C}$ be the following module. For every $b\in P$,

$X^{\uparrow a}(b):=\begin{cases}X&\text{ if $a\leq b$}\\\ 0&\text{
otherwise}\end{cases}$

and for every $b\leq c$ the morphism $X^{\uparrow a}(b\leq c)$ is the identity
when $a\leq b$ and zero otherwise. A $P$-module is _free_ if it is the direct
sum of modules of the form $X^{\uparrow a}$.

######  Definition 6.1:

A free presentation of a $P$-module $M:P\to\mathcal{C}$ is a natural
transformation $\phi:F\Rightarrow M$ from a free $P$-module $F$ such that for
all $a\in P$ the morphism $\phi(a):F(a)\to M(a)$ is an epimorphism.

The construction of a free presentation below appeared for ${\mathrm{Vec}}$ in
[7]. Here we present the same construction in an abelian category for
completeness.

######  Proposition 6.2:

Every $P$-module $M$ has a free presentation.

###### Proof.

Let $F$ be the free $P$-module $\bigoplus_{a\in P}M(a)^{\uparrow a}$. Define
the natural transformation $\phi:F\Rightarrow M$ summand-wise as follows. For
$a\in P$, let $\phi_{a}:M(a)^{\uparrow a}\Rightarrow M$ be the following
natural transformation:

$\phi_{a}(b):=\begin{cases}M(a\leq b)&\text{if $a\leq b$}\\\
0&\text{otherwise.}\end{cases}$

Note that $\phi_{a}(a)$ is an isomorphism. Then $\phi$ is the direct sum
$\oplus_{a\in P}\phi_{a}$. ∎

Every module $M$ has a free presentation, but it is not unique. For now, we
develop persistence for a fixed free presentation and show it is well defined
in the next subsection.

######  Definition 6.3:

The interval poset of a poset $P$, denoted ${\mathrm{Int}}\,P$, is the set of
all (closed) intervals of $P$ with the following ordering: $[a,b]\preceq[c,d]$
if $a\leq c$ and $b\leq d$. See Figure 6.

${b}$${d}$${a}$${c}$
(a) $P$

${{[d,d]}}$${{[b,d]}}$${{[c,d]}}$${{[b,b]}}$${{[a,d]}}$${{[c,c]}}$${{[a,b]}}$${{[a,c]}}$${{[a,a]}}$
(b) ${\mathrm{Int}}\,P$

Figure 6: Poset and its interval poset

Fix a free presentation $\phi:F\Rightarrow M$ and consider the following
pullback diagram for an interval $[a,b]\in{\mathrm{Int}}\,P$:

${F(a)\cap{\mathrm{ker}}\,\phi_{b}}$${{\mathrm{ker}}\,\phi(b)}$${F(a)}$${F(b).}$

We think of this pullback object as generators that appear by the start of the
interval and become relations by the end of the interval. For every
$[a,b]\preceq[c,d]$, the following monomorphisms induce a canonical
monomorphism $F(a)\cap{\mathrm{ker}}\,\phi_{b}\hookrightarrow
F(c)\cap{\mathrm{ker}}\,\phi_{d}$:

${F(a)}$${F(b)}$${{\mathrm{ker}}\,\phi(b)}$${F(c)}$${F(d)}$${{\mathrm{ker}}\,\phi(d).}$

######  Definition 6.4:

The birth-death module of a free presentation $\phi:F\Rightarrow M$ is the
module $\mathrm{BD}\phi:{\mathrm{Int}}\,P\to\mathcal{C}$ that assigns to every
interval $[a,b]$ the birth-death object $F(a)\cap{\mathrm{ker}}\,\phi_{b}$ and
to every $[a,b]\preceq[c,d]$ the canonical monomorphism.

The birth-death module satisfies an interesting pullback property that will be
useful later.

######  Proposition 6.5:

Let $\mathrm{BD}\phi$ be the birth-death module of a free presentation
$\phi:F\Rightarrow M$. For every $a\leq b\leq c\leq d$ in $P$, the following
subdiagram of $\mathrm{BD}\phi$ is a pullback diagram:

${F(b)\cap{\mathrm{ker}}\,\phi(c)}$${F(b)\cap{\mathrm{ker}}\,\phi(d)}$${F(a)\cap{\mathrm{ker}}\,\phi(c)}$${F(a)\cap{\mathrm{ker}}\,\phi(d).}$

###### Proof.

The statement follows from an easy chase on the following diagram:

${F}$${F(a)}$${F(b)}$${F(c)}$${F(d)}$${M}$${M(a)}$${M(b)}$${M(c)}$${M(d).}$$\scriptstyle{\phi}$

∎

Denote by $\mathrm{bd}\phi:{\mathrm{Int}}\,P\to K(\mathcal{C})$ the dimension
function of the birth-death module $\mathrm{BD}\phi$. When $P$ is totally
ordered, the Möbius inversion $\partial(\mathrm{bd}\phi)$ agrees with the
classical persistence diagram for all intervals $[a,b]$ where $a<b$ [7]. This
suggests the following generalization of the classical persistence diagram
proposed by Gülem and McCleary.

