homological Serre grading classes = draw = none

# The discrete flow category: structure and computation

Bjørnar Gullikstad Hem

###### Abstract.

In this article, we use concepts and methods from the theory of simplicial
sets to study discrete Morse theory. We focus on the discrete flow category
introduced by Vidit Nanda, and investigate its properties in the case where it
is defined from a discrete Morse function on a regular CW complex. We design
an algorithm to efficiently compute the Hom posets of the discrete flow
category in this case. Furthermore, we show that in the special case where the
discrete Morse function is defined on a simplicial complex, then each Hom
poset has the structure of a face poset of a regular CW complex. Finally, we
prove that the spectral sequence associated to the double nerve of the
discrete flow category collapses on page 2.

Email<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on simplicial sets
  3. 3 Discrete Morse theory
  4. 4 p-categories
  5. 5 The discrete flow category
  6. 6 Hom posets of discrete flow categories
  7. 7 Simplicial collapse on simplicial sets
  8. 8 Spectral sequences

##  1\. Introduction

Differentiable functions on smooth manifolds allow us to deduce properties of
the manifolds’ topology, through the use of Morse theory. For Morse theory to
apply, the differential function has to satisfy the _Morse condition_. Such
functions are called _Morse functions_ , and they are in many ways abundant. A
Morse function can be viewed as a “height function” on a manifold that allows
you to decompose the manifold into smaller, more manageable parts. Morse
theory has had vast applications, not only in topology, but also in other
areas, such as in the study of dynamical systems [18].

The combinatorial counterpart of Morse theory, discrete Morse theory, was
introduced by Robin Forman [6], and studies _discrete Morse functions_ on
_simplicial complexes_. Like Morse theory, discrete Morse theory allows us to
deduce information about the topology of a simplicial complex. As the name
suggests, the data of a discrete Morse function is discrete, making discrete
Morse theory suitable for computer applications. Discrete Morse theory has
seen much use in computer applications in later years, particularly in
topological data analysis (TDA). As an example, in [17, pp. 13–14], Scoville
explains how discrete Morse theory can be used to reduce the computational
complexity of homology computations in TDA.

One of the main foundations for this article is a paper by Nanda [14] that
introduces the _discrete flow category_ , an analogue in discrete Morse theory
for the _flow category_ of Cohen, Jones and Segal [4]. The flow category is a
category consisting of critical points and flow lines of a Morse function, and
whose classifying space captures the homotopy type of the manifold on which
the Morse function is defined. Likewise, the discrete flow category consists
of critical _cells_ and _gradient paths_ of a discrete Morse function, and has
a classifying space with the homotopy type of the simplicial complex on which
the discrete Morse function is defined. While the flow category is a
_topological category_ , meaning that the Hom sets are topological spaces, the
discrete flow category is a _p-category_ , meaning that the Hom sets are
posets. Constructing the classifying space of either of these categories
utilizes concepts from simplicial sets, inspiring the further study of
discrete Morse theory from the perspective of simplicial sets.

Another inspiration for this article is a paper by Vaupel, Hermansen and
Trygsland on section complexes of simplicial height functions [20]. In the
paper, a bisimplicial set is constructed from a _height function_ on a
simplicial set. This bisimplicial set gives a spectral sequence that can be
used to compute the homology of the simplicial set, and it is shown that under
certain conditions the spectral sequence collapses on the second page.

In this article we explore Nanda’s discrete flow category (in the special case
where it is defined from a discrete Morse function), and prove several
properties of it. In particular, we show that in the case where the discrete
Morse function is defined on a simplicial complex (not a general CW complex),
then the Hom posets in the discrete flow category are CW posets, meaning that
they have the structure of a face poset of a regular CW complex. We will refer
to this as Theorem A.

######  Theorem A.

Let $C$ be the discrete flow category of a discrete Morse function on a
simplicial complex. Then the for all objects $w$ and $z$ in $C$, the poset
$\operatorname{Hom}_{C}(w,z)^{\operatorname{op}}$ is the face poset of some
regular CW complex.

This result gives a simpler way of realizing the Hom posets as topological
spaces: instead of taking the geometric realization of the nerve, one can take
the corresponding regular CW complex.

Furthermore, we construct an algorithm, Algorithm 1, to compute the Hom posets
of the discrete flow category, where the input is a discrete Morse function
defined on a regular CW complex. For the algorithm, we use several of the
results needed for proving A. We provide a Python implementation for the
algorithm, which can be found at https://github.com/bjornarhem/discrete_flow .

Finally, we investigate the spectral sequence associated to the _double nerve_
of the discrete flow category, a bisimplicial set whose realization is the
classifying space of the category. As in the paper by Vaupel et al. [20], this
spectral sequence computes the homology of the regular CW complex from which
we define the discrete flow category. Classifying spaces of p-categories are
often complicated and hard to compute, and this is also the case for the
discrete flow category. However, as we prove in this article, the spectral
sequence associated to its double nerve has a particularly nice structure that
causes it to collapse on page 2. We will refer to this as Theorem B.

######  Theorem B.

The spectral sequence associated to the double nerve of a discrete flow
category collapses on page 2.

The definition of the mentioned spectral sequence will be made formal in the
text (in particular, there are two possible choices). We also show how you can
ignore degeneracies when computing the spectral sequence, which makes the
computation relatively simple. Finally, we provide several examples of
computing the spectral sequence, in which we make use of Algorithm 1 to
compute the Hom sets. We also provide a Python implementation of the spectral
sequence computation at https://github.com/bjornarhem/discrete_flow , that
outputs the spectral sequence pages given any discrete Morse function.

To prove B, we generalize the concept of _simplicial collapse_ to simplicial
sets. We then introduce _unique factorization categories_ , a class of
categories in which all morphisms have a unique decomposition into
“indecomposable” morphisms. Finally, we apply our generalization of simplicial
collapse to prove the following result.

######  Theorem C.

Let $C$ be a unique factorization category. Then the nerve of $C$ deformation
retracts to a simplicial set whose $n$–simplices are all degenerate for $n>1$.

This theorem is then used in the proof of B.

### Outline

Section 2 and 3 are dedicated to reviewing the theory of simplicial sets and
of discrete Morse theory, respectively, which will be needed for the following
sections. Section 4 contains the necessary preliminaries on p-categories, and
Section 5 is a review of the discrete flow category of a discrete Morse
function. In Section 6, we prove several results on the Hom posets of the
discrete flow category, and conclude with constructing Algorithm 1 and proving
A. In Section 7, we introduce simplicial collapse on simplicial sets and
unique factorization categories, and prove C. Finally, in Section 8, we
explain how the discrete flow category gives rise to a spectral sequence, and
prove B.

### Acknowledgments

This work is based on my master thesis, that I wrote for NTNU in Spring 2023.
I would like to thank my master thesis supervisors, Marius Thaule and Melvin
Vaupel, who have both been incredibly helpful and supportive. They have gone
above and beyond, both in encouraging me to explore my ideas and in helping me
seek answers to the countless questions I have asked them. Their feedback,
which has consistently been both thorough and constructive, has been
invaluable in this work. I would also like to thank my PhD supervisor, Kathryn
Hess Bellwald, for providing feedback in the final steps of writing this
article.

##  2\. Preliminaries on simplicial sets

In this section we define our notation related to simplicial sets, and state
some results that we will need. For a thorough exposition of simplicial sets,
see e.g., [8] or [7].

A _simplicial set_ is a functor

$\Delta^{\operatorname{op}}\to\mathsf{Set},$

where $\Delta$ is the simplex category and $\mathsf{Set}$ is the category of
sets. A morphism of simplicial sets is a natural transformation between two
such functors, and we denote the category of simplicial sets by
$\mathsf{sSet}$. The $i$th face map is denoted by $d_{i}$, and the $i$th
degeneracy map is denoted by $s_{i}$. For a simplicial set $X$, we denote its
geometric realization by $|X|$. For a small category $C$, we denote its nerve
by $\mathcal{N}(C)$.

###  2.1. Barycentric subdivision

We here define what we mean by the barycentric subdivision of a regular CW
complex. Recall that a regular CW complex is a CW complex where each attaching
map is a homeomorphism onto its image.

######  Definition 2.1.

Let $X$ be a regular CW complex. The barycentric subdivision of $X$, denoted
$T(X)$, is a simplicial complex whose vertices are the cells of $X$ and whose
$n$–simplices are sequences of distinct cells $(\sigma_{0},\dots,\sigma_{n})$
such that $\sigma_{0}\subseteq\dots\subseteq\sigma_{n}$.

An example of a barycentric subdivision is given in Figure 1.

Figure 1. The barycentric subdivision of the simplicial complex $\Delta^{2}$.

######  Theorem 2.2.

If $X$ is a regular CW complex, then the geometric realization of the
barycentric subdivision, $|T(x)|$, is homeomorphic to $X$.

For a proof, see [10, pp. 80–81].

###  2.2. Simplicial homology

In this section we state some results on simplicial homology that will be
needed in later sections. We get the following result by combining Theorem 2.1
and Theorem 2.4 in [8, Chapter III].

######  Theorem 2.3.

Let $C$ be the (unnormalized) simplicial chain complex associated to a
simplicial set, and $D$ its subcomplex of degeneracies. Then, for all $n$
there is an isomorphism

$H_{n}(C)\cong H_{n}(C/D),$

induced by the projection map $p\colon C\to C/D$.

The following corollary follows immediately from applying the long exact
sequence of homology to the short exact sequence of chain complexes $0\to D\to
C\to C/D\to 0$.

######  Corollary 2.4.

Let $C$ be the (unnormalized) simplicial chain complex associated to a
simplicial set, and $D$ its subcomplex of degeneracies. Then,

$H_{n}(D)\cong 0,$

for all $n$.

###  2.3. Bisimplicial sets

A _bisimplicial set_ is a functor

$\Delta^{\operatorname{op}}\times\Delta^{\operatorname{op}}\to\mathsf{Set},$

or equivalently,

$\Delta^{\operatorname{op}}\to\mathsf{sSet}.$

For a bisimplicial set $X$, the _diagonal_ of $X$, denoted
$\operatorname{diag}X$, is the simplicial set given by:

  * •

$(\operatorname{diag}X)([n])=X([n],[n])$,

  * •

For $\theta\colon[m]\to[n]$,
$(\operatorname{diag}X)(\theta)=X(\theta,\theta)$.

We define the _geometric realization_ of a bisimplicial set $X$, denoted
$|X|$, as the realization of the diagonal, i.e.,
$|X|\coloneq|\operatorname{diag}X|$.

##  3\. Discrete Morse theory

In this section, we briefly summarize the core ideas and definitions from
discrete Morse theory.

In Morse theory, one studies certain differentiable functions, called _Morse
functions_ , on smooth manifolds. These Morse functions allow us to deduce
information about the topology of the manifold. For more information about
Morse theory, see e.g. [13]. Discrete Morse theory is a discrete analogue of
this, where one studies real-valued functions on CW complexes. The functions
assign a single real value to each cell of the CW complex, and hence we get a
discrete set of values.

We will only define discrete Morse functions for regular CW complexes. For the
general definition, see [6]. Note also that one often consider only discrete
Morse functions on simplicial complexes, which can be considered a special
class of regular CW complexes.

######  Definition 3.1.

Let $X$ be a regular CW complex. A _discrete Morse function_ on $X$ is a
function $f\colon X\to\mathbb{R}$ that assigns a real value to each cell of
$X$ such that for each $k$–cell $x$, the following conditions hold.

  1. (1)

There is at most one $(k+1)$–cell $y$ such that $x\subseteq y$ and $f(x)\geq
f(y)$.

  2. (2)

There is at most one $(k-1)$–cell $y$ such that $y\subseteq x$ and $f(y)\geq
f(x)$.

######  Definition 3.2.

An $k$–cell $x$ is called _critical_ with respect to a discrete Morse function
$f\colon X\to\mathbb{R}$ if the following conditions hold.

  1. (1)

There are no $(k+1)$–cells $y$ such that $x\subseteq y$ and $f(x)\geq f(y)$,
and

  2. (2)

There are no $(k-1)$–cells $y$ such that $y\subseteq x$ and $f(y)\geq f(x)$.

Cells that are not critical are called _regular_. Note that for a discrete
Morse function, it is impossible for both of these conditions to fail to hold
for a single cell (this is often called the exclusion lemma) [6, Lemma 2.5].
This also implies that if $x\subseteq y$ and $\dim y>\dim x+1$, then
$f(y)>f(x)$. An example of a discrete Morse function is given in Figure 2.

$\mathbf{\underline{7}}$$\mathbf{\underline{6}}$$\mathbf{2}$$\mathbf{4}$$\mathbf{\underline{1}}$$\mathbf{3}$$\mathbf{5}$
Figure 2. An example of a discrete Morse function. The critical simplices are
underlined and colored red.

###  3.1. Gradient vector fields

In discrete Morse theory, the essential information in a discrete Morse
function is the specification of those pairs such that one of the conditions
in 3.2 fails to hold, its _regular pairs_. Therefore, instead of talking about
a discrete Morse function $f$, we often talk about its induced _gradient
vector field_ $V_{f}$.

######  Definition 3.3.

Let $f\colon X\to\mathbb{R}$ be a discrete Morse function, let $x$ be a
$k$–cell and let $y$ be a $(k+1)$–cell. Then $\\{x,y\\}$ is called a _regular
pair_ if $x\subseteq y$ and $f(x)\geq f(y)$.

######  Definition 3.4.

Let $f\colon X\to\mathbb{R}$ be a discrete Morse function. The _induced
gradient vector field_ of $f$, denoted $V_{f}$, is defined as the set of
_regular pairs_ , i.e.,

$V_{f}=\\{\\{x,y\\}:x\subsetneq y,f(x)\geq f(y)\\}.$

An example of a gradient vector field is given in Figure 3. Two discrete Morse
functions that induce the same gradient vector field are said to be _Forman
equivalent_.

Figure 3. The gradient vector field for the discrete Morse function in Figure
2.

Given a regular CW complex $X$, a _discrete vector field_ on $X$ is a set $V$
of mutually disjoint pairs $\\{x,y\\}$ such that $\dim y=\dim x+1$. Given a
discrete vector field $V$, a _$V$ –path_ is a sequence of simplices
$(x_{0},y_{0},x_{1},\dots,y_{m},x_{m+1})$ such that $x_{i}\in X_{k},y_{i}\in
X_{k+1}$, $x_{i+1}\subseteq y_{i}$, $\\{x_{i},y_{i}\\}\in V$ and $x_{i+1}\neq
x_{i}$. Discrete vector fields and gradient vector fields are related in the
following way.

######  Theorem 3.5.

[6, Theorem 9.3] A discrete vector field $V$ is a gradient vector field of
some discrete Morse function if and only if there are no nontrivial $V$–paths
that start and end on the same cell.

###  3.2. Simplicial collapse

In this section, we restrict our attention to simplicial complexes, and define
what is known as a _simplicial collapse_. A _free face_ in a simplicial
complex is a $k$–simplex that is the face of exactly one $(k+1)$–simplex. If
$x\in X^{k}$ is a free face, and $y$ is its $(k+1)$–dimensional coface, then
$\\{x,y\\}$ is called a _free pair_. Removing a free pair from a simplicial
complex is called an _elementary collapse_. A sequence of elementary collapses
is called a _compound collapse_ or _simplicial collapse_. If the simplicial
complex $Y$ can be produced from a simplicial complex $X$ through a compound
collapse, we say that $X$ _collapses_ to $Y$ and write $X\searrow Y$. Of
particular importance is the fact that elementary collapses, and as a
consequence compound collapses, yield a deformation retract. That is, if
$X\searrow Y$, then $X$ deformation retracts to $Y$.

An example of an elementary collapse is given in Figure 4.

Figure 4. An elementary collapse.

##  4\. p-categories

A p-category is a category enriched in the category of posets. Thus, the Hom
sets all admit a poset structure with respect to which composites is
compatible, i.e., if $f\Rightarrow f^{\prime}$ and $g\Rightarrow g^{\prime}$
then $f\circ g\Rightarrow f^{\prime}\circ g^{\prime}$. Here we have used the
symbol $\Rightarrow$ to denote a partial order relation between two morphisms,
which we often will.

A _strict p-functor_ $F\colon C\to D$ maps objects in $C$ to objects in $D$,
and maps each Hom poset $\operatorname{Hom}_{C}(x,y)$ monotonically to
$\operatorname{Hom}_{D}(F(x),F(y))$, such that

  1. (1)

$F(\operatorname{id}_{x})=\operatorname{id}_{F(x)}$ for all objects $x$ in
$C$, and

  2. (2)

$F(f\circ g)=F(f)\circ F(g)$ for all composable $f$ and $g$.

A p-category can be viewed as a special case of a strict 2–category, where the
2–morphisms are partial order relations between morphisms. One can easily
verify that this satisfies all the axioms of a strict 2–category. In
particular, for a p-category viewed as a 2–category, there can only be zero or
one 2–morphisms between two 1–morphism (either there is a partial order
relation, or there is not).

There are several ways to define the classifying space of a p-category (and
more generally, a 2–category), but they are all homotopy equivalent ([2],
[3]). In the rest of this section, we define the _double nerve_ of a
p-category and let its geometric realization define the classifying space of a
p-category.

###  4.1. Simplicial categories

A simplicial category is a simplicial object in $\mathsf{Cat}$, i.e., a
functor

$\Delta^{\operatorname{op}}\to\mathsf{Cat}.$

Composing such a functor $F\colon\Delta^{\operatorname{op}}\to\mathsf{Cat}$
with the nerve functor $\mathcal{N}\colon\mathsf{Cat}\to\mathsf{sSet}$ gives
rise to a bisimplicial set $(\mathcal{N}\circ F)$. You can further take the
diagonal of this bisimplicial set to get the simplicial set
$\operatorname{diag}(\mathcal{N}\circ F)$.

######  Example 4.1.

Consider the following simplicial category $\mathscr{S}$.

  * •

$\mathscr{S}[0]$ is a category with two objects, $a$ and $b$, and two non-
identity morphisms, $f\colon a\to b$ and $g\colon a\to b$.

  * •

$\mathscr{S}[1]$ consists of degeneracies of $\mathscr{S}[0]$ and a single
non-degenerate morphism $F\colon s_{0}a\to s_{0}b$, with $d_{1}F=f$ and
$d_{0}F=g$.