######  Definition 6.6:

Let $\phi:F\Rightarrow M$ a free presentation of a $P$-module $M$. The
persistence diagram of $\phi$ is the Möbius inversion
$\partial(\mathrm{bd}\phi)$.

There is a drawback to this generalization of the persistence diagram.
Bottleneck stability, as introduced by Cohen-Steiner, Edelsbrunner, and Harer
[5], is an important property of the classical persistence diagram, but it is
unlikely such a theorem exists for this generalization. The big hurdle here is
the fact that $\partial(\mathrm{BD}\phi)$ may take negative values for $P$ not
totally ordered. We hope the order cosheaf of $\mathrm{BD}\phi$ will offer a
path forward since, by Theorem 3.13, $\partial(\mathrm{bd}\phi)$ is the Euler
charactersictic of $H^{\downarrow}_{\ast}(\mathrm{BD}\phi)$. In other words,
there is now a lift of $\partial(\mathrm{bd}\phi)$ to a topological object and
we hope this will inspire a topological bottleneck stability.

######  Definition 6.7:

Let $\phi:F\Rightarrow M$ be a free presentation of a $P$-module $M$. The
persistent Möbius homology of $\phi$ is the Möbious homology module
$H^{\downarrow}_{\ast}(\mathrm{BD}\phi):{\mathrm{Int}}\,P\to\mathcal{C}$.

The following example considers two similar modules over the same poset. Their
persistence diagrams are the same, but their persistent Möbius homologies are
different. Persistent Möbius homology is a finer invariant than the
persistence diagram.

######  Example 6.8:

Let $P$ be the poset in Figure 6 and consider Figure 7 for an example of two
$P$-modules, their free presentations, and their birth-death modules. Note
that the dimension functions of both birth-death modules are the same and
therefore their persistence diagrams are the same. However, their persistent
Möbius homology does see a difference. The table below lists the Möbius
homology of the two birth-death modules at the interval $[a,b]$.

$[a,d]$ | $\phi$ | $\psi$
---|---|---
$H^{\downarrow}_{0}$ | ${\mathsf{k}}$ | $0$
$H^{\downarrow}_{1}$ | ${\mathsf{k}}$ | $0$
$H^{\downarrow}_{2}$ | $0$ | $0$

A third invariant worth considering is the total homology
$H_{\ast}\big{(}\Delta P;\underline{\mathrm{BD}\phi}\big{)}$ of the birth-
death cosheaf. By Corollary 3.15,

$\sum_{[a,b]\in{\mathrm{Int}}\,P}\partial(\mathrm{BD}\phi)[a,b]=\sum_{d\geq
0}(-1)^{d}\Big{[}H_{d}\big{(}\Delta({\mathrm{Int}}\,P);\underline{\mathrm{BD}\phi}\big{)}\Big{]}.$

${{\mathsf{k}}}$${0}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$$\scriptstyle{0}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$
(a) $M:P\to{\mathrm{Vec}}$

${{\mathsf{k}}}$${0}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$$\scriptstyle{0}$$\scriptstyle{\footnotesize{\begin{pmatrix}0&1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$
(b) $N:P\to{\mathrm{Vec}}$

${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}}$${0}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{0}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$
(c) $\phi:F\Rightarrow M$

${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}}$${0}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$${{\mathsf{k}}\oplus{\mathsf{k}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0&1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{0}$$\scriptstyle{\footnotesize{\begin{pmatrix}0&1\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1&0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$
(d) $\psi:G\Rightarrow N$

${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}}$${{\mathsf{k}}}$${0}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$
(e) $\mathrm{BD}\psi:{\mathrm{Int}}\,P\to{\mathrm{Vec}}$

${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}^{2}}$${{\mathsf{k}}}$${{\mathsf{k}}}$${{\mathsf{k}}}$${0}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}1\\\
0\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}0\\\
1\end{pmatrix}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$
(f) $\mathrm{BD}\psi:{\mathrm{Int}}\,P\to{\mathrm{Vec}}$

Figure 7: Two $P$-modules $M$ and $N$, their free presentations $\phi$ and
$\psi$, and their birth-death modules $\mathrm{BD}\phi$ and $\mathrm{BD}\psi$

###  6.1 Independence of Free Presentation

Our definition of persistent Möbius homology depends on the choice of a free
presentation. In this section, we show that the persistent Möbius homology at
intervals $[a,b]$ off the diagonal, i.e., $a<b$, is independent of this
choice. Traditionally, persistence is not concerned about intervals on the
diagonal as they have zero persistence.