  * •

For $n\geq 2$, all objects and morphisms in $\mathscr{S}[n]$ are degenerate.

Figure 5 illustrates the simplicial set
$\operatorname{diag}(\mathcal{N}\circ\mathscr{S})$ (omitting unimportant
degenerate simplices). Observe that the geometric realization is a suspension
of $|\Delta^{1}|$, i.e., homeomorphic to a disk $D^{2}$.

$a$$a$$b$$b$$s_{0}\operatorname{id}_{a}$$s_{0}\operatorname{id}_{b}$$s_{0}f$$s_{0}g$$F$$(s_{1},s_{0})F$$(s_{0},s_{1})F$
Figure 5. The simplicial set
$\operatorname{diag}(\mathcal{N}\circ\mathscr{S})$. Note that the top and
bottom edges are degenerate, as $s_{0}\operatorname{id}_{a}=(s_{0},s_{0})a$,
and similar for $s_{0}\operatorname{id}_{b}$. The left and right edges are,
however, not degenerate as 1–simplices in
$\operatorname{diag}(\mathcal{N}\circ\mathscr{S})$.

###  4.2. The double nerve of a p-category

Given a p-category $C$, one can construct a simplicial category
$\operatorname{\overline{N}}C$ as follows:

  * •

The objects in $(\operatorname{\overline{N}}C)[0]$ are the objects in $C$.

  * •

For $n\geq 1$, the objects in $(\operatorname{\overline{N}}C)[n]$ are
degeneracies of the objects in $(\operatorname{\overline{N}}C)[0]$.

  * •

The morphisms $a\to b$ in $(\operatorname{\overline{N}}C)[0]$ are the
morphisms $a\to b$ in $C$.

  * •

For $n\geq 1$, the morphisms $(s_{0})^{n}a\to(s_{0})^{n}b$ in
$(\operatorname{\overline{N}}C)[n]$ are $n$–simplices in
$\mathcal{N}\left(\operatorname{Hom}_{C}(a,b)\right)$ (where we consider the
poset $\operatorname{Hom}_{C}(a,b)$ as an ordinary category and take its
nerve).

In other words, morphisms in
$\operatorname{Hom}_{(\operatorname{\overline{N}}C)[n]}\left((s_{0})^{n}a,(s_{0})^{n}b\right)$
are sets of morphisms $\\{f_{0},\dots,f_{n}\\}$ in
$\operatorname{Hom}_{C}(a,b)$ such that

$f_{0}\Rightarrow f_{1}\Rightarrow\dots\Rightarrow f_{n}$

(here $(s_{0})^{n}$ means $s_{0}$ applied $n$ times).

The face and degeneracy maps are as in
$\mathcal{N}\left(\operatorname{Hom}_{C}(a,b)\right)$.

  * •

The composition of $\\{f_{0},\dots,f_{n}\\}\colon(s_{0})^{n}a\to(s_{0})^{n}b$
with $\\{g_{0},\dots,g_{n}\\}\colon(s_{0})^{n}b\to(s_{0})^{n}c$ is
$\\{g_{0}\circ f_{0},\dots,g_{n}\circ f_{n}\\}$.

It’s easy to verify that $g_{0}\circ f_{0}\Rightarrow g_{1}\circ
f_{1}\Rightarrow\dots\Rightarrow g_{n}\circ f_{n}$, so that the composition
rule is well-defined. It’s also easily verified that this composition rule is
associative, and that
$\\{\operatorname{id}_{x},\dots,\operatorname{id}_{x}\\}$ acts as the identity
on $(s_{0})^{n}x$. To show that $\operatorname{\overline{N}}C$ is a well-
defined simplicial category, it remains to show that the face maps and
degeneracy maps are functors between small categories. It’s clear that the
face map
$d_{i}\colon(\operatorname{\overline{N}}C)[n]\to(\operatorname{\overline{N}}C)[n-1]$
sends a morphisms
$\\{f_{0},\dots,f_{n}\\}\in\operatorname{Hom}_{(\operatorname{\overline{N}}C)([n])}\left((s_{0})^{n}a,(s_{0})^{n}b\right)$
to
$\operatorname{Hom}_{(\operatorname{\overline{N}}C)([n-1])}\left(d_{i}(s_{0})^{n}a,d_{i}(s_{0})^{n}b\right)$,
as $d_{i}(s_{0})^{n}=(s_{0})^{n-1}$. Furthermore, $d_{i}$ commutes with
composition:

$\displaystyle d_{i}$
$\displaystyle\left(\\{g_{0},\dots,g_{n}\\}\circ\\{f_{0},\dots,f_{n}\\}\right)$
$\displaystyle=d_{i}\\{g_{0}\circ f_{0},\dots,g_{n}\circ f_{n}\\}$
$\displaystyle=\\{g_{0}\circ f_{0},\dots,g_{i-1}\circ f_{i-1},g_{i+1}\circ
f_{i+1},\dots,g_{n}\circ f_{n}\\}$
$\displaystyle=\\{g_{0},\dots,g_{i-1},g_{i+1},\dots,g_{n}\\}\circ\\{f_{0},\dots,f_{i-1},f_{i+1},\dots,f_{n}\\}$
$\displaystyle=\left(d_{i}\\{g_{0},\dots,g_{n}\\}\right)\circ\left(d_{i}\\{f_{0},\dots,f_{n}\\}\right)$

Hence, the face maps are functors. Similar computations for $s_{i}$ gives that
the degeneracy maps are functors. In conclusion,
$\operatorname{\overline{N}}C$ is a well-defined simplicial category.

######  Example 4.2.

Let’s consider the p-category $\mathscr{P}$ with two objects, $a$ and $b$, two
non-identity morphisms, $f\colon a\to b$ and $g\colon a\to b$, and a single
non-identity partial order relation $f\Rightarrow g$ (see also Figure 6 for an
illustration).

$f$$g$$a$$b$$\Downarrow$ Figure 6. The p-category $\mathscr{P}$.

Let $\operatorname{\overline{N}}\mathscr{P}$ be the corresponding simplicial
category as described above. Then the objects in
$(\operatorname{\overline{N}}\mathscr{P})[0]$ are $a$ and $b$, while the
morphisms in $(\operatorname{\overline{N}}\mathscr{P})[0]$ are $f$ and $g$,
together with identities. There is a single non-degenerate morphism in
$(\operatorname{\overline{N}}\mathscr{P})[1]$, namely $F:=\\{f,g\\}$, which
has the faces $d_{1}F=f$ and $d_{0}F=g$. All other objects and morphisms are
degenerate. Thus, $\operatorname{\overline{N}}\mathscr{P}$ is precisely the
simplicial category $\mathscr{S}$ from 4.1.

As mentioned in the previous section, we can concatenate the simplicial
category $\operatorname{\overline{N}}C$ with the nerve operation $\mathcal{N}$
to get a bisimplicial set. We will call this composite operation the _double
nerve_.

######  Definition 4.3.

Let $C$ be a p-category. The _double nerve_ of $C$, denoted
$\operatorname{N\overline{N}}C$, is the bisimplicial set
$\mathcal{N}\circ(\operatorname{\overline{N}}C)$. The _classifying space_ of
$C$ is the geometric realization of the double nerve, i.e.,
$|\operatorname{N\overline{N}}C|$.

Note that in [2] and [3], the intermediate simplicial category
$\operatorname{\overline{N}}C$ ($\underline{\text{N}}C$ in their papers) is
defined differently; there, the $n$–simplices are horizontal compositions, not
vertical compositions. The double nerve, however, is the same in both
definitions, except that the indices in the bisimplicial set are swapped.

##  5\. The discrete flow category

In this section we present the definition of the discrete flow category, as
defined by Nanda in [14]. Note that we only define the discrete flow category
of a discrete Morse function on a regular CW complex. For the general discrete
flow category, see [14].

###  5.1. The entrance path category

We now describe an example of a p-category, the _entrance path category_ of a
regular CW complex. It is similar to the face poset, but has more structure:
it includes data on _how_ a cell is a face of another cell.

######  Definition 5.1.

Let $X$ be a regular CW complex. The _entrance path category_
$\operatorname{Ent}[X]$ of $X$ is a p-category given by the following.

  1. (1)

The objects are the cells in $X$.

  2. (2)

The morphisms $x\to y$ are strictly descending sequences of cells
$(x=x_{0}>x_{1}>\dots>x_{k}=y)$, with the understanding that
$\operatorname{id}_{x}$ is the sequence $(x)$ with a single element.

  3. (3)

The partial order is defined so that $f\Rightarrow f^{\prime}$ if and only if
$f$ is a (not necessarily contiguous) subsequence of $f^{\prime}$.

  4. (4)

Composition of morphisms are given by concatenating sequences as follows.

$\left(z>x_{1}>\dots>x_{k}\right)\circ\left(y_{0}>\dots>y_{l-1}>z\right)=\left(y_{0}>\dots>y_{l-1}>z>x_{1}>\dots
x_{k}\right)$

A theorem, proved in [14, p. 11], states the following.

######  Theorem 5.2.

Let $X$ be a finite regular CW complex. Then $X$ is homotopy equivalent to
$|\mathcal{N}(\operatorname{Ent}[X])|$.

###  5.2. Localization

Given a category $C$ and a collection of morphisms $Q$ that contains all
identities and is closed under composition, one can construct a new category
$C[Q^{-1}]$, called the _localization_ of $C$ at $Q$. This category is the
minimal category containing $C$ where all morphisms in $Q$ have inverses. The
localization comes with a functor $L\colon C\to C[Q^{-1}]$ that sends objects
in $C$ to their copies in $C[Q^{-1}]$ and morphisms in $C$ to their
equivalence classes in $C[Q^{-1}]$.

We now describe how to localize a p-category at a special class of morphisms.

######  Definition 5.3.

A morphism $f\colon x\to y$ in a p-category $C$ is called an _atom_ if

  1. (1)

$f\Rightarrow f^{\prime}$ holds for any $f^{\prime}\in C(x,y)$,

  2. (2)

$x=y$ implies $f=\operatorname{id}_{x}$, and

  3. (3)

Solutions to $h\circ g\Rightarrow f$ for morphisms $g\colon x\to z$ and
$h\colon z\to y$ only exist when

     * •

$z=x$, in which case $(g,h)=(\operatorname{id}_{x},f)$, or

     * •

$z=y$, in which case $(g,h)=(f,\operatorname{id}_{y})$.

As an example, in the entrance path category of a regular CW complex, the
atoms are precisely the sequences of length 2, i.e., those on the form
$(x>y)$, together with the identities.

######  Definition 5.4.

A collection $\Sigma$ of morphisms in a p-category is called _directed_ if

  1. (1)

all morphisms in $\Sigma$ are atoms,

  2. (2)

if $f\colon x\to y$ is in $\Sigma$, then $x\neq y$, and

  3. (3)

if $f\colon x\to y$ is in $\Sigma$, then there are no morphisms $y\to x$ in
$\Sigma$.

We again give an example in the entrance path category. Let $f\colon
X\to\mathbb{R}$ be a discrete Morse function. The gradient vector field
$V_{f}=\\{(x_{i},y_{i})\\}$ gives a collection $\Sigma=\\{(y_{i}>x_{i})\\}$ of
morphisms in $\operatorname{Ent}[X]$. It’s easily verified that $\Sigma$ is
directed.

Now, given a p-category $C$ and a directed collection $\Sigma$ of morphisms in
$C$ so that the union $\Sigma^{+}$ with all identities is closed under
composition, we can define the _localization of $C$ at $\Sigma$_. We write
this as $\operatorname{Loc}_{\Sigma}C$, and define it as follows.

######  Definition 5.5.

Let $C$, $\Sigma$ and $\Sigma^{+}$ be as above. Then
$\operatorname{Loc}_{\Sigma}C$ is a p-category given by the following.

  * •

The objects are the same as the objects in $C$.

  * •

The morphisms $w\to z$ are equivalence classes of zigzags of the form

${w}$${y_{0}}$${x_{0}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k}$$g_{k+1}$

where the $f_{i}$’s are in $\Sigma^{+}$, and the $g_{i}$’s are arbitrary, and
the equivalence relation is generated by the following relations. Two zigzags
are related _horizontally_ if they differ by intermediate identity maps, and
_vertically_ if they form the rows of a commutative diagram of the form

${w}$${y_{0}}$${x_{0}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$${w}$${y_{0}^{\prime}}$${x_{0}^{\prime}}$${y_{1}^{\prime}}$${\cdots}$${x_{k}^{\prime}}$${z}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k}$$g_{k+1}$$g_{0}^{\prime}$$f_{0}^{\prime}$$g_{1}^{\prime}$$f_{1}^{\prime}$$f_{k}^{\prime}$$g_{k+1}^{\prime}$$\operatorname{id}_{w}$$u_{0}$$v_{0}$$u_{1}$$v_{k}$$\operatorname{id}_{z}$

where the $u_{i}$’s and $v_{i}$’s are in $\Sigma^{+}$.

  * •

The partial order is defined so that $\gamma^{\prime}\Rightarrow\gamma$ if and
only if there exist representatives of $\gamma$ and $\gamma^{\prime}$ that fit
into the top and bottom row, respectively, of a (not necessarily commutative)
diagram of the form

${w}$${y_{0}}$${x_{0}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$${w}$${y_{0}^{\prime}}$${x_{0}^{\prime}}$${y_{1}^{\prime}}$${\cdots}$${x_{k}^{\prime}}$${z}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k}$$g_{k+1}$$g_{0}^{\prime}$$f_{0}^{\prime}$$g_{1}^{\prime}$$f_{1}^{\prime}$$f_{k}^{\prime}$$g_{k+1}^{\prime}$$\operatorname{id}_{w}$$u_{0}$$v_{0}$$u_{1}$$v_{k}$$\operatorname{id}_{z}$

where, again, $u_{i}$ and $v_{i}$ are in $\Sigma^{+}$.

  * •

Composition of morphisms are given by concatenating representatives as follows

$\displaystyle\left[w\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}\dots\
\xleftarrow{f_{k}}x_{k}\xrightarrow{g_{k+1}}z\right]\circ\left[v\xrightarrow{g_{0}^{\prime}}y_{0}^{\prime}\xleftarrow{f_{0}^{\prime}}\dots\
\xleftarrow{f_{k}^{\prime}}x_{k}^{\prime}\xrightarrow{g_{k+1}^{\prime}}w\right]$
$\displaystyle\quad=\left[v\xrightarrow{g_{0}^{\prime}}y_{0}^{\prime}\xleftarrow{f_{0}^{\prime}}\dots\
\xleftarrow{f_{k}^{\prime}}x_{k}^{\prime}\xrightarrow{g_{0}\circ
g_{k+1}^{\prime}}y_{0}\xleftarrow{f_{0}}\dots\
\xleftarrow{f_{k}}x_{k}\xrightarrow{g_{k+1}}z\right].$

One also gets a functor (or equivalently, a strict p-functor) in this case,
which we write $L_{\Sigma}\colon C\to\operatorname{Loc}_{\Sigma}C$. The
functor sends objects to themselves and morphisms to their respective
equivalence classes.

As mentioned above, a gradient vector field $V_{f}=\\{(x_{i},y_{i})\\}$ gives
a directed collection $\Sigma=\\{(y_{i}>x_{i})\\}$ on the entrance path
category. Thus, given a discrete Morse function $f\colon X\to\mathbb{R}$, we
can localize $\operatorname{Ent}[X]$ at $\Sigma$, which will give us a
category where the morphisms corresponding to regular pairs have inverses. We
will call this $\Sigma$ the _Morse system_ induced by $f$.

###  5.3. The discrete flow category

We are now ready to define the discrete flow category of a discrete Morse
function.

######  Definition 5.6.

Let $f\colon X\to\mathbb{R}$ be a discrete Morse function, and let $\Sigma$ be
its induced Morse system. Let $\operatorname{Loc}_{\Sigma}[X]$ be the
p-category localization of $\operatorname{Ent}[X]$ at $\Sigma$. The _discrete
flow category_ of $f$, denoted $\operatorname{Flo}_{\Sigma}[X]$, is the full
subcategory of $\operatorname{Loc}_{\Sigma}[X]$ generated by the critical
cells of $f$.

A special case of the main result of [14] states the following.

######  Theorem 5.7.

Let $f\colon X\to\mathbb{R}$ be a discrete Morse function, and let $\Sigma$ be
its induced Morse system. Then the classifying space of the discrete flow
category of $f$ is homotopy equivalent to $X$.

######  Example 5.8.

As an example, consider the discrete Morse function on $X\cong S^{1}$
illustrated in Figure 7.

$x$$y$$z$$a$$b$$c$ Figure 7. A gradient vector field of a discrete Morse
function $f\colon X\to\mathbb{R}$.

The objects of $\operatorname{Flo}_{\Sigma}[X]$ are the critical cells of $f$,
which are $x$ and $c$. There are two morphisms from $x$ to $c$, corresponding
to the two zigzag paths $x>b<y>c$ and $x>a<z>c$. There are no partial order
relation between these morphisms. The other Hom posets are
$\operatorname{Hom}(x,x)=\\{\operatorname{id}_{x}\\}$,
$\operatorname{Hom}(c,c)=\\{\operatorname{id}_{c}\\}$ and
$\operatorname{Hom}(c,x)=\emptyset$. Thus, the classifying space
$|\mathcal{N}(\operatorname{Flo}_{\Sigma}[X])|$ becomes $S^{1}$, as
illustrated in Figure 8.

$[x>a<z>c]$$[x>b<y>c]$$x$$c$$\mathcal{N}$$[x>a<z>c]$$[x>b<y>c]$$x$$c$ Figure
8. The discrete flow category and its nerve (the identity morphisms are
omitted).

##  6\. Hom posets of discrete flow categories

In this section we study the structure of the Hom posets of the discrete flow
category of a discrete Morse function. We first develop some technical results
on the Hom posets, and in particular show that the posets are _graded_. We
then use these results to construct an algorithm to compute said posets.
Finally, we prove A: for a discrete Morse function on a simplicial complex
$X$, the opposite poset of each Hom poset in its discrete flow category is a
_CW poset_ , i.e., it has the structure of a face poset of a regular CW
complex.

###  6.1. Preliminaries on posets

In [1], Björner describes sufficient conditions for a poset to be the face
poset of a regular CW complex. Note that he defines face posets to have an
added least element $\hat{0}$, as opposed to our definition of
$\operatorname{Fac}[X]$. Björner’s definition of CW posets is as follows.

######  Definition 6.1.

A poset $P$ is said to be a _CW poset_ if

  1. (1)

$P$ has a least element $\hat{0}$,

  2. (2)

$P$ is nontrivial, i.e., has more than one element,

  3. (3)

For all $x\in P\setminus\\{\hat{0}\\}$ the realization of the open interval
$(\hat{0},x)$ is homeomorphic to a sphere (i.e., to some $S^{k}$, where $k$
depends on $x$).

We now formalize the statement that CW posets are the face posets of regular
CW complexes (augmented with a least element).

######  Definition 6.2.

Let $X$ be a regular CW complex. Let the _face poset_ of $X$, denoted
$\operatorname{Fac}[X]$, be the poset where the elements are the cells in $X$
and $x\geq y$ whenever $y$ is contained in $x$.

Denote by $\operatorname{Fac}^{+}[X]$ the face poset $\operatorname{Fac}[X]$
augmented with a least element $\hat{0}$. In other words,
$\operatorname{Fac}^{+}[X]=\operatorname{Fac}[X]\cup\\{\hat{0}\\}$, where
$\hat{0}$ is defined to be smaller than all other elements.

The following statement is proved in [1, p. 11].

######  Theorem 6.3.

A poset $P$ is a CW poset if and only if it is isomorphic to
$\operatorname{Fac}^{+}[X]$ for some regular CW complex $X$.

A poset is _bounded_ if it has a least and greatest element. Furthermore, we
say that $x$ _covers_ $y$ if $x>y$ and there exists no element $z$ such that
$x>z>y$.

######  Definition 6.4.

A graded poset is a poset $P$ equipped with a _rank function_ , which is a
function $\rho\colon P\to\mathbb{Z}$ satisfying the following two properties.

  1. (1)

The function $\rho$ is compatible with the partial order, meaning that $x>y$
implies $\rho(x)>\rho(y)$.

  2. (2)

The function $\rho$ is compatible with the covering relation, meaning that if
$x$ covers $y$ then $\rho(x)=\rho(y)+1$.

For an element $x$, we will call $\rho(x)$ the _rank_ of $x$.

###  6.2. Hom posets of a discrete flow category

In this section, we let $X$ be a finite regular CW complex, $f\colon
X\to\mathbb{R}$ a discrete Morse function, and $\Sigma$ the Morse system on
$\operatorname{Ent}[X]$ consisting of the regular pairs of $f$. Furthermore,
we let $w$ and $z$ be arbitrary objects in $\operatorname{Flo}_{\Sigma}[X]$
and consider the Hom poset $\operatorname{Hom}(w,z)$ (note that whenever we
write $\operatorname{Hom}(w,z)$, we mean
$\operatorname{Hom}_{\operatorname{Flo}_{\Sigma}[X]}(w,z)$).

First, we construct an algebraic invariant for morphisms in the discrete flow
category, which we will use when proving that two different morphisms are not
equal.

Let $G$ be the free Abelian group on all morphisms in $\operatorname{Ent}[X]$
that are atoms, modulo all identities (i.e., we define identities to be 0).
The algebraic invariant will be an element of this group. Given a
representative

$\gamma=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right)$

of some $[\gamma]\in\operatorname{Hom}(w,z)$, we let

$\alpha(g_{i})=\left(g_{i}^{0}+g_{i}^{1}+\dots+g_{i}^{N_{i}}\right)\in G,$

where $g_{i}=g_{i}^{N_{i}}\circ\dots\circ g_{i}^{0}$ is the atom decomposition
of $g_{i}$. We now define the algebraic invariant $I$ as follows.

######  Definition 6.5.

Let $[\gamma]\in\operatorname{Hom}(w,z)$. Define $I([\gamma])$ as

$I([\gamma])=\sum_{i=0}^{k}\alpha(g_{i})-\sum_{i=0}^{k-1}f_{i}.$ (1)

######  Lemma 6.6.

The function $I$, as defined in 6.5, is well-defined.

###### Proof.

To show that $I$ is well-defined, we must show that it is preserved under
horizontal and vertical relations.

(H)

To see that it is preserved under horizontal relations, it is enough to check
that $I([\gamma\circ\gamma^{\prime}])=I([\gamma])+I([\gamma^{\prime}])$, and
that

$I([x_{i}\xrightarrow{g_{i}}y_{i}\xleftarrow{\operatorname{id}}x_{i+1}\xrightarrow{g_{i+1}}y_{i+1}])=I([x_{i}\xrightarrow{g_{i+1}\circ
g_{i}}y_{i+1}]),$

which follows from the fact that identities are defined to be 0 in $G$, and
that $\alpha(g\circ h)=\alpha(g)+\alpha(h)$.

(V)

To see that $I$ is preserved under vertical relations, consider a diagram:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${\cdots}$${x_{k}^{\prime}}$${z}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$g_{0}^{\prime}$$f_{0}^{\prime}$$g_{1}^{\prime}$$f_{1}^{\prime}$$f_{k-1}^{\prime}$$g_{k}^{\prime}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k-1}$$g_{k}$$u_{0}$$v_{1}$$u_{1}$$v_{k}$

We have $\alpha(g_{i}^{\prime})=\alpha(g_{i})+v_{i}-u_{i}$, and
$f_{i}^{\prime}=f_{i}+u_{i}-v_{i+1}$ (as elements of $G$). Putting this
together gives

$\displaystyle I([\gamma])$
$\displaystyle=\sum_{i=0}^{k}\alpha(g_{i}^{\prime})-\sum_{i=0}^{k-1}f_{i}^{\prime}$
$\displaystyle=\sum_{i=0}^{k}\alpha(g_{i})+\sum_{i=0}^{k}v_{i}-\sum_{i=0}^{k}u_{i}-\left(\sum_{i=0}^{k-1}f_{i}+\sum_{i=0}^{k-1}u_{i}-\sum_{i=0}^{k-1}v_{i+1}\right)$
$\displaystyle=\sum_{i=0}^{k}\alpha(g_{i})-\sum_{i=0}^{k-1}f_{i},$

where we use that $v_{0}=\operatorname{id}_{w}=0$ and
$u_{k}=\operatorname{id}_{z}=0$. Hence, $I$ is preserved under vertical
relations.∎

######  Theorem 6.7.

Let $[\gamma]$ and $[\gamma^{\prime}]$ be morphisms in
$\operatorname{Hom}(w,z)$ such that $[\gamma]\Rightarrow[\gamma^{\prime}]$.
Let

$\tau=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right)$

be a representative of $[\gamma]$. Then it is possible to choose a
representative $\tau^{\prime}$ of $[\gamma^{\prime}]$, such that

$\tau^{\prime}=\left(w=x_{0}\xrightarrow{g_{0}^{\prime}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}^{\prime}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}^{\prime}}y_{k}=z\right),$

for some $g_{i}^{\prime}$, such that $g_{i}\Rightarrow g_{i}^{\prime}$ holds
for all $i$.

Furthermore, the $g_{i}^{\prime}$ are unique.

Note that a partial order $g_{i}\Rightarrow g_{i}^{\prime}$ means that
$g_{i}=(z_{0}>z_{1}>\dots>z_{m})$ is a subsequence of
$g_{i}^{\prime}=(z_{0}^{\prime}>z_{1}^{\prime}>\dots>z_{n}^{\prime})$.
Therefore, this theorem tells us that a partial order
$[\gamma]\Rightarrow[\gamma^{\prime}]$ corresponds to picking a representative
of $[\gamma]$ and adding elements to the sequences that constitutes the right-
pointing arrows (the $g_{i}$).

###### Proof.

We prove this in two parts. First we show that there exists a diagram

${w}$${\bar{y}_{0}}$${\bar{x}_{1}}$${\bar{y}_{1}}$${\cdots}$${\bar{x}_{k}}$${z}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$\bar{g}_{0}$$\bar{f}_{0}$$\bar{g}_{1}$$\bar{f}_{1}$$\bar{f}_{k-1}$$\bar{g}_{k}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k-1}$$g_{k}$$u_{0}$$v_{1}$$u_{1}$$v_{k}$
(2)

where the bottom row is $\tau$ and the top row is a representative of
$[\gamma^{\prime}]$. Then we show that we can modify this diagram so that the
top row is of the form

$\tau^{\prime}=\left(w=x_{0}\xrightarrow{g_{0}^{\prime}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}^{\prime}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}^{\prime}}y_{k}=z\right),$

where $\tau^{\prime}$ is still a representative of $[\gamma^{\prime}]$.

To show the first part, we show that given a diagram representing a partial
order relation (as the one in (2)), we can replace the bottom row by either

(V)

a vertically related representative, or

(H)

a horizontally related representative,

and modify the top row appropriately to get a new diagram.

(V)

The case with vertically related representatives is simple; you simply compose
the partial order diagram with the vertical relation diagram.

(H)

For horizontal relations, we must show that you can both add intermediate
identity maps, and remove them.

The case with adding intermediate identity maps is also simple, you simply add
identity maps to both the top and bottom rows and copy the vertical map
(clearly, the top rows is also horizontally related). For example, for adding
an identity map at an $x_{i}$, you replace

${\cdots}$${x_{i}^{\prime}}$${\cdots}$${\cdots}$${x_{i}}$${\cdots}$$v_{i}$

with

${\cdots}$${x_{i}^{\prime}}$${x_{i}^{\prime}}$${x_{i}^{\prime}}$${\cdots}$${\cdots}$${x_{i}}$${x_{i}}$${x_{i}}$${\cdots}$$\operatorname{id}$$\operatorname{id}$$\operatorname{id}$$\operatorname{id}$$v_{i}$$v_{i}$$v_{i}$

Removing intermediate identity maps is more complicated (in fact, it only
works on the bottom row, not the top row). Consider a diagram

${x_{i}^{\prime}}$${y_{i}^{\prime}}$${x_{i+1}^{\prime}}$${y_{i+1}^{\prime}}$${x_{i}}$${y_{i}}$${x_{i+1}}$${y_{i+1}}$$g_{i}^{\prime}$$f_{i}^{\prime}$$g_{i+1}^{\prime}$$g_{i}$$g_{i+1}$$v_{i}$$u_{i}$$v_{i+1}$$u_{i+1}$

If $f_{i}^{\prime}=\operatorname{id}$, we can clearly remove the identity maps
from both rows. Suppose not. Then $u_{i}$ must be $\operatorname{id}$ and
$v_{i+1}=f_{i}^{\prime}$. As $g_{i+1}\circ v_{i+1}\Rightarrow u_{i+1}\circ
g_{i+1}^{\prime}$, and $v_{i+1}$ is an atom on the form
$(x_{i+1}^{\prime}>x_{i+1})$ with $\dim x_{i+1}^{\prime}=\dim x_{i+1}+1$, we
must have that $v_{i+1}\circ g_{i+1}^{\prime}$ is sequence that starts with
$(x_{1}^{\prime}>x_{1})$. There are now two cases: either
$g_{i+1}^{\prime}=\hat{g}_{i+1}\circ v_{i+1}$ for some $\hat{g}_{i+1}$, or
$g_{i+1}^{\prime}=\operatorname{id}$ and $u_{i+1}=v_{i+1}$.

In the first case, the top row is equivalent to

$x_{i}^{\prime}\xrightarrow{\hat{g}_{i+1}\circ g_{i}^{\prime}}y_{i+1},$

as $g_{i+1}^{\prime}=\hat{g}_{i}\circ v_{i+1}=\hat{g}_{i}\circ
f_{i}^{\prime}$, and we can replace the diagram with

${x_{i}^{\prime}}$${y_{i+1}^{\prime}}$${x_{i}}$${y_{i+1}}$$\hat{g}_{i+1}\circ
g_{i}^{\prime}$$g_{i+1}\circ g_{i}$$v_{i}$$u_{i+1}$

The reader can verify that in the case $g_{i+1}^{\prime}=\operatorname{id}$,
$f_{i+1}^{\prime}$ will be forced to equal $\operatorname{id}$, which allows
us to make a similar rewrite to remove the identity map from the bottom row.

Now, we have a diagram

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${\cdots}$${x_{k}^{\prime}}$${z}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$g_{0}^{\prime}$$f_{0}^{\prime}$$g_{1}^{\prime}$$f_{1}^{\prime}$$f_{k-1}^{\prime}$$g_{k}^{\prime}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k-1}$$g_{k}$$u_{0}$$v_{1}$$u_{1}$$v_{k}$

such that the bottom row is a $[\tau]$ and the top row is a representative of
$[\gamma^{\prime}]$.

Assume either $x_{1}\neq x_{1}^{\prime}$ or $y_{0}\neq y_{0}^{\prime}$. There
are three cases to consider:

(I)

$x_{1}\neq x_{1}^{\prime}$ and $y_{0}\neq y_{0}^{\prime}$

(II)

$x_{1}=x_{1}^{\prime}$ and $y_{0}\neq y_{0}^{\prime}$

(III)

$x_{1}\neq x_{1}^{\prime}$ and $y_{0}=y_{0}^{\prime}$.

Case I

In this case, the exhaustion axiom gives that $y_{0}=x_{1}$ and
$y_{0}^{\prime}=x_{1}^{\prime}$, which implies that $v_{1}=u_{0}$. The first
three squares in the diagram is then:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$g_{0}^{\prime}$$g_{1}^{\prime}$$g_{0}$$g_{1}$$u_{0}$$u_{0}$$u_{1}$

As $g_{1}\circ u_{0}\Rightarrow u_{1}\circ g_{1}^{\prime}$, and $u_{0}$ is an
atom on the form $(x_{1}^{\prime}>x_{1})$ with $\dim x_{1}^{\prime}=\dim
x_{1}+1$, we must have that $u_{1}\circ g_{1}^{\prime}$ is sequence that
starts with $(x_{1}^{\prime}>x_{1})$. There are now two cases: either
$g_{1}^{\prime}=\hat{g}_{1}\circ u_{0}$ for some $\hat{g}_{1}$, or
$g_{1}^{\prime}=\operatorname{id}_{x_{1}^{\prime}}$ and $u_{1}=u_{0}=g_{1}$.

In the first case, the top row is vertically related to

$w\xrightarrow{u_{0}\circ
g_{0}^{\prime}}y_{0}\xleftarrow{\operatorname{id}}x_{1}\xrightarrow{\hat{g}_{1}}y_{1}^{\prime},$

so we can replace the diagram with:

${w}$${y_{0}}$${x_{1}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$u_{0}\circ
g_{0}^{\prime}$$\hat{g}_{1}$$g_{0}$$g_{1}$$u_{1}$

In the second case, we must have $g_{1}=\operatorname{id}$ and
$u_{1}=\operatorname{id}$. Hence, the first four squares in the diagram:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${x_{2}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${x_{2}}$$g_{0}^{\prime}$$g_{0}$$f_{1}$$u_{0}$$u_{0}$$u_{0}$$v_{2}$

Now, the top row is vertically related to

$w\xrightarrow{u_{0}\circ
g_{0}^{\prime}}y_{0}\xleftarrow{\operatorname{id}}x_{1}\xrightarrow{\operatorname{id}}y_{1}^{\prime}\xleftarrow{u_{0}}x_{2}^{\prime},$

so we can replace the diagram with:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${x_{2}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${x_{2}}$$u_{0}\circ
g_{0}^{\prime}$$u_{0}$$g_{0}$$f_{1}$$v_{2}$

Case II

In this case the exhaustion axiom gives $y_{0}^{\prime}=x_{1}^{\prime}$ and
$f_{0}=u_{0}$, so the first three squares of the diagram is:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$g_{0}^{\prime}$$g_{1}^{\prime}$$g_{0}$$u_{0}$$g_{1}$$u_{0}$$u_{1}$

The top row is vertically related to $x_{0}\xrightarrow{u_{0}\circ
g_{0}^{\prime}}y_{0}\xleftarrow{u_{0}}x_{1}\xrightarrow{g_{1}}y_{1}^{\prime}$
(see [14, Remark 2.9]). We can thus replace this part of the diagram with:

${w}$${y_{0}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$u_{0}\circ
g_{0}^{\prime}$$u_{0}$$g_{1}^{\prime}$$g_{0}$$u_{0}$$g_{1}$$u_{1}$

Case III

In this case the exhaustion axiom gives $y_{0}=x_{1}$ and
$f_{0}^{\prime}=v_{1}$, so the first three squares of the diagram is:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$g_{0}^{\prime}$$v_{1}$$g_{1}^{\prime}$$g_{0}$$g_{1}$$v_{1}$$u_{1}$

As in the first case, either $g_{1}^{\prime}=\operatorname{id}$ or
$g_{1}^{\prime}=\hat{g}_{1}\circ v_{1}$ for some $\hat{g}_{1}$. In the latter
case, the top row is vertically related to

$w\xrightarrow{g_{0}^{\prime}}y_{0}^{\prime}\xleftarrow{\operatorname{id}}x_{1}\xrightarrow{\hat{g}_{1}}y_{1}^{\prime},$

and we can replace the diagram with

${w}$${y_{0}}$${x_{1}}$${y_{1}^{\prime}}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$$g_{0}^{\prime}$$\hat{g}_{1}$$g_{0}$$g_{1}$$u_{1}$

If, instead, $g_{1}^{\prime}=\operatorname{id}$, we must have
$g_{1}=\operatorname{id}$ and $u_{1}=v_{1}$. As in case 1, we look at the
first four squares of the diagram, and see that we can replace the top row, in
this case with

$w\xrightarrow{g_{0}^{\prime}}y_{0}^{\prime}\xleftarrow{\operatorname{id}}x_{1}\xrightarrow{\operatorname{id}}y_{1}\xleftarrow{v_{1}}x_{2}^{\prime}.$

Continuing this process on the right until you reach the end of the diagram,
you end up with a new diagram on the form

${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$g_{0}^{\prime\prime}$$f_{0}$$g_{1}^{\prime\prime}$$f_{1}$$f_{k-1}$$g_{k}^{\prime\prime}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k-1}$$g_{k}$

where the and bottom row is $\tau$ and the top row is a representative of
$[\gamma^{\prime}]$. It follows from the diagram that $g_{i}\Rightarrow
g_{i}^{\prime\prime}$ for all $i$, and hence the top row is our desired
representative $\tau^{\prime}$.