######  Proposition 6.9:

Let $\phi:F\Rightarrow M$ and $\phi:G\Rightarrow M$ be two free presentations
of a $P$-module $M$. For every $a<b$,
$H^{\downarrow}_{\ast}(\mathrm{BD}\phi)[a,b]\cong
H^{\downarrow}_{\ast}(\mathrm{BD}\psi)[a,b]$.

The proof involves three intermediate modules.

  * •

Let $K_{M}:{\mathrm{Int}}\,P\to\mathcal{C}$ be the module that assigns to
every interval $[a,b]$ the kernel of the morphism $M(a\leq b)$ and to every
$[a,b]\preceq[c,d]$ the restriction (dashed arrow) of the morphism $M(a\leq
c)$ below:

${{\mathrm{ker}}\,M(a\leq b)}$${{\mathrm{ker}}\,M(c\leq
d)}$${M(a)}$${M(c).}$$\scriptstyle{M(a\leq c)}$

  * •

Let $D_{\phi}:{\mathrm{Int}}\,P\to\mathcal{C}$ be the module that assigns to
every interval $[a,b]$ the birth-death object $\mathrm{BD}\phi[a,a]$ and to
every $[a,b]\preceq[c,d]$ the canonical monomorphism
$\mathrm{BD}\phi\big{(}[a,a]\preceq[c,c]\big{)}$. Define
$D_{\psi}:{\mathrm{Int}}\,P\to\mathcal{C}$ similarly.

######  Lemma 6.10:

For all $a<b$ in $P$, both $H^{\downarrow}_{\ast}D_{\phi}[a,b]$ and
$H^{\downarrow}_{\ast}D_{\psi}[a,b]$ are zero.

###### Proof.

We give the proof for $D_{\phi}$ but the same argument applies to $D_{\psi}$.
Let $f:P\to{\mathrm{Int}}\,P$ be the monotone function $f(a):=[a,a]$ and
$g:{\mathrm{Int}}\,P\to P$ the monotone function $g[b,c]:=b$. For every $a\in
P$ and $[b,c]\in{\mathrm{Int}}\,P$ the following holds:

$f(a)=[a,a]\leq[b,c]\text{ iff }a\leq g[b,c]=b.$

This makes $f:P\leftrightarrows{\mathrm{Int}}\,P:g$ a Galois connection. Let
$B:P\to\mathcal{C}$ be the module that assigns to every $a\in P$ the object
$\mathrm{BD}\phi[a,a]$ and to every $a\leq b$ the canonical monomorphism
between the two birth-death objects. Note $D_{\phi}=B\circ g$ and
$f^{-1}[a,b]$ is empty unless $a=b$. The desired result follows from Theorem
5.3. ∎

###### Proof of Proposition 6.9.

The $P$-modules $D_{\phi}$, $\mathrm{BD}\phi$, and $K_{M}$ fit into a short
exact sequence:

${0}$${D_{\phi}}$${\mathrm{BD}\phi}$${K_{M}}$${0.}$

This short exact sequence of modules gives rise to a short exact sequence of
Möbius chain complexes:

${0}$${C^{\downarrow}_{\bullet}D_{\phi}[a,b]}$${C^{\downarrow}_{\bullet}(\mathrm{BD}\phi)[a,b]}$${C^{\downarrow}_{\bullet}K_{M}[a,b]}$${0.}$

This short exact sequence of Möbius chain complexes gives rise to a long exact
sequence of Möbius homology objects:

${\cdots}$${H^{\downarrow}_{n+1}K_{M}[a,b]}$${H^{\downarrow}_{n}D_{\phi}[a,b]}$${H^{\downarrow}_{n}(\mathrm{BD}\phi)[a,b]}$${\cdots}$${H^{\downarrow}_{n-1}(\mathrm{BD}\phi)[a,b]}$${H^{\downarrow}_{n-1}D_{\phi}[a,b]}$${H^{\downarrow}_{n}K_{M}[a,b].}$$\scriptstyle{i}$