Finally, we prove that the $g_{i}^{\prime}$ are unique. We do this by using
the algebraic invariant $I$, as defined in 6.5, and showing that given $I$,
the $g_{i}^{\prime}$ are decided by the $x_{i}$, $y_{i}$ and $f_{i}$.

We know that $I$ is an algebraic invariant for morphisms in
$\operatorname{Hom}(w,z)$. Thus, given a representation $\tau^{\prime}$ as in
the theorem statement, where the $x_{i}$, $y_{i}$ and $f_{i}$ are given, we
know what all the atoms in the $g_{i}^{\prime}$ are. It remains to show that
these atoms can only appear in one specific order, which will mean that all
the $g_{i}^{\prime}$ are uniquely defined.

For this, recall that $f\colon X\to\mathbb{R}$ is the Morse function from
which the Morse system is induced. We can assume without loss of generality
that $f$ is injective on the cells of $X$. Now, for an atom $(x>y)$, let
$\beta((x>y))=(\dim x,f(x),f(y))\in\mathbb{R}^{3}$. Then
$\beta((x>y))=\beta((x^{\prime}>y^{\prime}))$ if and only if
$(x>y)=(x^{\prime}>y^{\prime})$ (as $f$ is injective). Furthermore, using the
lexicographical ordering on $\mathbb{R}^{3}$, $\beta$ is decreasing on the
atoms in the right-pointing arrows of a representation, in the order they
appear:

  * •

If $(x^{\prime}>y^{\prime})$ directly succeeds $(x>y)$ (possibly with an
intermediate identity maps), then $x^{\prime}=y$, and $\dim x^{\prime}>\dim
y^{\prime}=\dim x>\dim y$.

  * •

If there is a non-identity left-pointing map $f_{i}$ directly between two
atoms $(x>y)$ and $(x^{\prime}>y^{\prime})$, then $f_{i}=(x^{\prime}>y)$, and
$\dim x=\dim x^{\prime}$. Then, we have $f(y)>f(x^{\prime})$, and either
$x=x^{\prime}$ or $f(x)>f(y)$, so $f(x)\geq f(x^{\prime})$. Likewise, either
$y=y^{\prime}$ or $f(x^{\prime})>f(y^{\prime})$ so $f(y)\geq f(y^{\prime})$.
In conclusion, this gives $\beta((x>y))\geq\beta((x^{\prime}>y^{\prime}))$.

Thus, given the $x_{i}$, $y_{i}$ and $f_{i}$ in a representation
$\tau^{\prime}$ as in the theorem statement, the atoms in the
$g_{i}^{\prime}$, and their order, is uniquely determined, and hence the
$g_{i}^{\prime}$ themselves are uniquely determined, which proves the last
part of the theorem. ∎

Now, let $\operatorname{Hom}(w,z)^{\operatorname{op}}$ be the _opposite_ of
the poset $\operatorname{Hom}(w,z)$, i.e., the poset with elements equal to
the elements of $\operatorname{Hom}(w,z)$ and with the partial order reversed.
Define further $P_{w,z}$ as $\operatorname{Hom}(w,z)^{\operatorname{op}}$
augmented with a least element $\hat{0}$. In other words,

$P_{w,z}=\operatorname{Hom}(w,z)^{\operatorname{op}}\cup\\{\hat{0}\\},$ (3)

where $\hat{0}$ is defined to be smaller than all other elements.

Next, we show that $P_{w,z}$ is graded by defining a rank function on it. To
do this, we first define the following rank function $r$ on
$\operatorname{Hom}_{\operatorname{Ent}[X]}(a,b)^{\operatorname{op}}$, for
arbitrary objects $a,b\in\operatorname{Ent}[X]$.

$r\left(a=x_{0}>x_{1}>\dots>x_{k}=b\right)=k$ (4)

Here, we take $r(\operatorname{id}_{x})$ to be $0$.

It’s easy to verify that $r$ is, in fact, a rank function. Also, observe that
$r$ satisfies the following identity.

$r(\gamma\circ\gamma^{\prime})=r(\gamma)+r(\gamma^{\prime})$ (5)

We now define a rank function on $P_{w,z}$.

######  Definition 6.8.

Define the function $\rho\colon P_{w,z}\to\mathbb{Z}$ as follows.

Let $x\in P_{w,z}$. If $x=\hat{0}$, then $\rho(x)=-1$. Else, let $x$ be
represented by

$\tau=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right).$

Then,

$\rho(x)=N+\sum_{i=0}^{k-1}r(f_{i})-\sum_{i=0}^{k}r(g_{i}),$

where $N=\dim w-\dim z$.

The rank function $\rho$ can be interpreted as the number of cells that are
“skipped” in the $g_{i}$’s in the representative.

######  Theorem 6.9.

The function $\rho$, as defined in 6.8, is well-defined and a rank function.

###### Proof.

First, to prove well-definedness of $\rho$, we need to show that equivalent
representations give the same value of $\rho$. To do this, we first show that
horizontally related representations give the same value of $\rho$, and then
we show that vertically related representations give the same value of $\rho$.

  1. (1)

Let $\tau$ and $\tau^{\prime}$ be horizontally related representations. Then
$\tau$ can be transformed to $\tau^{\prime}$ by making substitutions of the
form

$\left(x_{i}\xrightarrow{g_{i}}y_{i}\xleftarrow{\operatorname{id}_{y_{i}}}y_{i}\xrightarrow{g_{i+1}}y_{i+1}\right)\leftrightarrow\left(x_{i}\xrightarrow{g_{i+1}\circ
g_{i}}y_{i+1}\right).$

As $r(g_{i+1}\circ g_{i})=r(g_{i+1})+r(g_{i})$ and
$r(\operatorname{id}_{y_{i}})=0$, these substitutions preserve the value of
$\rho$, so both $\tau$ and $\tau^{\prime}$ give the same value of $\rho$.

  2. (2)

Let $\tau$ and $\tau^{\prime}$ be vertically related representations. Then
they form the bottom and top rows of a commutative diagram:

${w}$${y_{0}^{\prime}}$${x_{1}^{\prime}}$${y_{1}^{\prime}}$${\cdots}$${x_{k}^{\prime}}$${z}$${w}$${y_{0}}$${x_{1}}$${y_{1}}$${\cdots}$${x_{k}}$${z}$$g_{0}^{\prime}$$f_{0}^{\prime}$$g_{1}^{\prime}$$f_{1}^{\prime}$$f_{k-1}^{\prime}$$g_{k}^{\prime}$$g_{0}$$f_{0}$$g_{1}$$f_{1}$$f_{k-1}$$g_{k}$$u_{0}$$v_{1}$$u_{1}$$v_{k}$

We adopt the notation that $v_{0}=\operatorname{id}_{w}$ and
$u_{k}=\operatorname{id}_{z}$. Then $r(v_{0})=r(u_{k})=0$. Now, commutativity
of the diagram gives that $r(u_{i})+r(f_{i}^{\prime})=r(u_{i}\circ
f_{i}^{\prime})=r(f_{i}\circ v_{i+1})=r(f_{i})+r(v_{i+1})$. Similarly,
$r(u_{i})+r(g_{i}^{\prime})=r(g_{i})+r(v_{i})$. Applying this, we get

$\displaystyle N$
$\displaystyle+\sum_{i=0}^{k-1}r(f_{i}^{\prime})-\sum_{i=0}^{k}r(g_{i}^{\prime})$
$\displaystyle=N+\sum_{i=0}^{k-1}\left(r(f_{i})+r(u_{i})-r(v_{i+1})\right)-\sum_{i=0}^{k}\left(r(g_{i})+r(u_{i})-r(v_{i})\right)$
$\displaystyle=N+r(u_{k})-r(v_{0})+\sum_{i=0}^{k-1}r(f_{i})-\sum_{i=0}^{k}r(g_{i})$
$\displaystyle=N+\sum_{i=0}^{k-1}r(f_{i})-\sum_{i=0}^{k}r(g_{i}).$

Thus, $\tau$ and $\tau^{\prime}$ give the same value of $\rho$.

Now, we prove that $\rho$ is a rank function. To prove this, we must prove
that it satisfies the two conditions in 6.4.

  1. (1)

We prove that $\rho$ is compatible with the partial order.

First, let $x$ and $y$ be elements in $P_{w,z}$ different from $\hat{0}$ such
that $x>y$. Then $x\Rightarrow y$ when considered as elements of
$\operatorname{Hom}(w,z)$. We can thus pick representatives $\tau$ and
$\tau^{\prime}$ as in Theorem 6.7, and we get that

$\rho(x)-\rho(y)=\sum_{i=0}^{k}r(g_{i}^{\prime})-\sum_{i=0}^{k}r(g_{i})=\sum_{i=0}^{k}\left(r(g_{i}^{\prime})-r(g_{i})\right)>0,$

where the last inequality follows from the fact that $g_{i}\Rightarrow
g_{i}^{\prime}$ and that $g_{i}\neq g_{i}^{\prime}$ for at least one $i$
(otherwise, $x$ would equal $y$).

Now, we must show that $\rho(x)>\rho(\hat{0})$ for $x\neq 0$. For this, let

$\tau=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right)$

be a representative of $x$. Observe that $r(g_{i})\leq\dim x_{i}-\dim y_{i}$.
Furthermore, $r(f_{i})=\dim x_{i+1}-\dim y_{i}$ (as $\dim x_{i+1}-\dim y_{i}$
is either 0 or 1, due to the fact that the Morse system $\Sigma$ consists of
regular pairs of a discrete Morse function). Hence,

$\displaystyle\rho(x)=$ $\displaystyle
N+\sum_{i=0}^{k-1}r(f_{i})-\sum_{i=0}^{k}r(g_{i})$ (6) $\displaystyle\geq$
$\displaystyle N+\sum_{i=0}^{k-1}\left(\dim x_{i+1}-\dim
y_{i}\right)-\sum_{i=0}^{k}\left(\dim x_{i}-\dim y_{i}\right)$
$\displaystyle=$ $\displaystyle N+\dim y_{k}-\dim x_{0}=N+\dim z-\dim w=0.$

Thus, $\rho(x)\geq 0>-1=\rho(\hat{0})$, as desired.

  2. (2)

We prove that $\rho$ is compatible with the covering relation.

First, let $x$ and $y$ be elements in $P_{w,z}$ different from $\hat{0}$ such
that $x$ covers $y$. We again choose representatives $\tau$ and
$\tau^{\prime}$ as in Theorem 6.7. It follows that $g_{i}=g_{i}^{\prime}$ for
all but one $i$, or else we could choose a representative $\gamma$ so that
$x>[\gamma]>y$. Similarly, it follows that for this $i$, $g_{i}^{\prime}$
covers $g_{i}$, or else we could find a $\hat{g}$ such that
$g_{i}^{\prime}>\hat{g}>g_{i}$, which again would let us choose a
representative $\gamma$ so that $x>[\gamma]>y$. As $g_{i}^{\prime}$ covers
$g_{i}$, we have that $r(g_{i}^{\prime})=r(g_{i})+1$, and thus

$\rho(x)-\rho(y)=\sum_{i=0}^{k}r(g_{i}^{\prime})-\sum_{i=0}^{k}r(g_{i})=1,$

as desired.

Now, we must show that if $x$ covers $\hat{0}$, then
$\rho(x)=\rho(\hat{0})+1=0$. An element $x$ covers $\hat{0}$ if and only if
there are no $z\in\operatorname{Hom}(w,z)$ such that $x\Rightarrow z$. Let the
following be a representative of such a $x$.

$\tau=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right)$

Then, all $g_{i}$ must be maximal in
$\operatorname{Hom}_{\operatorname{Ent}[X]}(x_{i},y_{i})$, or else we could
replace this $g_{i}$ to get a representative $\tau^{\prime}$ with
$[\tau]\Rightarrow[\tau^{\prime}]$. It is easy maximal elements in
$\operatorname{Hom}_{\operatorname{Ent}[X]}(x_{i},y_{i})$ are sequences on the
form $(x_{i}=a_{0}>a_{1}>\dots>a_{m}=y_{i})$ with $m=\dim x_{i}-\dim y_{i}$,
and hence $r(g_{i})=\dim x_{i}-\dim y_{i}$. We get that the inequality in (6)
is an equality in this case, and $\rho(x)=0$, as desired.∎

###  6.3. Computing the Hom posets

In this section, we describe an algorithm for computing the Hom posets in the
discrete flow category. We still consider the case of a Morse system induced
by a discrete Morse function on a regular CW complex, and use the same
notation as the previous section. We first state some results that are needed
for the algorithm.

First, we prove a useful lemma. Recall that the regular pairs of a discrete
Morse function are the pairs of simplices $(x,y)$, where $x\in X^{k}$ and
$y\in X^{k+1}$, such that $x<y$ and $f(x)\geq f(y)$ (the set of these is the
induced gradient vector field, $V_{f}$). The following lemma essentially tells
us that we can always assume, without loss of generality, that this inequality
is an equality for all regular pairs.

######  Lemma 6.10.

Let $f\colon X\to\mathbb{R}$ be a discrete Morse function. There exists a
Forman equivalent discrete Morse function $\tilde{f}$ such that
$\tilde{f}(x)=\tilde{f}(y)$ for each regular pair $(x,y)$ of $f$.

###### Proof.

Define $\tilde{f}\colon X\to\mathbb{R}$ as follows.

$\tilde{f}(x)=\begin{cases}f(y),&\quad\text{if }(x,y)\in V_{f}\text{ for some
}y,\\\ f(x),&\quad\text{ otherwise.}\\\ \end{cases}$

It’s clear that $\tilde{f}(x)=\tilde{f}(y)$ for all regular pairs of $f$, so
it remains to show that $\tilde{f}$ is a discrete Morse function, and that
$\tilde{f}$ and $f$ are Forman equivalent. We prove both of these things by
showing that their induced gradient vector fields, $V_{\tilde{f}}$ and $V_{f}$
are the same.

Clearly, $V_{f}\subseteq V_{\tilde{f}}$. We need to show that
$V_{\tilde{f}}\subseteq V_{f}$. First, observe that $\tilde{f}(x)\leq f(x)$
for all $x$.

Now, suppose $(x,y)\in V_{\tilde{f}}$. Then $x<y$ and
$\tilde{f}(x)\geq\tilde{f}(y)$. If $\tilde{f}(y)=f(y)$, then
$f(y)=\tilde{f}(y)\leq\tilde{f}(x)\leq f(x)$, so $(x,y)\in V_{f}$. Suppose
$\tilde{f}(y)\neq f(y)$. Then $(y,z)\in V_{f}$ for some $z$, and
$\tilde{f}(y)=f(z)$. Let $w\neq y$ be such that $x<w<z$. If $(x,w)$ is a
regular pair, then $\tilde{f}(x)=f(w)<f(z)=\tilde{f}(y)$, which is a
contradiction. If $(x,w)$ is not a regular pair, then $\tilde{f}(x)\leq
f(x)<f(w)<f(z)=\tilde{f}(y)$; also a contradiction. Hence, we cannot have that
$f(y)\neq\tilde{f}(y)$, which completes the proof. ∎

######  Theorem 6.11.

A morphism $[\gamma]\in\operatorname{Hom}(w,z)$ is represented by a unique
sequence $(w=x_{0},x_{1},\dots,x_{k}=z)$ of cells, such that

  * •

no two elements in the sequence are equal, and

  * •

for all $i<k$, either $x_{i+1}<x_{i}$ or $\\{x_{i},x_{i+1}\\}$ is a regular
pair.

It is clear that such a sequence always defines a morphism in
$\operatorname{Hom}(w,z)$, so this tells us that we can count the morphisms in
$\operatorname{Hom}(w,z)$ by counting the simple paths from $w$ to $z$ in a
directed graph with face relations and regular pairs as edges.

###### Proof.

We get a sequence representation from a representation

$\gamma=\left(x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}\right)$

by concatenating all the $g_{i}$’s. For example,

$\left(x_{0}\xrightarrow{g_{0}=(x_{0}>x_{0}^{1}>y_{0})}y_{0}\xleftarrow{f_{0}=(x_{1}>y_{0})}x_{1}\xrightarrow{g_{1}=(x_{1}>x_{1}^{1}>x_{1}^{2}>y_{1})}y_{1}\right)$

becomes

$(x_{0},x_{0}^{1},y_{0},x_{1},x_{1}^{1},x_{1}^{2},y_{1}).$

To get a sequence without repeating elements, we choose an appropriate
representation. To see that this is possible, first observe that we can
eliminate successive repeating elements by removing intermediate identity maps
(horizontal relation). Furthermore, by 6.10, given a sequence
$(x_{0},\dots,x_{k})$, there is a discrete Morse function $\tilde{f}$ such
that $\tilde{f}(x_{0})\geq\tilde{f}(x_{1})\geq\dots\geq\tilde{f}(x_{k})$.
Thus, if a cell $x^{\prime}$ repeats in the sequence, the only other cell that
can be between the two $x^{\prime}$ is $y^{\prime}$ where
$\\{x^{\prime},y^{\prime}\\}$ form a regular pair. Now, we can simplify the
representation, as per [14, Remark 2.9], to turn
$(\dots,x_{i},x^{\prime},y^{\prime},x^{\prime},x_{i+4},\dots)$ into
$(\dots,x_{i},x^{\prime},x_{i+4},\dots)$.

It remains to show that for a given morphism, this sequence representation is
unique. For this, we show that the sequence representation is decided by the
algebraic invariant $I$, as defined in 6.5.

Let $(w=x_{0},x_{1},\dots,x_{k}=z)$ and $(w=y_{0},y_{1},\dots,y_{l}=z)$ be
sequence representations of $[\tau]$ and $[\sigma]$, such that
$I([\tau])=I([\sigma])$. Now, as $x_{0}$ appears only once in the first
sequence, the coefficient before the atom $(x_{0}>x_{1})$ (or $(x_{1}>x_{0})$)
in $I([\tau])$ is nonzero. Hence, as $y_{0}=x_{0}$ appears only once in the
second sequence, we must have $y_{1}=x_{1}$ for $I([\sigma])$ to equal
$I([\tau])$. Similarly, we must have $y_{2}=x_{2}$, and so on, so the
sequences must be equal in all places. This completes the proof of uniqueness.
∎

For the following theorem, recall the definitions of $r$ (4) and $\rho$ (6.8)
from the previous section.

######  Theorem 6.12.

Let $[\gamma]\in\operatorname{Hom}(w,z)$ be a morphism represented by the
sequence $(x_{0},\dots,x_{k})$. Let $I$ be the set of indices
$i\in\\{0,\dots,k-1\\}$ such that $x_{i}>x_{i+1}$. Then,

$\rho([\gamma])=\sum_{i\in I}\left(\dim x_{i}-\dim x_{i+1}-1\right)$

###### Proof.

The morphism $[\gamma]$ is represented by

$\gamma=\left(x_{j_{0}}\xrightarrow{g_{0}}x_{j_{1}-1}\xleftarrow{f_{0}}x_{j_{1}}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{j_{m}}\xrightarrow{g_{k}}x_{j_{m+1}-1}\right)$

for some sequence $(j_{0},\dots,j_{m+1})$ (with $j_{0}=0$ and $j_{m+1}=k+1$).
Then,

$\displaystyle\sum_{i\in I}$ $\displaystyle\left(\dim x_{i}-\dim
x_{i+1}-1\right)=\sum_{i=0}^{m}\sum_{l=j_{i}}^{j_{i+1}-2}\left(\dim x_{l}-\dim
x_{l+1}-1\right)$ $\displaystyle=\sum_{i=0}^{m}\left(\dim x_{j_{i}}-\dim
x_{j_{i+1}-1}-(j_{i+1}-j_{i}-1)\right)$ $\displaystyle=\dim x_{j_{0}}-\dim
x_{j_{m+1}-1}+\sum_{i=1}^{m}\left(\dim x_{j_{i}}-\dim
x_{j_{i}-1}\right)-\sum_{i=0}^{m}\left(j_{i+1}-j_{i}-1\right)$
$\displaystyle=\dim x_{0}-\dim
x_{k}+\sum_{i=0}^{m-1}r(f_{i})-\sum_{i=0}^{m}r(g_{i})=\rho([\gamma]).\qed$

######  Theorem 6.13.

Let $[\gamma]$ and $[\tau]$ be morphisms in $\operatorname{Hom}(w,z)$, such
that $\rho([\tau])=\rho([\gamma])+1$. Then, $[\tau]$ covers $[\gamma]$ if and
only if the sequence representation of $[\gamma]$ equals the sequence
representation of $[\tau]$ with exactly one element added or exactly one
element removed.

###### Proof.

First, suppose $[\tau]$ covers $[\gamma]$. By Theorem 6.7, you can get
$[\gamma]$ from $[\tau]$ by adding an element $x^{\prime}$ to the sequence
representation of $[\tau]$ and possibly reducing to a non-repeating sequence.
There are two cases to consider,

(I)

The element $x^{\prime}$ was not in the sequence representation of $[\tau]$.

(II)

The element $x^{\prime}$ was in the sequence representation of $[\tau]$.

Case I

In this case, the sequence representation of $[\gamma]$ is simply the sequence
representation of $[\tau]$ with $x^{\prime}$ added.

Case II

In this case, adding $x^{\prime}$ to the sequence representation of $[\tau]$
gives something of the form
$(x_{0},\dots,x_{i},x^{\prime},x_{i+1},x^{\prime},x_{i+2},\dots,x_{k})$, which
is equivalent to $(x_{0},\dots,x_{i},x^{\prime},x_{i+2},\dots,x_{k})$. Hence,
the sequence representation of $[\gamma]$ is the sequence representation of
$[\tau]$ with $x_{i+1}$ removed.

Now, for the other direction, there are again two cases to consider.

(I)

The sequence representation of $[\gamma]$ equals that of $[\tau]$ with one
element added.

(II)

The sequence representation of $[\gamma]$ equals that of $[\tau]$ with one
element removed.

Case I

Let the sequence representations of $[\tau]$ and $[\gamma]$ be
$(x_{0},\dots,x_{k})$ and
$(x_{0},\dots,x_{i},x^{\prime},x_{i+1},\dots,x_{k})$, respectively. We use
Theorem 6.12 and the fact that $\rho([\tau])=\rho([\gamma])+1$ to conclude
that $\dim x_{i}>\dim x^{\prime}>\dim x_{i+1}$. Hence, $[\gamma]$ equals
$[\tau]$ with the element $x^{\prime}$ added to a right-pointing arrow, so
$[\gamma]\Rightarrow[\tau]$.

Case II

Let the sequence representations of $[\tau]$ and $[\gamma]$ be
$(x_{0},\dots,x_{k})$ and $(x_{0},\dots,x_{i},x_{i+2},\dots,x_{k})$,
respectively. We use Theorem 6.12 and the fact that
$\rho([\tau])=\rho([\gamma])+1$ to conclude that either $\dim x_{i+1}>\dim
x_{i}$ or $\dim x_{i+2}>\dim x_{i+1}$. In the first case,
$[x_{i}>x_{i+2}]=[x_{i}<x_{i+1}>x_{i}>x_{i+2}]$. In the second case,
$[x_{i}>x_{i+2}] =[x_{i}>x_{i+2}>x_{i+1}<x_{i+2}]$. In any case, $[\gamma]$
can be constructed by adding a single element (either $x_{i}$ or $x_{i+2}$) to
a right-pointing arrow of $[\tau]$, so $[\gamma]\Rightarrow[\tau]$.

When $[\gamma]\Rightarrow[\tau]$, $[\gamma]$ covers $[\tau]$, as their rank
differ by one, and this concludes the proof. ∎

Algorithm 1 Compute Hom poset

1:A regular CW complex $X$, a gradient vector field $V_{f}$, a source $w$, and
a target $z$.

2:The Hom poset $\operatorname{Hom}(w,z)$ of the discrete flow category.

3:Construct a directed graph $G$ with cells as vertices and an edge $u\to v$
if $v<u$ or if $\\{u,v\\}$ is a regular pair

4:In $G$, find all paths with non-repeating vertices from $w$ to $z$

5:For all paths, compute the rank with Theorem 6.12.

6:for $i=1..(\dim w-\dim z)$ do

7: Compute the covering relations between paths of rank $i-1$ and paths of
rank $i$, using Theorem 6.13.

8:end for

Note that the only data needed for the CW complex $X$ is the graph of its
covering relations. Only the covering relations of $\operatorname{Hom}(w,z)$
are output with this algorithm, but the full set of partial order relations is
easily computed from this.

Regarding complexity of this algorithm, the main computational cost is
comparing the covering relations. Comparing two paths to determine if one
covers the other takes $\mathcal{O}(l)$ time, where $l$ is the length of the
longest path. For each path $\gamma$, the algorithm makes one comparison to
each path of rank $\rho(\gamma)+1$. Hence, the algorithm makes
$\mathcal{O}(n\cdot W)$ comparisons, where $n$ is the number of morphisms in
$\operatorname{Hom}(w,z)$, and $W$ is the maximum number of morphisms of a
given rank (the maximal “width” of a rank layer in the Hasse diagram). In
total, the complexity of the algorithm is $\mathcal{O}(l\cdot n\cdot W)$ (it’s
easy to verify that steps 1.-3. have smaller complexity than this). As $W$ is
clearly smaller than $n$, we could write this more simply as
$\mathcal{O}(l\cdot n^{2})$. For comparison, listing the sequence
representations of all morphisms in $\operatorname{Hom}(w,z)$ takes
$\Theta(l\cdot n)$ time (and this is thus a lower bound of how fast an
algorithm to compute $\operatorname{Hom}(w,z)$ can be).

###  6.4. The Hom posets are CW posets

In this section, we prove A: that in the case where $X$ is a simplicial
complex, then for any $w,z\in\operatorname{Flo}_{\Sigma}[X]$ with
$\operatorname{Hom}(w,z)$ nonempty, the poset $P_{w,z}$, as defined in (3), is
a CW poset. (Note that when $\operatorname{Hom}(w,z)$ is empty, then $P_{w,z}$
is just the poset with one element.)

######  Lemma 6.14.

Let $P$ and $Q$ be posets with greatest elements $\hat{1}_{P}$ and
$\hat{1}_{Q}$, respectively. Suppose that $|P\setminus\\{\hat{1}_{P}\\}|\cong
S^{m}$ and $|Q\setminus\\{\hat{1}_{Q}\\}|\cong S^{n}$. Then the realization of
the poset $(P\times Q)\setminus\\{(\hat{1}_{P},\hat{1}_{Q})\\}$ is
homeomorphic to $S^{m+n+1}$.

###### Proof.

From [16, Proposition 1.9], the realization of $(P\times
Q)\setminus\\{(\hat{1}_{P},\hat{1}_{Q})\\}$ is the join of $S^{m}$ and
$S^{n}$, which is $S^{m+n+1}$. ∎

######  Definition 6.15.

Let $X$ be a simplicial complex, and let $\sigma\in X$. We define the _coface
complex_ as the set

$\operatorname{cof}\sigma=\\{\tau\setminus\sigma:\sigma\subsetneq\tau\\}$

In other words, the coface complex of $\sigma$ contains all the proper cofaces
of $\sigma$, with $\sigma$ subtracted as a set from the complexes111Note that
the coface complex is the same as the _link_ of a simplex, defined in e.g.,
[15]..

######  Lemma 6.16.

Let $X$ be a simplicial complex, and let $\sigma\in X$. The coface complex
$\operatorname{cof}\sigma$ is a simplicial complex.

###### Proof.

We need to prove that for each $\gamma\in\operatorname{cof}\sigma$, each
nonempty subset $\gamma^{\prime}\subseteq\gamma$ is also an element of
$\operatorname{cof}\sigma$. Suppose $\gamma\in\operatorname{cof}\sigma$. Then
$\gamma=\tau\setminus\sigma$ for some proper coface $\tau$ of $\sigma$. Then,
$(\gamma^{\prime}\cup\sigma)\subseteq\tau$, so $\gamma^{\prime}\cup\sigma$ is
a simplex in $X$, and $(\gamma^{\prime}\cup\sigma)$ is a proper coface of
$\sigma$, so $\gamma^{\prime}=(\gamma^{\prime}\cup\sigma)\setminus\sigma$ is
in $\operatorname{cof}\sigma$. ∎

######  Lemma 6.17.

Let $X$ be a simplicial complex, and let $\sigma\in X^{n}$ and $\tau\in X^{m}$
with $\tau\subseteq\sigma$. Then the realization of
$\operatorname{Hom}_{\operatorname{Ent}[X]}(\sigma,\tau)\setminus\\{(\sigma>\tau)\\}$
is homeomorphic to $S^{n-m-2}$.

###### Proof.

The poset
$\operatorname{Hom}_{\operatorname{Ent}[X]}(\sigma,\tau)\setminus\\{(\sigma>\tau)\\}$
contains all sequences $(\sigma>\upsilon_{1}>\dots>\upsilon_{k}>\tau)$, with
$k\geq 1$, and the partial order is given by inclusion. This poset is
isomorphic to the poset of descending sequences
$(\upsilon_{1}>\dots>\upsilon_{k})$ of simplices in
$\partial(\sigma\setminus\tau)\subseteq\operatorname{cof}\tau$, i.e., the
boundary of $\sigma\setminus\tau$ in the coface complex of $\tau$ (where the
ordering is still given by inclusion). This is again isomorphic to the
barycentric subdivision $T(\partial(\sigma\setminus\tau))$.

Now,

$|T(\partial(\sigma\setminus\tau))|\cong|\partial(\sigma\setminus\tau)|\cong|\partial\Delta^{n-m-1}|\cong|S^{n-m-2}|,$

as desired (here we use that $\sigma\setminus\tau$ is a set of cardinality
$(n+1)-(m+1)=n-m$, and hence an $(n-m-1)$–simplex). ∎

In the proof of the following lemma, we use the fact that for a poset $P$,
$|P^{\operatorname{op}}|\cong|P|$, which is a special case of the fact that
$|\mathcal{N}(C^{\operatorname{op}})|\cong|\mathcal{N}(C)|$ for a general
category $C$.

######  Lemma 6.18.

Let $X$ be a simplicial complex. Let
$P=\operatorname{Hom}_{\operatorname{Ent}[X]}(w,z)^{\operatorname{op}}\cup\\{\hat{0}\\}$,
where $\hat{0}$ is a least element. Then, for any $g\in
P\setminus\\{\hat{0}\\}$, the realization of the open interval $(\hat{0},g)$
is homeomorphic to a sphere.

###### Proof.

Let $g=(x_{0},\dots,x_{k})$. Then,

$(\hat{0},g]\quad\cong\quad\operatorname{Hom}(x_{0},x_{1})^{\operatorname{op}}\times\operatorname{Hom}(x_{1},x_{2})^{\operatorname{op}}\times\dots\times\operatorname{Hom}(x_{k-1},x_{k})^{\operatorname{op}},$

where the $\operatorname{Hom}(x_{i},x_{i+1})$ are Hom posets in
$\operatorname{Ent}[X]$. Now, observe that the atom $(x_{i}>x_{i+1})$ is the
greatest element in $\operatorname{Hom}(x_{i},x_{i+1})^{\operatorname{op}}$.
Furthermore, by 6.17, the realization of
$\operatorname{Hom}(x_{i},x_{i+1})^{\operatorname{op}}\setminus\\{(x_{i}>x_{i+1})\\}$
is homeomorphic to a sphere. Thus, by applying 6.14 to the equation above, we
get that the realization $|(\hat{0},g)|=|(\hat{0},g]\setminus\\{g\\}|$ is
homeomorphic to a sphere. ∎

######  Theorem 6.19 (Theorem A).

In the case that $X$ is a simplicial complex, the poset $P_{w,z}$, as defined
in (3), is a CW poset.

###### Proof.

We use 6.1. Conditions (1.) and (2.) are clearly fulfilled, so we need only to
show that for all $x\in P_{w,z}\setminus\\{\hat{0}\\}$, the realization of the
open interval $(\hat{0},x)$ is homeomorphic to a sphere.

Let $x$ be represented by

$\tau=\left(w=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=z\right).$

Applying Theorem 6.7, we see that

$(\hat{0},x]\quad\cong\quad(\hat{0},g_{0}]\times(\hat{0},g_{1}]\times\dots\times(\hat{0},g_{k}].$

By 6.18, the realization of $(0,g_{i})$ is a sphere. Now, as in the proof for
6.18, we apply 6.14 to the equation above and get that $|(\hat{0},x)|$ is
homeomorphic to a sphere. ∎

In conclusion, what this result tells us is that in the discrete flow
category, the opposite poset of a Hom set is the face poset of a regular CW
complex. The nerve of the face poset of a regular CW complex is its
barycentric subdivision, and hence has a geometric realization which is
homeomorphic to the CW complex. Furthermore, the nerve of the opposite of a
poset is isomorphic to the nerve of the poset. Thus, we get a simpler way to
describe the geometric realization of the nerve of a Hom set in the discrete
flow category: instead of taking the nerve and realizing, we can construct the
CW complex of which it is a face poset, which results in a description with
fewer cells. To construct this CW complex, we need only compute the rank
values and the covering relations, which we can do with Algorithm 1.

######  Example 6.20.

We illustrate the observations in the last paragraph with an example.

Consider a regular CW decomposition of the filled sphere $D^{3}$, as
illustrated in Figure 9. We define the gradient vector field
$V_{f}=\\{(w>y),(b>z)\\}$, and let $\Sigma$ be its corresponding Morse system.
We compute $\operatorname{Hom}(f,x)$ in the discrete flow category.

$y$$x$$w$$z$$t$$b$$f$ Figure 9. A CW decomposition of $D^{3}$. The 0–cells are
$x$ and $y$, the 1–cells are $w$ and $z$, the 2–cells are $t$ and $b$ and the
single 3–cell is $f$.

As we have shown that $\operatorname{Hom}(f,x)$ is a graded poset, we can
represent it by a Hasse diagram. The highest rank is $\dim f-\dim x-1=2$. All
rank 2 elements are gradient paths on the form

$f=x_{0}\xrightarrow{g_{0}}y_{0}\xleftarrow{f_{0}}x_{1}\xrightarrow{g_{1}}\cdots\
\xleftarrow{f_{k-1}}x_{k}\xrightarrow{g_{k}}y_{k}=x,$

where all $g_{i}$ are atoms. There are four of these:

  * •

$f>x$

  * •

$f>z<b>x$

  * •

$f>y<w>x$

  * •

$f>z<b>y<w>x$

All elements of lower rank can be constructed from these by inserting
intermediate cells, and all Hasse diagram edges correspond to inserting an
intermediate cell in a sequence. For example, inserting $b$ between $f$ and
$x$ in $f>x$ gives $f>b>x$, and inserting $b$ between $f$ and $z$ in $f>z<b>x$
gives $f>b>z<b>x=f>b>x$. We now draw the Hasse diagram, which is illustrated
in Figure 10.

Figure 10. The Hasse diagram for the Hom poset $\operatorname{Hom}(f,x)$ in
the discrete flow category.

The corresponding regular CW complex is illustrated in Figure 11, and it’s
easy to see that this is a contractible space.

Figure 11. The face poset of this regular CW complex is
$\operatorname{Hom}(f,x)$.

######  Example 6.21.

We illustrate that Theorem 6.19 may fail to hold when $X$ is not a simplicial
complex. In fact, 6.18, which may be viewed as a special case of Theorem 6.19
where the Morse system is empty (so that
$\operatorname{Flo}_{\Sigma}[X]=\operatorname{Ent}[X]$), fails to hold in our
example.