By Lemma 6.10, both $H^{\downarrow}_{n}D_{\phi}[a,b]$ and
$H^{\downarrow}_{n-1}D_{\phi}[a,b]$ are zero. This makes $i$ is an
isomorphism. The same argument constructs a canonical isomorphism between
$H^{\downarrow}_{n}(\mathrm{BD}\psi)[a,b]$ and $H^{\downarrow}_{n}K_{M}[a,b]$.
∎

###  6.2 Galois Connections and Persistence

The homological version of Rota’s Galois connection theorem applies to
persistent Möbius homology. This follows immediately from the observation that
a Galois connection between posets induces a Galois connection between their
intervals posets.

A monotone function $f:P\to Q$ induces a monotone function
${\mathrm{Int}}\,f:{\mathrm{Int}}\,P\to{\mathrm{Int}}\,Q$ defined as
${\mathrm{Int}}\,f[a,b]=\big{[}f(a),f(b)\big{]}$. Similarly, a monotone
function $g:Q\to P$ induces a monotone function
${\mathrm{Int}}\,g:{\mathrm{Int}}\,Q\to{\mathrm{Int}}\,P$.

######  Lemma 6.11:

If $f:P\leftrightarrows Q:g$ is a Galois connection, then
${\mathrm{Int}}\,f:{\mathrm{Int}}\,P\leftrightarrows{\mathrm{Int}}\,Q:{\mathrm{Int}}\,g$
is a Galois connection.

###### Proof.

The following statement is to be verified: for all $[a,b]\in{\mathrm{Int}}\,P$
and $[x,y]\in{\mathrm{Int}}\,Q$, ${\mathrm{Int}}\,f[a,b]\leq[x,y]$ iff
$[a,b]\leq{\mathrm{Int}}\,g[x,y]$. Unwind the definition of $\preceq$ to get

$f(a)\leq x\text{ and }f(b)\leq y\Leftrightarrow a\leq g(x)\text{ and }b\leq
g(y).$

In other words, $f$ and $g$ must form a Galois connection which is true by
assumption. ∎

Consider a $P$-module $M:P\to\mathcal{C}$ and a free presentation
$\phi:F\Rightarrow M$. Given a Galois connection $f:P\leftrightarrows Q:g$,
let $N:=M\circ g:Q\to\mathcal{C}$. The free presentation $\phi$ restricts to a
free presentation $\psi:G\Rightarrow N$ where $G:=F\circ g$ and
$\psi(x):=(\phi\circ g)(x)$. The following corollary follows from Theorem 5.3.

######  Corollary 6.12:

Let $f:P\leftrightarrows Q:g$ be a Galois connection. For a module
$M:P\to\mathcal{C}$ and a free presentation $\phi:F\Rightarrow M$, let $N$ be
the $Q$-module $M\circ g$ and $\psi:F\circ g\Rightarrow N$ the free
presentation $\phi\circ g$. For every interval $[x,y]\in{\mathrm{Int}}\,Q$,

$H_{\ast}(\mathrm{BD}\psi)[x,y]\cong
H_{\ast}\Big{(}\Delta({\mathrm{Int}}\,f)^{-1}_{\leq[x,y]},\Delta({\mathrm{Int}}\,f)^{-1}_{<[x,y]};\mathrm{BD}\phi\Big{)}.$

###  6.3 Bounding Persistent Homological Dimension

When underlying poset of a $P$-module $M$ is a finite distributive lattice,
there is a bound on the highest non-trivial dimension of its persistent Möbius
homology. Since $P$ is distributive, ${\mathrm{Int}}\,P$ is also distributive.
If $a\in P$ covers $m$ elements and $b\in P$ covers $n$ elements, then the
interval $[a,b]\in{\mathrm{Int}}\,P$ covers $m\cdot n$ elements. The following
corollary follows immediately from Corollary 5.6.

######  Corollary 6.13:

Let $P$ be a distributive lattice and suppose that for $a\leq b$, $a$ covers
$m$ elements and $b$ covers $n$ elements. Then for any free presentation
$\phi:F\Rightarrow M$ of a $P$-module $M$, $H^{\downarrow}_{>m\cdot
n}(\mathrm{BD}\phi)[a,b]=0$.

We believe the bound of $m\cdot n$ can be improved. For example, when $P$ is
totally ordered, as in the case of classical persistence, $m\cdot n\leq 1$ but
$H^{\downarrow}_{>1}(\mathrm{BD}\phi)=0$ as follows.