First, we describe how the suspension of a simplicial complex is also a
simplicial complex. Let $X$ be a simplicial complex. Its cone $CX$ can be
viewed as a simplicial complex as follows:

$CX=X\cup\\{\\{0\\}\\}\cup\\{\sigma\cup\\{0\\}:\sigma\in X\\}.$

The suspension $\Sigma X$, which is just the union of two cones $CX$ along
$X$, can then be viewed as the simplicial complex:

$\Sigma X=X\cup\\{\\{0\\}\\}\cup\\{\sigma\cup\\{0\\}:\sigma\in
X\\}\cup\\{\\{1\\}\\}\cup\\{\sigma\cup\\{1\\}:\sigma\in X\\}.$ (7)

Our counterexample uses the following fact: there exists a simplicial complex
$Y$ such that the suspension $\Sigma Y$ is homeomorphic to a sphere, but $Y$
is not homeomorphic to $S^{k}$ for any $k$. For this, we will consider the
Poincaré sphere, which is a simplicial complex and a homology 3–sphere, but
not homeomorphic to a sphere. Its suspension is not homeomorphic to a sphere,
but its double suspension is (for more details, see [19, Example 3.2.11]).
Hence, letting $Y$ be the suspension of the Poincaré sphere, we get our
desired properties.

Now, let $Y$ be the suspension of the Poincaré sphere. As $\Sigma Y$ is a
simplicial complex and homeomorphic to $S^{5}$, it is also a regular CW
complex of dimension 4. Hence we can construct a new regular CW complex $Z$ by
attaching a 5–cell $e$ along $\Sigma Y$. The CW complex $Z$ is illustrated in
Figure 12.

$1$$0$$e$$Y$ Figure 12. An illustration of the CW complex $Z$.

Now, consider the Hom poset

$\operatorname{Hom}_{\operatorname{Ent}[Z]}(e,0)\setminus\\{(e>0)\\},$ (8)

where 0 is a 0–simplex in $\Sigma Y$ as per the notation in (7). This poset
consists of all descending chains $(e>\tau_{1}>\dots>\tau_{k}>0)$, with $k\geq
1$, ordered by inclusion. The set of $\tau_{i}$ satisfying $e>\tau_{i}>0$ is
precisely $\\{\\{0\\}\cup\sigma:\sigma\in Y\\}$. Hence, the poset in (8) is
isomorphic to the barycentric subdivision of $Y$. Thus,

$|\operatorname{Hom}_{\operatorname{Ent}[Z]}(e,0)\setminus\\{(e>0)\\}|\cong|T(Y)|\cong|Y|,$

which is not homeomorphic to a sphere. Thus, the interval $(\hat{0},(e>0))\in
P_{e,0}$ (which is just the opposite of the poset in (8)) is not a sphere, and
hence $P_{e,0}$ is not a sphere.

##  7\. Simplicial collapse on simplicial sets

In this section we define _regular_ simplicial sets, and generalize the
concept of _simplicial collapse_ to these simplicial sets. We further prove
how the nerves of certain categories are regular, and apply simplicial
collapse to a subclass of these categories.

###  7.1. Collapses on simplicial complexes

In Section 3.2, we defined _free faces_ and _collapses_.

We now provide a more general definition of free faces, that allows for
codimension greater than 1.

######  Definition 7.1.

Let $X$ be a simplicial complex and let $\tau$ and $\sigma$ be simplices in
$X$ such that

  1. (1)

$\tau\subsetneq\sigma$, and

  2. (2)

all cofaces of $\tau$ are faces of $\sigma$.

Then $\tau$ is called a _free face_ and $\\{\tau,\sigma\\}$ is a _free pair_.

Recall that in our old definition of free pairs, removing a free pair of $X$
from a simplicial complex constitutes an elementary collapse. Recall that we
write $X\searrow Y$ if $Y$ can be reached through a series of elementary
collapses on $X$. The following theorem justifies our generalization of the
definition of free pairs.

######  Theorem 7.2.

[9, Proposition 9.18] Let $X$ be a simplicial complex and let
$\\{\tau,\sigma\\}$ be a free pair according to 7.1. Let
$Y\setminus[\tau,\sigma]$ be the simplicial complex where all simplices
$\gamma$ satisfying $\tau\subseteq\gamma\subseteq\sigma$ have been removed
from $X$. Then $Y$ is a simplicial complex, and $X\searrow Y$.

Note that in the previous theorem, we might as well have written
$\tau\subseteq\gamma$ instead of $\tau\subseteq\gamma\subseteq\sigma$, as all
cofaces of $\tau$ are faces of $\sigma$.

An example of a simplicial collapse of the kind described in Theorem 7.2 is
given in Figure 13.

$a$$b$$c$$b$$c$ Figure 13. A simplicial collapse given by the free pair
$\\{\\{a\\},\\{a,b,c\\}\\}$.

###  7.2. Collapses on simplicial sets

We generalize free pairs and collapses to simplicial sets. We will restrict
ourselves to a certain class of simplicial sets, which we will name _regular_
simplicial sets.

######  Definition 7.3.

Let $X$ be a simplicial set. Then $X$ is called _regular_ if for all
$n\in\mathbb{N}$ (including 0) and all nondegenerate $x\in X_{n}$, the map

$\displaystyle\operatorname{Hom}_{\Delta}([0],[n])$ $\displaystyle\to X_{0}$
$\displaystyle\theta$ $\displaystyle\mapsto X(\theta)(x)$

is injective.

This definition says that for a nondegenerate $n$–simplex, all of the $(n+1)$
0–dimensional faces are different. This also implies that for each
nondegenerate $n$–simplex $x$ and all $m<n$, all of the $m$–dimensional faces
of $x$ are different222More precisely, the map
$\operatorname{Hom}_{\Delta}([m],[n])\to X_{m}$ that maps $\theta$ to
$X(\theta)(x)$ is injective., because no two $m$–dimensional faces have the
same 0–dimensional faces. In other words, for $x\in X_{n}$ nondegenerate, the
simplicial set consisting of $x$, all the faces of $x$, and all their
degeneracies, is isomorphic to $\Delta^{n}$. This last part is the motivation
behind the definition. Just as for simplicial complexes, the sub-simplicial
set generated by a nondegenerate $n$–simplex $x$ looks like $\Delta^{n}$,
which will allow us to define free pairs and collapses, just as for simplicial
complexes.

######  Example 7.4.

An example of a regular simplicial set is given in Figure 14. Observe how
there are two 1–simplices with $b$ and $c$ as faces, something that is not
possible for simplicial complexes. However, just as for simplicial complexes,
all 1–simplices lie between two different 0–simplices, and the 2–simplex $A$
looks like a 2–simplex in a simplicial complex. To make the last point more
precise, the simplicial set consisting of $A$ and all its faces and
degeneracies, is isomorphic to $\Delta^{2}$.

$A$$\alpha$$\delta$$\beta$$\gamma$$a$$b$$c$ Figure 14. A regular simplicial
set (degeneracies omitted).

######  Lemma 7.5.

The realization of a regular simplicial set is a regular CW complex.

###### Proof.

Let $X$ be a regular simplicial set. Recall that a CW complex is regular if
all attaching maps are homeomorphisms onto their image. From [11, Theorem
14.1], we know that $|X|$ is a CW complex with one $n$–cell for each
nondegenerate simplex. Furthermore, for an $n$–cell corresponding to a
nondegenerate $n$–simplex $x$, the attaching map is given by sending
$|\partial\Delta^{n}|$ to $|\partial x|$ (in the canonical way). Now, from the
observations in the above paragraph, we know that this map is a homeomorphism,
so we are finished. ∎

We introduce some new notation. We say that $\tau\in X_{m}$ is a _face_ of
$\sigma\in X_{n}$ if $\tau=X(\theta)(\sigma)$ for some
$\theta\in\operatorname{Hom}_{\Delta}([m],[n])$. Note that in this definition,
degeneracies are also considered faces, but the nondegenerate faces are just
as for simplicial complexes. We say that $\sigma$ is a coface of $\tau$ when
$\tau$ is a face of $\sigma$. We write $\tau\subseteq\sigma$ when $\tau$ is a
face of $\sigma$, and we write $\tau\subsetneq\sigma$ when $\tau$ is a
_proper_ face of $\sigma$, i.e., when $\tau\subseteq\sigma$ and
$\tau\neq\sigma$. It’s clear that this relation is transitive, as if
$\tau\subseteq\tau^{\prime}$ and $\tau^{\prime}\subseteq\tau^{\prime\prime}$,
then $\tau=X(\theta)(\tau^{\prime})$ and
$\tau^{\prime}=X(\theta^{\prime})(\tau^{\prime\prime})$ for some
$\theta,\theta^{\prime}$, so
$\tau=X(\theta\circ\theta^{\prime})(\tau^{\prime\prime})$. It’s not, however,
antisymmetric, as all simplices are faces of all their degeneracies.

######  Example 7.6.

In the simplicial set in Figure 14, the faces of $A$ are

  * •

$A$ itself,

  * •

the degeneracies of $A$,

  * •

the nondegenerate simplices $a$, $b$, $c$, $\alpha$, $\beta$ and $\delta$,

  * •

the degeneracies of the above mentioned simplices.

The simplex $\gamma$, on the other hand, is _not_ a face of $A$.

######  Definition 7.7.

Let $X$ be a regular simplicial set, and let $\tau$ and $\sigma$ be
nondegenerate simplices such that

  1. (1)

$\tau\subsetneq\sigma$, and

  2. (2)

all cofaces of $\tau$ are faces of $\sigma$.

Then $\tau$ is called a _free face_ and $\\{\tau,\sigma\\}$ is a _free pair_.

Note that the fact that $\tau$ and $\sigma$ are nondegenerate and that
$\tau\subsetneq\sigma$ implies that $\dim\tau<\dim\sigma$.

######  Theorem 7.8.

Let $X$ be a simplicial set and let $\\{\tau,\sigma\\}$ be a free pair
according to 7.1. Let $Y\setminus[\tau,\sigma]$ be the simplicial set where
all simplices $\gamma$ satisfying $\tau\subseteq\gamma\subseteq\sigma$ have
been removed from $X$. Then $Y$ is a simplicial set, and the realization $|Y|$
is a deformation retract of $|X|$.

###### Proof.

We first prove that $Y$ is a simplicial set. We must show that for all $x\in
Y$, all faces (including degeneracies) of $x$ are contained in $Y$. Let $x$ be
an $n$–simplex in $Y$, and suppose $y$ is a face of $x$ not in $Y$. Then $y$
is a coface of $\tau$. But then $x$ is also a coface of $\tau$ (and hence also
a face of $\sigma)$, so $x\notin Y$, which is a contradiction. This proves
that $Y$ is a well-defined simplicial set.

We now show that $|Y|$ is a deformation retract of $|X|$. From 7.5, we know
that $|X|$ is a regular CW complex, and so is $|Y|$. Let $e_{\tau}$ and
$e_{\sigma}$ denote the cells in $|X|$ corresponding to $\tau$ and $\sigma$,
respectively. Then $\overline{e_{\sigma}}$ (i.e., the closure of $e_{\sigma}$)
is homeomorphic to $|\Delta^{n}|$ (for some $n$), and under this
homeomorphism, $e_{\tau}$ maps to some face of $|\Delta^{n}|$. Thus,
$\overline{e_{\sigma}}$ deformation retracts to
$\left(\overline{e_{\sigma}}\setminus\\{e^{\prime}:e_{\tau}\subseteq
e^{\prime}\\}\right)$ (i.e., $\overline{e_{\sigma}}$ with all the cofaces of
$e_{\tau}$ removed), just as for simplicial complexes. This deformation
retract then extends to a deformation retract from $|X|$ to $|Y|$ by setting
it to be identity everywhere else. ∎

When $X$ deformation retracts to $Y$ in this way, we call it a _collapse_ ,
and say that $X$ _collapses_ to $Y$.

######  Example 7.9.

In Figure 15, $\\{a,A\\}$ is a free pair in the regular simplicial set on the
left hand side. The right hand side shows the simplicial set after the
corresponding collapse. Observe how the collapse gives a deformation retract
on the closure of the 2–cell corresponding to $A$, which extends to a
deformation retract on the entire realization by defining it to be identity
everywhere else.

$A$$\alpha$$\delta$$\beta$$\gamma$$a$$b$$c$$\beta$$\gamma$$b$$c$ Figure 15. A
collapse of a regular simplicial set.

###  7.3. Unique factorization categories

We now define a certain class of categories, which we will call _unique
factorization categories_. In these categories, all morphism can be written as
a composition of one or more _indecomposable_ morphisms in a unique way. We
shall see that these categories have nerves that are regular simplicial sets,
and with many free faces.

######  Definition 7.10.

Let $C$ be a category and $f\colon A\to B$ a morphism in $C$. Then $f$ is
called _indecomposable_ if it cannot be written as a composite of non-identity
morphisms, i.e., if there exist no morphisms $h\neq\operatorname{id}_{B}$ and
$g\neq\operatorname{id}_{A}$ such that $f=h\circ g$.

We say that a category is _finite_ if both its set of objects and set of
morphisms is finite.

######  Definition 7.11.

A _finite directed category_ is a finite category such that:

  * •

For all objects $a$, $\operatorname{Hom}(a,a)$ contains only the identity
$\operatorname{id}_{a}$.

  * •

For all objects $a\neq b$, if $\operatorname{Hom}(a,b)$ is nonempty, then
$\operatorname{Hom}(b,a)=\emptyset$.

Any finite poset (viewed as a category) is an example of a finite directed
category.

It’s clear that for a finite directed category, each morphism $f$ can be
written as a composition $f=f_{k}\circ\cdots\circ f_{1}$ of indecomposable
morphisms. However, its possible that there are several different ways to do
this. For example, consider the face poset of the simplicial complex
$\Delta^{2}$. Here, the morphism $(\\{0\\}\leq\\{0,1,2\\})$ has two
decompositions: $(\\{0,1\\}\leq\\{0,1,2\\})\circ(\\{0\\}\leq\\{0,1\\})$ and
$(\\{0,2\\}\leq\\{0,1,2\\})\circ(\\{0\\}\leq\\{0,2\\})$

######  Proposition 7.12.

Let $C$ be a finite directed category. Then the nerve of $C$ is a regular
simplicial set.

###### Proof.

A nondegenerate $n$–simplex in $\mathcal{N}(C)$ is a set of composable non-
identity morphisms $\\{f_{i}\colon x_{i}\to x_{i+1}:0\leq i<n\\}$. Suppose
$x_{i}=x_{j}$ for some $i<j$. If $j=i+1$, then $f_{i}$ is a non-identity
morphism in $\operatorname{Hom}(x_{i},x_{i})$, which contradicts 7.11. If
$j>i+1$, then both $\operatorname{Hom}(x_{i},x_{i+1})$ and
$\operatorname{Hom}(x_{i+1},x_{i})$ are nonempty, which also contradicts 7.11.

Hence, any nondegenerate $n$–simplex in $\mathcal{N}(C)$ has $n+1$ different
0–dimensional faces, and so $\mathcal{N}(C)$ is regular. ∎

######  Definition 7.13.

A _unique factorization category_ is a finite directed category $C$ such that
all non-identity morphisms $f$ in $C$ can be written as a composition of non-
identity indecomposable morphisms in a unique way.

######  Example 7.14.

Let $C$ be the entrance path category of a finite CW complex $X$ considered as
an ordinary category, i.e., with the poset structure of the Hom sets removed.
Then $C$ is a unique factorization category. To see this, observe that each
non-identity morphism goes to a cell of strictly lower dimension, so $C$ is
finite directed. Furthermore, an entrance path $(x_{0}>\dots>x_{k})$ factors
uniquely to $(x_{k}>x_{k-1})\circ\dots\circ(x_{0}>x_{1})$.

Similarly, we can show that a discrete flow category of a finite CW complex,
with the poset structure of the Hom sets removed, is a unique factorization
category. The factorization of a gradient path is given by subsequences
between critical cells. As each non-identity morphism goes from a critical
cell to a critical cell of lower dimension, this gives a well-defined
factorization into indecomposable morphisms. Now, to see that this
factorization is unique, suppose we have $f,f^{\prime}\colon a\to b$ and
$g,g^{\prime}\colon b\to c$ with $g\circ f=g^{\prime}\circ f^{\prime}$. Then
the sequence representations of $g\circ f$ and $g^{\prime}\circ f^{\prime}$
are equal, and as $b$ is critical, both sequence representations contain $b$.
Now, the subsequences starting at $a$ and ending at $b$ have to be equal, and
these are precisely the sequence representations of $f$ and $f^{\prime}$, so
$f=f^{\prime}$. Similarly, $g=g^{\prime}$. This shows that factorizations are
unique.

To give an example, in the discrete flow category in 6.20, the morphism
$[f>t>z<b>x]$ has the unique factorization $[t>z<b>x]\circ[f>t]$.

We now prove C.

######  Theorem 7.15 (Theorem C).

Let $C$ be a unique factorization category. Then the nerve of $C$ deformation
retracts to a simplicial set whose $n$–simplices are all degenerate for $n>1$.

###### Proof.

The main idea of the proof is to collapse $\mathcal{N}(C)$ by removing
1–simplices represented by higher compositions.

We collapse $\mathcal{N}(C)$ inductively. Let $k$ be the largest integer such
that there is a 1–simplex in $\mathcal{N}(C)$ represented by a morphism $f$
that decomposes into $k$ indecomposable morphisms, i.e.,
$f=f_{k}\circ\dots\circ f_{1}$. We claim that $f$ is a free face, and $f$ is
in a free pair with the $k$–simplex $\\{f_{1},\dots,f_{k}\\}$. To see why this
is true, let $\\{g_{1},\dots,g_{m}\\}$ be a coface of $f$. Then
$g=g_{m}\circ\dots\circ g_{1}$ is a 1–simplex that factors through $f$, and
thus $g=f$, as otherwise, $g$ would factor into more than $k$ indecomposable
morphisms. It follows that each $g_{i}$ is a composite
$g_{i}=f_{j_{i+1}-1}\circ\dots\circ f_{j_{i}}$, and hence,
$\\{g_{1},\dots,g_{m}\\}$ is a face of $\\{f_{1},\dots,f_{k}\\}$.

Now, we can remove any $k$–composable 1–simplex $f$ from the simplicial set
through a series of collapses. Now, all remaining 1–simplices decomposes into
$k-1$ or fewer indecomposable factors. We can repeat the process with $k-1$,
and so on, until all remaining 1–simplices are indecomposable morphisms. Now,
all remaining nondegenerate simplices are of dimension 0 or 1, as any higher
simplex $\\{f_{1},\dots,f_{k}\\}$ has the decomposable morphism
$f_{k}\circ\dots\circ f_{1}$ as a face. ∎

By combining this theorem with Theorem 2.3, we get the following corollary.

######  Corollary 7.16.

Let $C$ be a unique factorization category. Then

$H_{n}(\mathcal{N}(C))\cong 0$

for $n\geq 2$.

##  8\. Spectral sequences

We define a spectral sequence as a collection of bigraded differential abelian
groups $\\{E^{r}_{*,*},\partial^{r}:r=0,1,\dots\\}$, where the bidegree of
$\partial^{r}$ is $(-r,r-1)$. For a double complex $C$ we denote its
horizontal differentials with $\partial_{h}\colon C_{*,*}\to C_{*-1,*}$ and
its vertical differentials with $\partial_{v}\colon C_{*,*}\to C_{*,*-1}$.
Furthermore, we let $\operatorname{Tot}C$ denote the total complex of $C$,
with the convention that its differentials are given by
$\partial^{\operatorname{Tot}}_{n}=\partial_{v}+(-1)^{\text{vertical
degree}}\partial_{h}$.

We will use the two spectral sequences in the following theorem, which is the
homological version of Theorem 2.15 in [12] (see also [21, pp. 141–143] for
the homological formulation).

######  Theorem 8.1.

Let $C$ be a double complex such that $C_{p,q}=0$ whenever $p<0$ or $q<0$.
Then there are two spectral sequences ${}_{I}E$ and ${}_{II}E$ with

${}_{I}E^{2}_{p,q}\cong
H_{I}H_{II}(C)_{p,q},\quad\text{and}\quad_{II}E^{2}_{p,q}\cong
H_{II}H_{I}(C)_{p,q},$

that both converge to $H_{*}(\operatorname{Tot}C)$.

###  8.1. The spectral sequence of a bisimplicial set

To a bisimplicial set $X$, we can associate a double complex $FX$ with
$(FX)_{p,q}=\mathbb{Z}X_{p,q}$. The horizontal and vertical differentials are
induced by the horizontal and vertical face maps as for the chain complex of a
simplicial set, i.e.,

$\displaystyle\partial_{h}$
$\displaystyle=\sum_{i}(-1)^{i}(d_{i},\operatorname{id}),$
$\displaystyle\partial_{v}$
$\displaystyle=\sum_{i}(-1)^{i}(\operatorname{id},d_{i}).$

A simple calculation shows that $\partial_{h}^{2}=0$, $\partial_{v}^{2}=0$ and
$\partial_{h}\partial_{v}=\partial_{v}\partial_{h}$.

The following theorem of Dold and Puppe tells us that the homology of
$\operatorname{Tot}(FX)$ equals the homology of $\operatorname{diag}X$ [5,
Theorem 2.9] (see also [8, Theorem 2.5]). Hence, we can compute the homology
of $\operatorname{diag}X$ with either of the spectral sequences in Theorem
8.1.

######  Theorem 8.2 (Dold–Puppe).

Let $X$ be a bisimplicial set and $FX$ its associated double complex. Then

$H_{*}\operatorname{Tot}(FX)\cong H_{*}(\operatorname{diag}X)$

As a bisimplicial set has an infinite amount of degenerate simplices,
computing the spectral sequence associated to $\operatorname{Tot}(FX)$ can be
impractical. However, we will show that, just as for the chain complex of a
simplicial set, we can in fact ignore the degenerate simplices (both the
vertical and horizontal ones). For the proof, we will need the following
lemma.

######  Lemma 8.3.

Let $C_{*}$ be a chain complex, and let $D_{*}$ and $D^{\prime}_{*}$ be
subcomplexes of $C_{*}$. Suppose

$\displaystyle H_{n}(D)$ $\displaystyle\cong 0,$ $\displaystyle
H_{n}(D^{\prime})$ $\displaystyle\cong 0,\text{ and }$ $\displaystyle
H_{n}(D\cap D^{\prime})$ $\displaystyle\cong 0$

for all $n\in\mathbb{Z}$. Then,

$H_{n}(D+D^{\prime})\cong 0$

for all $n\in\mathbb{Z}$.

Here, $D_{*}\cap D^{\prime}_{*}$ and $D_{*}+D^{\prime}_{*}$ are the
subcomplexes of $C_{*}$ defined by 333Note the difference between
$D_{*}+D^{\prime}_{*}$ and the direct sum $D_{*}\oplus D^{\prime}_{*}$.

$\displaystyle(D\cap D^{\prime})_{n}$ $\displaystyle=D_{n}\cap
D^{\prime}_{n},\text{ and}$ $\displaystyle(D+D^{\prime})_{n}$
$\displaystyle=D_{n}+D^{\prime}_{n}=\\{x+y:x\in D_{n},y\in D^{\prime}_{n}\\}.$

###### Proof.

There is a short exact sequence of chain complexes

$0\rightarrow D_{*}\cap D^{\prime}_{*}\rightarrow D_{*}\oplus
D^{\prime}_{*}\rightarrow D_{*}+D^{\prime}_{*}\rightarrow 0,$

where the first map is $x\mapsto(x,-x)$, and the second map is $(x,y)\mapsto
x+y$. The induced long exact sequence in homology then proves the lemma. ∎

Now, let $D^{h}_{p,q}$ be the set of horizontal degeneracies in $X_{p,q}$,
i.e., the simplices on the form $(s_{i},\operatorname{id})x$ for some $x\in
X_{p-1,q}$ and some $i$. Likewise, let $D^{v}_{p,q}$ be the set of vertical
degeneracies in $X_{p,q}$. Then $\mathbb{Z}D^{h}_{p,q}$ and
$\mathbb{Z}D^{v}_{p,q}$ are subgroups of $(FX)_{p,q}$. Now, as $D^{h}_{*,q}$
are just the degeneracies in the simplicial set $X_{*,q}$, we have that
$\partial_{h}(D^{h}_{p,q})\subseteq D^{h}_{p-1,q}$ (as stated in Section 2.2).
Furthermore, for $(s_{i},\operatorname{id})x\in D^{h}_{p,q}$,

$\displaystyle\partial_{v}(s_{i},\operatorname{id})x$
$\displaystyle=\left(\sum_{i}(-1)^{i}(\operatorname{id},d_{i})\right)(s_{i},\operatorname{id})x$
(9)
$\displaystyle=\sum_{i}(-1)^{i}(s_{i},d_{i})x=(s_{i},\operatorname{id})\left(\sum_{i}(-1)^{i}(\operatorname{id},d_{i})\right)x,$

so $\partial_{v}(D^{h}_{p,q})\subseteq D^{h}_{p,q-1}$. Hence, $FD^{h}$ given
by $(FD^{h})_{p,q}=\mathbb{Z}D^{h}_{p,q}$ is a well defined sub-double complex
of $FX$, and we can form the quotient double complex $FX/FD^{h}$. Likewise, we
have well-defined double complexes $FD^{v}$ and $FX/FD^{v}$. We then also have
a sub-double complex $F(D^{h}\cup D^{v})$, again given by $F(D^{h}\cup
D^{v})_{p,q}=\mathbb{Z}(D^{h}_{p,q}\cup D^{v}_{p,q})$, and we can form the
quotient $FX/F(D^{h}\cup D^{v})$.

######  Proposition 8.4.

Let $X$ be a bisimplicial set. Then

$H_{*}\operatorname{Tot}(FX)\cong H_{*}\operatorname{Tot}\left(FX/F(D^{h}\cup
D^{v})\right).$

###### Proof.

From 2.4, we get that $H_{I}(FD^{h})\cong 0$ everywhere. Theorem 8.1 then
gives us that $H_{*}(\operatorname{Tot}FD^{h})\cong 0$. By the same argument
for $D^{v}$, we get that $H_{*}(\operatorname{Tot}FD^{v})\cong 0$.

Now, the degenerate simplices in the diagonal $\operatorname{diag}X$ are
precisely the elements of $D^{h}\cap D^{v}$. Applying 2.4 again, we get that
$H_{*}(\operatorname{diag}(D^{h}\cap D^{v}))=0$, and applying Theorem 8.2, we
get that $H_{*}(\operatorname{Tot}F(D^{h}\cap D^{v}))\cong 0$.

Note now that $\operatorname{Tot}F(D^{h}\cup
D^{v})=\operatorname{Tot}(FD^{h}+FD^{v})=(\operatorname{Tot}FD^{h})+(\operatorname{Tot}FD^{v})$
and that $\operatorname{Tot}F(D^{h}\cap
D^{v})=(\operatorname{Tot}FD^{h})\cap(\operatorname{Tot}FD^{v})$. It now
follows from 8.3 that $H_{*}(\operatorname{Tot}F(D^{h}\cup D^{v}))\cong 0$.
Hence, $H_{*}(\operatorname{Tot}FX/F(D^{h}\cup D^{v}))\cong
H_{*}\left((\operatorname{Tot}FX)/(\operatorname{Tot}F(D^{h}\cup
D^{v}))\right)\cong H_{*}(\operatorname{Tot}FX)$. ∎

###  8.2. The spectral sequence of the discrete flow category

From the discrete flow category $\operatorname{Flo}_{\Sigma}[X]$, we get a
bisimplicial set
$S:=\operatorname{N\overline{N}}\operatorname{Flo}_{\Sigma}[X]$, as described
in Section 4.2. As the realization of the diagonal, $|\operatorname{diag}S|$,
is (homotopy equivalent to) the classifying space of
$\operatorname{Flo}_{\Sigma}[X]$, we can compute the homology of this
classifying space through simplicial homology on $\operatorname{diag}S$. This
is again equal to the homology of the total complex, $\operatorname{Tot}FS$,
by Theorem 8.2. Finally, we can compute the homology of $\operatorname{Tot}FS$
with one of two the spectral sequences in Theorem 8.1. We will use the
spectral sequence ${}_{I}E$, i.e., we will take vertical homology first. As we
shall show, this spectral sequence collapses on page 2 (B in the
introduction).

Recall that an element of $S_{p,q}$ consists of $q$ horizontally composable
sets of $p$ vertically composable 2–morphisms. For p-categories, this is a set
$\\{f_{0,j}\Rightarrow\dots\Rightarrow f_{p,j}:f_{i,j}\colon x_{j}\to
x_{j+1},0\leq j<q\\}$. The face and degeneracy maps, both in the vertical and
in the horizontal direction, are as for the ordinary nerve.

First, we compute some examples.

######  Example 8.5.

We consider the discrete flow category of a discrete Morse function on
$S^{2}$, as described in [14, Section 6.1]. Let $S$ be the double nerve of the
discrete flow category. By 8.4, we need only consider the elements of $S$ that
are not in the image of any degeneracy map (either vertical or horizontal)
when constructing the spectral sequence.

${f_{7}}$${f_{0}}$${f_{1}}$${f_{6}}$ ${f_{2}}$${f_{5}}$${f_{4}}$${f_{3}}$
Figure 16. The Hom poset $\operatorname{Hom}(t,w)$.

As the discrete flow category has only two objects, $t$ and $w$, there are no
nondegenerate horizontal compositions, and thus $E^{0}_{0,q}=0$ for $q\geq 2$.
In the bottom row, we have $E^{0}_{0,0}=\mathbb{Z}^{2}$, corresponding to the
two objects, and $E^{0}_{p,0}=0$ for $p\geq 1$. Furthermore,
$\operatorname{Hom}(t,w)$ (illustrated in Figure 16) has 8 nondegenerate
1-morphisms, 8 nondegenerate 2-morphisms (i.e., partial orders) and no
nondegenerate compositions of morphisms, so
$E^{0}_{0,1}=E^{0}_{1,1}=\mathbb{Z}^{8}$ and $E^{0}_{p,1}=0$ for $p\geq 2$.

[title = Page 0] [”Z^2”](0,0) [”Z^8”](0,1) [”Z^8”](1,1) 0̣ (0,1) [title = Page
1] [”Z”](0,0) [”Z^7”](0,1) [”Z^8”](1,1) 1̣ (1,1) [title = Page 2] [”Z”](0,0)
[”Z”](1,1)

Figure 17. The first three pages of the spectral sequence for the discrete
flow category of a discrete Morse function on $S^{2}$.

The 0th page, $E^{0}$, is illustrated in Figure 17, together with the two
succeeding pages. To compute $E^{1}$, we need only find the differential
$\partial^{0}_{0,1}\colon E^{0}_{0,1}\to E^{0}_{0,0}$. This image takes
$f_{i}\colon t\to w$ to $(w-t)$, and hence its image is
$\mathbb{Z}(1,-1)\cong\mathbb{Z}$. This gives us $E^{1}$, as illustrated.

To compute $E^{2}$, we need the differential $\partial^{1}_{1,1}\colon
E^{1}_{1,1}\to E^{1}_{0,1}$. Observe that that $E^{1}_{0,1}$ is generated by
$[f_{i}-f_{0}],i\in{1,\dots,7}$. From this, it’s clear that
$\partial^{1}_{1,1}$ is surjective (for example, $[f_{3}-f_{0}]$ is
$\partial^{1}\left[(f_{2}\Rightarrow f_{3})-(f_{2}\Rightarrow
f_{1})+(f_{0}\Rightarrow f_{1})\right])$. Hence, we get $E^{2}$ as
illustrated. The resulting homology of the total complex, and hence $S^{2}$,
is then as follows.

$H_{n}(S^{2})=\begin{cases}\mathbb{Z},&\quad n=0,2,\\\ 0,&\quad\text{
otherwise.}\\\ \end{cases}$

######  Example 8.6.

We compute a slightly more complicated example, the discrete flow category of
the discrete Morse function on $D^{3}$ from 6.20. There are three critical
cells: $f$, $t$ and $x$. We computed the Hom poset $\operatorname{Hom}(f,x)$
in 6.20. The Hom poset $\operatorname{Hom}(t,x)$ is as for the $S^{2}$
example, i.e., the one illustrated in Figure 16. Finally,
$\operatorname{Hom}(f,t)$ consists of just one morphism, $(f>t)$.

[title = Page 0] [”Z^3”](0,0) [”Z^30”](0,1) [”Z^8”](0,2) [”Z^60”](1,1)
[”Z^8”](1,2) [”Z^32”](2,1) 0̣ (0, 2) 0̣ (0, 1) 0̣ (1, 2) [title = Page 1]
[”Z”](0,0) [”Z^20”](0,1) [”Z^52”](1,1) [”Z^32”](2,1) 1̣ (1, 1) 1̣ (2, 1) (2,2)
[title = Page 2] [”Z”](0,0) (2,2)

Figure 18. The first three pages of the spectral sequence for the discrete
flow category of a discrete Morse function on $D^{3}$.

We count the nondegenerate simplices to get $E^{0}$. In $E_{p,0}$, as in the
previous example, we get the $E^{0}_{p,0}=0$ for $p\geq 1$, and
$E^{0}_{0,0}=\mathbb{Z}^{3}$, one copy of $\mathbb{Z}$ for each of the
critical cells. There are 30 non-identity morphisms, so
$E^{0}_{0,1}=\mathbb{Z}^{30}$. There are 60 non-identity partial order
relations, so $E^{0}_{1,1}=\mathbb{Z}^{60}$. The only Hom poset with
composable partial orders is $\operatorname{Hom}(f,x)$, and there are 32
possible nondegenerate pairs of composable partial orders, so
$E^{0}_{2,1}=\mathbb{Z}^{32}$. Finally, there are no nondegenerate triples of
composable partial orders, so $E^{0}_{p,1}=0$ for $p\geq 3$.

The only non-identity (horizontally) composable morphisms are $(f>t)$ composed
with one of the eight morphisms in $\operatorname{Hom}(t,x)$, so
$E^{0}_{0,2}=\mathbb{Z}^{8}$. Similarly, the only nondegenerate horizontally
composable partial orders are $\left((f>t)\Rightarrow(f>t)\right)$ composed
with one of the eight non-identity partial orders in
$\operatorname{Hom}(t,x)$444Even though the partial order
$\left((f>t)\Rightarrow(f>t)\right)$ is degenerate, its horizontal composition
with a nondegenerate partial order is not.. Thus,
$E^{0}_{1,2}=\mathbb{Z}^{8}$. Now, all elements in $S_{2,2}$ includes either a
vertical identity (on $(f>t)$) or a horizontal identity (on $f$, $t$ or $x$),
and is thus degenerate, so $E^{0}_{2,2}=0$, and the same holds for
$E^{0}_{p,2}$ for $p\geq 3$.

Finally, any horizontal composition of 3 or more must include an identity (as
we only have 3 critical cells), so $E^{0}_{p,q}=0$ for $q\geq 3$. We can now
draw the 0th page of the spectral sequence. The first three pages are shown in
Figure 18.

To get $E^{1}$ from $E^{0}$, me must inspect the three differentials

$\displaystyle\partial^{0}_{0,1}\colon E^{0}_{0,1}\to E^{0}_{0,0},$
$\displaystyle\partial^{0}_{0,2}\colon E^{0}_{0,2}\to E^{0}_{0,1},\text{ and
}$ $\displaystyle\partial^{0}_{1,2}\colon E^{0}_{1,2}\to E^{0}_{1,1}.$

The image of $\partial^{0}_{0,1}$ is the span of $\\{f-t,f-x,t-x\\}$, so its
cokernel is $\mathbb{Z}$. Now, $\partial^{0}_{0,2}$ takes a composable pair
$\\{f,g\\}$ to $(f+g-f\circ g)$. Now, as $f\circ g$ is different for each of
the composable pairs $\\{f,g\\}$ in $E^{0}_{0,2}$, the kernel of
$\partial^{0}_{0,2}$ is 0. Furthermore, we see that computing the cokernel of
$\partial^{0}_{0,2}$ amounts to identifying $f\circ g$ with $f+g$ for each of
these 8 compositions, so the cokernel is torsion-free and
$E^{1}_{0,1}=\mathbb{Z}^{30-8-2}=\mathbb{Z}^{20}$. By a similar argument, the
kernel of $\partial^{0}_{1,2}$ is 0 and its cokernel is $\mathbb{Z}^{52}$.

Computing $E^{2}$ from $E^{1}$ can be done through tedious computation.
Working out this computation gives that $\partial^{1}_{1,1}$ is surjective and
that $\operatorname{im}\partial^{1}_{2,1}=\ker\partial^{1}_{1,1}$, so
$E^{2}_{0,1},E^{2}_{1,1}$ and $E^{2}_{2,1}$. The spectral sequence now
collapses, and we get that the computed homology $\mathbb{Z}$ in degree 0 and
0 everywhere else, as expected from $D^{3}$.