######  Proposition 6.14:

Let $P$ be a totally ordered poset and $M$ a $P$-module. For all intervals
$[b,d]\in{\mathrm{Int}}\,P$, $H^{\downarrow}_{>0}M[b,d]=0.$

###### Proof.

Let $a\in P$ be the element one less than $b$, if it exists, otherwise let
$a=b$. Similarly, let $c\in P$ be the element one less than $d$, if it exists,
otherwise let $c=d$. Note that $({\mathrm{Int}}\,P)_{\leq[b,d]}$ is a
distributive lattice and $[b,d]$ covers $[a,d]$ and $[b,c]$. By Corollary 5.6,
$H^{\downarrow}_{\ast}(\mathrm{BD}\phi)[b,d]$ is isomorphic to the Möbius
homology at the top element, denoted $1$, of the following much smaller
submodule of $\mathrm{BD}\phi$, denoted $N$:

${\mathrm{BD}\phi[a,d]}$${\mathrm{BD}\phi{[b,d]}}$${\mathrm{BD}\phi{[a,c]}}$${\mathrm{BD}\phi{[b,c]}.}$

By Proposition 6.5, $N$ is a pullback diagram. Now let $D$ be the following
module over the same smaller poset:

${\mathrm{BD}\phi[a,c]}$${\mathrm{BD}\phi{[a,c]}}$${\mathrm{BD}\phi{[a,c]}}$${\mathrm{BD}\phi{[a,c]}.}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$$\scriptstyle{{\mathbbm{1}}}$

The Möbius chain complex of $D$ at $1$ is

${\cdots}$${\mathrm{BD}\phi[a,c]}$${\mathrm{BD}\phi[a,c]\oplus\mathrm{BD}\phi[a,c]}$${\mathrm{BD}\phi[a,c]}$${0}$$\scriptstyle{\partial_{3}}$$\scriptstyle{\footnotesize{\begin{pmatrix}{\mathbbm{1}}\\\
{\mathbbm{1}}\end{pmatrix}}}$$\scriptstyle{\footnotesize{\begin{pmatrix}{\mathbbm{1}}&{\mathbbm{1}}\end{pmatrix}}}$$\scriptstyle{\partial_{0}}$

implying $H^{\downarrow}_{\ast}D(1)=0$. Note $D\hookrightarrow N$. Let
$Q:=N/D$ be the following quotient module:

${\dfrac{\mathrm{BD}\phi[a,d]}{\mathrm{BD}\phi[a,c]}}$${\dfrac{\mathrm{BD}\phi{[b,d]}}{\mathrm{BD}\phi[a,c]}}$${0=\dfrac{\mathrm{BD}\phi{[a,c]}}{\mathrm{BD}\phi[a,c]}}$${\dfrac{\mathrm{BD}\phi{[b,c]}}{\mathrm{BD}\phi[a,c]}.}$

Since $Q$ is the quotient of a pullback diagram by a pullback subdiagram, it
is also a pullback diagram.

We are now ready to finish the proof. The short exact sequence $0\to D\to N\to
Q\to 0$ gives rise to a long exact sequence of Möbius homology objects:

${\cdots}$${H^{\downarrow}_{n+1}D(1)}$${H^{\downarrow}_{n}N(1)}$${H^{\downarrow}_{n}Q(1)}$${\cdots}$${H^{\downarrow}_{n-1}Q(1)}$${H^{\downarrow}_{n-1}N(1)}$${H^{\downarrow}_{n}D(1).}$$\scriptstyle{i}$

Since $H^{\downarrow}_{\ast}D(1)=0$, $i$ must be an isomorphism. The object
$H^{\downarrow}_{n}Q(1)$ is the homology of the following chain complex where
$\partial_{1}$ is the direct sum of the two monomorphisms:

${\cdots}$${0}$${\dfrac{\mathrm{BD}\phi[a,d]}{\mathrm{BD}\phi[a,c]}\oplus\dfrac{\mathrm{BD}\phi[b,c]}{\mathrm{BD}\phi[a,c]}}$${\dfrac{\mathrm{BD}\phi[b,d]}{\mathrm{BD}\phi[a,c]}}$${0}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

Of course, $H^{\downarrow}_{2}Q(1)=0$. The morphism $\partial_{1}$ must be a
monomorphism because $Q$ is a pullback diagram with the pullback being $0$.
Thus $H^{\downarrow}_{1}Q(1)=0$. ∎

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