The key takeaway from this example is that all rows above $q=1$ became 0
everywhere on the $E^{1}$–page. As we shall now show, this always happens,
which causes the spectral sequence to collapse on the second page. For the
rest of the section, when we write a _discrete flow category_ , we shall mean
a discrete flow category induced from a discrete Morse function on a regular
CW complex, as defined in Section 5.

For the following result, recall that for a p-category $C$,
$\operatorname{\overline{N}}C$ is the simplicial category described in Section
4.2.

######  Theorem 8.7.

Let $X$ be a regular CW complex and let $\Sigma$ be a Morse system induced
from a discrete Morse function. Then
$(\operatorname{\overline{N}}\operatorname{Flo}_{\Sigma}[X])([n])$ is a unique
factorization category for all $n\geq 0$.

###### Proof.

As observed in 7.14,
$(\operatorname{\overline{N}}\operatorname{Flo}_{\Sigma}[X])([0])$, which is
just $\operatorname{Flo}_{\Sigma}[X]$ with the poset structure removed, is a
unique factorization category.

Now, let $n\geq 1$. The morphisms in
$(\operatorname{\overline{N}}\operatorname{Flo}_{\Sigma}[X])([n])$ are
represented by $f_{0}\Rightarrow\dots\Rightarrow f_{n}$ with $f_{i}$ morphisms
in $\operatorname{Flo}_{\Sigma}[X]$. It is clear that
$(\operatorname{\overline{N}}\operatorname{Flo}_{\Sigma}[X])([n])$ is a finite
directed category. Furthermore, given a representative
$f_{0}\Rightarrow\dots\Rightarrow f_{n}$, $f_{0}\colon x\to y$ has a unique
factorization through critical cells $x=c_{0},c_{1},\dots,c_{k}=y$. By Theorem
6.7, then all $f_{i}$ factors through these critical cells. Let
$f_{i}=f_{i}^{1}\circ\dots\circ f_{i}^{k}$ be the the decomposition of $f_{i}$
through $c_{0},\dots,c_{k}$. We then have a decomposition
$\\{f_{0},\dots,f_{n}\\}=\\{f_{0}^{1},\dots,f_{n}^{1}\\}\circ\dots\circ\\{f_{0}^{k},\dots,f_{n}^{k}\\}$.
The sets on the right hand side are indecomposable, as $f_{0}^{i}$ is
indecomposable for all $i$. Furthermore, the decomposition is unique, as
$f_{0}^{1}\circ\dots f_{0}^{k}$ is the unique decomposition of $f_{0}$, and
all other $f_{i}^{j}$ are decided uniquely by this decomposition. ∎

######  Corollary 8.8.

Let $\\{E^{r}_{*,*}\\}$ be the spectral sequence associated to a discrete flow
category. Then $E^{1}_{p,q}=0$ for $q\geq 2$

###### Proof.

Let $C$ be the discrete flow category. By definition, $E^{1}_{p,q}$ is the
$q$th homology of the simplicial set
$E^{0}_{p,*}=(\operatorname{N\overline{N}}C)_{p,*}=\mathcal{N}\left((\operatorname{\overline{N}}C)([p])\right)$.
By combining Theorem 8.7 and 7.16, we get that this is 0 when $q\geq 2$. ∎

######  Lemma 8.9.

Let $\\{E^{r}_{*,*}\\}$ be the spectral sequence associated to a discrete flow
category. Then $E^{2}_{p,0}=0$ for $p\geq 1$.

###### Proof.

Let $C$ be the discrete flow category. We have that $E^{1}_{p,0}$ is the free
abelian group on the connected components of
$\mathcal{N}\left((\operatorname{\overline{N}}C)([p])\right)$. Now, if $a$ and
$b$ are connected in
$\mathcal{N}\left((\operatorname{\overline{N}}C)([p])\right)$, then they are
connected in $\mathcal{N}\left((\operatorname{\overline{N}}C)([i])\right)$ for
all $i$. Hence, $E^{1}_{p,0}=\\{(s_{0})^{p}[x]:[x]\in E^{1}_{0,0}\\}$. Now, as
$\partial_{h}(s_{0})^{p}[x]$ is $[x]$ when $p$ is odd and 0 when $p$ is even,
taking the horizontal homology gives us 0 everywhere except in $E^{1}_{0,0}$.
∎

With the previous two results, we are finally ready to prove B.

######  Theorem 8.10 (Theorem B).

The spectral sequence associated to a discrete flow category collapses on the
second page.

###### Proof.

As $E^{1}_{p,q}=0$ for $q\geq 2$, the same applies to $E^{2}$. Hence, the only
potential nonzero entries on page 2 are $E^{2}_{0,0}$ and $E^{2}_{p,1},p\geq
0$. Hence, there can be no nonzero differentials from page 2 and onward. ∎

Note that the above results applies to the ordinary spectral sequence
associated to a bisimplicial set, _not_ the one where degenerate simplices
have been removed (as in 8.4). However, for computational purposes it is
useful to remove degenerate simplices from the spectral sequence, as there are
always infinitely many degenerate simplices. Therefore, we now show that the
spectral sequence also collapses on page 2 when removing degeneracies.

######  Theorem 8.11.

Let $X$ be the double nerve of a discrete flow category. Then the spectral
sequence associated to the double complex

$FX/F(D^{h}\cup D^{v})$

collapses at the second page.

###### Proof.

Let $\\{E^{r}_{*,*}\\}$ be the associated spectral sequence. Firstly, there
are no nondegenerate simplices in $X_{p,0}$ for $p\geq 1$, so
$E^{0}_{p,0}\cong 0$ for $p\geq 1$ (and the same holds for the succeeding
pages).

We now show that $E^{1}_{p,q}\cong 0$ for $q\geq 2$. Recall that
$E^{1}_{p,q}=H_{II}(FX/F(D^{h}\cup D^{v}))_{p,q}$ First, we show that
$H_{II}(FX/FD^{h})_{p,q}\cong 0$ for $q\geq 2$. Suppose that
$[x]\in(FX/FD^{h})_{p,q}$, with $q\geq 2$, such that $\partial_{v}[x]=0$. Then
$x\in FX_{p,q}$ is such that $\partial_{v}x\in FD^{h}$, so
$\partial_{v}x=s^{h}_{0}y_{0}+\dots+s^{h}_{p-1}y_{p-1}$ for some
$y_{0},\dots,y_{p-1}\in FX_{p-1,q}$, where
$s_{i}^{h}=(s_{i},\operatorname{id})$ is the $i$th horizontal degeneracy map.
Now, let $x^{\prime}=x-s^{h}_{p-1}d^{h}_{p}x$. Then $[x^{\prime}]=[x]$ in
$(FX/FD^{h})_{p,q}$, and

$\displaystyle\partial_{v}x^{\prime}$
$\displaystyle=\partial_{v}x-\partial_{v}(s^{h}_{p-1}d^{h}_{p})x=\partial_{v}x-(s^{h}_{p-1}d^{h}_{p})\partial_{v}x$
$\displaystyle=\left(s^{h}_{0}y_{0}+\dots+s^{h}_{p-1}y_{p-1}\right)-\left(s^{h}_{p-1}d^{h}_{p}s^{h}_{0}y_{0}+\dots+s^{h}_{p-1}d^{h}_{p}s^{h}_{p-1}y_{p-1}\right).$

We now utilize the fact that $d_{p}s_{p-1}=\operatorname{id}$ and that when
$i<p-1$, $s_{p-1}d_{p}s_{i}=s_{p-1}s_{i}d_{p-1}=s_{i}s_{p-2}d_{p-1}$. Thus,
the sum above rewrites to

$\displaystyle s^{h}_{0}$
$\displaystyle(y_{0}-s^{h}_{p-2}d^{h}_{p-1}y_{0})+\dots+s^{h}_{p-2}(y_{p-2}-s^{h}_{p-2}d^{h}_{p-1}y_{p-2})+s^{h}_{p-1}(y_{p-1}-y_{p-1})$
$\displaystyle=s^{h}_{0}y_{0}^{\prime}+\dots s^{h}_{p-2}y_{p-2}^{\prime}.$

Now, continuing this process by letting
$x^{\prime\prime}=x^{\prime}-s^{h}_{p-2}d^{h}_{p-1}x^{\prime}$, and so on, we
end up with a $\hat{x}\in FX_{p,q}$ such that
$[\hat{x}]=[x]\in(FX/FD^{h})_{p,q}$ and $\partial_{v}\hat{x}=0$. But as
$H_{II}(FX)_{p,q}$ is 0 when $q\geq 2$, then $\hat{x}$ is in the image of
$\partial_{v}$, and thus $[x]$ is in the image of $\partial_{v}$. Hence,
$H_{II}(FX/FD^{h})_{p,q}\cong 0$ when $q\geq 2$.

It remains to show that $H_{II}(FX/F(D^{h}\cup D^{v}))_{p,q}\cong 0$ when
$q\geq 2$. For this, we use that $G/(AB)=(G/A)/(B/(A\cap B))$ for groups
$A,B<G$, so we have a short exact sequence of chain complexes

$0\rightarrow FD^{v}/(FD^{h}\cap FD^{v})\rightarrow FX/FD^{h}\rightarrow
FX/F(D^{h}\cup D^{v})\rightarrow 0.$ (10)

We know from 2.4 that $H_{II}(FD^{v})\cong 0$. Furthermore, one can show that
$D^{h}_{p,*}$ together with the vertical face and degeneracy maps is a
simplicial set555This follows from the fact that if $x$ is a horizontal
degeneracy, then so is $d^{v}_{i}x$ and $s^{v}_{i}x$, as
$d^{v}_{i}s^{h}_{j}=s^{h}_{j}d^{v}_{i}$ and
$s^{v}_{i}s^{h}_{j}=s^{h}_{j}s^{v}_{i}$.. The degeneracies in this simplicial
set are $D^{h}\cap D^{v}$, so, again by 2.4, $H_{II}(F(D^{h}\cap
D^{v}))=H_{II}(FD^{h}\cap FD^{v}))\cong 0$. In then follows (from the long
exact sequence in homology of short exact sequences of chain complexes) that
$H_{II}(F^{v}/(FD^{h}\cap FD^{v}))=0$.

Finally, we can use the long exact sequence in homology from the short exact
sequence in (10) to conclude that $H_{II}(FX/F(D^{h}\cup D^{v}))_{p,q}\cong
H_{II}(FX/FD^{h})_{p,q}$ (for all $p$ and $q$). In particular,
$H_{II}(FX/F(D^{h}\cup D^{v}))_{p,q}\cong 0$ for $q\geq 2$.

Hence, $E^{1}_{p,q}$ is 0 when $q\geq 2$, and this, together with the fact
that $E^{1}_{p,0}\cong 0$ when $p\geq 1$, shows that the spectral sequence
collapses on the second page. ∎

Observe that due to this theorem, when computing the spectral sequence, we
never need to know $E^{0}_{p,q}$ for $q\geq 3$. In fact, the only things we
need to compute at page 1 is the row $E^{1}_{*,1}$ and the single square
$E^{1}_{0,0}$. The squares in $E^{1}_{*,1}$ also have a relatively simple
description; the differential $\partial_{v}E^{0}_{p,2}\to E^{0}_{p,1}$ sends a
composable pair $\\{f,g\\}$ to $f+g-g\circ f$. Hence, $E^{1}_{p,1}$ is the
same as $E^{0}_{p,1}$ where all morphisms have been identified with the sum of
the morphisms in their (unique) decomposition of indecomposable morphisms.

We now show another example of a spectral sequence computation, using Theorem
8.11 and the observations in the last example.

######  Example 8.12.

We compute the discrete flow category of a discrete Morse function on the
2–torus, illustrated in Figure 19. We compute the associated spectral sequence
(with degeneracies removed), with rational coefficients (to make the
computations easier).

$A$$B$$C$$D$$\alpha$$\beta$$\alpha$$\beta$$\gamma$$\delta$$\varepsilon$$\zeta$$\varepsilon$$\zeta$$\eta$$\theta$$a$$b$$a$$c$$d$$c$$a$$b$$a$
Figure 19. A discrete Morse function (represented by its gradient vector
field) on a regular CW decomposition of the 2–torus. The regular pairs are
given by red, dashed arrows.

The critical cells are $A$, $\alpha$, $\epsilon$ and $a$. We compute the Hom
posets with Algorithm 1, and get:

$\displaystyle\operatorname{Hom}(A,\alpha)$
$\displaystyle=\left\\{f_{1}=[A>\alpha],\ f_{2}=[A>\gamma<C>\alpha]\right\\},$
$\displaystyle\operatorname{Hom}(A,\varepsilon)$
$\displaystyle=\left\\{f_{1}^{\prime}=[A>\varepsilon],\
f_{2}^{\prime}=[A>\eta<B>\varepsilon]\right\\},$
$\displaystyle\operatorname{Hom}(\alpha,a)$
$\displaystyle=\left\\{g_{1}=[\alpha>a],\ g_{2}=[\alpha>b<\beta>a]\right\\},$
$\displaystyle\operatorname{Hom}(\varepsilon,a)$
$\displaystyle=\left\\{g_{1}^{\prime}=[\varepsilon>a],\
g_{2}^{\prime}=[\varepsilon>c<\zeta>a]\right\\}.$

The only Hom poset with nontrivial partial orders is
$\operatorname{Hom}(A,a)$, which consists of 36 morphisms in four connected
components, as illustrated in Figure 20.

$\displaystyle g_{1}\circ f_{1}\Rightarrow\ \bullet\ \Leftarrow
g_{1}^{\prime}\circ f_{1}^{\prime}$ $\displaystyle g_{2}\circ
f_{1}\Rightarrow\ \bullet\ \Leftarrow\quad\dots\quad\Rightarrow\ \bullet\
\Leftarrow g_{1}^{\prime}\circ f_{2}^{\prime}$ $\displaystyle g_{1}\circ
f_{2}\Rightarrow\ \bullet\ \Leftarrow\quad\dots\quad\Rightarrow\ \bullet\
\Leftarrow g_{2}^{\prime}\circ f_{1}^{\prime}$ $\displaystyle g_{2}\circ
f_{2}\Rightarrow\ \bullet\ \Leftarrow\quad\dots\quad\Rightarrow\ \bullet\
\Leftarrow g_{2}^{\prime}\circ f_{2}^{\prime}$ Figure 20. The four components
in the Hom poset $\operatorname{Hom}(A,a)$. The unnamed morphisms (represented
by dots) are all indecomposable.

[title = Page 0] [”Q^4”](0,0) [”Q^44”](0,1) [”Q^8”](0,2) [”Q^32”](1,1) 0̣ (0,
2) 0̣ (0, 1) [title = Page 1] [”Q”](0,0) [”Q^33”](0,1) [”Q^32”](1,1) 1̣ (1, 1)
(1,2) [title = Page 2] [”Q”](0,0) [”Q^2”](0,1) [”Q”](1,1) (1,2)

Figure 21. The first three pages of the spectral sequence for a discrete flow
category of the 2–torus.

The discrete flow category has 4 objects, 44 morphisms, 8 nondegenerate
horizontal compositions of morphisms, 32 non-identity partial orders, and no
other nondegenerate compositions. Hence, the $E^{0}$ page is as illustrated in
Figure 21.

On page 1, $E^{1}_{0,0}$ is $\mathbb{Q}$, as the nerve of the category has one
connected component. Hence, $E^{1}_{0,1}=\mathbb{Q}^{44-8-3}=\mathbb{Q}^{33}$.
Furthermore, $E^{1}_{0,1}$ can be described as the cycles in $E^{0}_{0,1}$,
where all compositions $f\circ g$ has been identified with $f+g$.

Now, to compute $E^{2}$, we need only determine the map
$\partial^{2}_{1,1}\colon E^{1}_{1,1}\to E^{1}_{0,1}$. For this, observe that
for each of the four components in Figure 20, we can assign an alternating sum
of partial orders $x_{i}$, so that

$\displaystyle\partial^{2}x_{1}=g_{1}^{\prime}\circ f_{1}^{\prime}-g_{1}\circ
f_{1}=g_{1}^{\prime}+f_{1}^{\prime}-g_{1}-f_{1},$
$\displaystyle\partial^{2}x_{2}=g_{1}^{\prime}\circ f_{2}^{\prime}-g_{2}\circ
f_{1}=g_{1}^{\prime}+f_{2}^{\prime}-g_{2}-f_{1},$
$\displaystyle\partial^{2}x_{3}=g_{2}^{\prime}\circ f_{1}^{\prime}-g_{1}\circ
f_{2}=g_{2}^{\prime}+f_{1}^{\prime}-g_{1}-f_{2},$
$\displaystyle\partial^{2}x_{4}=g_{2}^{\prime}\circ f_{2}^{\prime}-g_{2}\circ
f_{2}=g_{2}^{\prime}+f_{2}^{\prime}-g_{2}-f_{2}.$

Now, $\partial^{2}(x_{1}-x_{2}-x_{3}+x_{4})=0$. One can verify (for example
through writing $\partial^{2}$ as a matrix and row reducing) that
$(x_{1}-x_{2}-x_{3}+x_{4})$ generates the kernel of $\partial^{2}$, so that
$E^{2}_{1,1}=\mathbb{Q}$ and $E^{2}_{0,1}=\mathbb{Q}^{2}$. The spectral
sequence now collapses, as predicted by Theorem 8.11, and the computed
homology of the 2–torus is:

$H_{n}(T^{2};\mathbb{Q})=\begin{cases}\mathbb{Q},&\quad n=0,2,\\\
\mathbb{Q}^{2},&\quad n=1,\\\ 0,&\quad\text{ otherwise.}\\\ \end{cases}$

Observe that in this case we got a nonzero group in $E^{2}_{1,1}$ even though
the homology of (the nerve of) the Hom poset $\operatorname{Hom}(A,a)$ is 0 in
degree 1. This happens because on page 1, the compositions $f\circ g$ becomes
identified with the sums $f+g$, so that the differential $\partial^{2}\colon
E^{1}_{1,1}\to E^{1}_{0,1}$ has a nonzero element in the kernel, even though
the differential $\partial_{h}\colon E^{0}_{1,1}\to E^{0}_{0,1}$ does not.

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