# Monotonicity of the period map for the equation
$-\varphi^{\prime\prime}+\varphi-\varphi^{k}=0$.

###### Abstract.

In this paper, we establish the monotonicity of the period map in terms of the
energy levels for certain periodic solutions of the equation
$-\varphi^{\prime\prime}+\varphi-\varphi^{k}=0$, where $k>1$ is a real number.
We present a new approach to demonstrate this property, utilizing spectral
information of the corresponding linearized operator around the periodic
solution and tools related to Floquet theory.

###### Key words and phrases:

Periodicity of the period map, Floquet theory, Periodic solutions.
∗Corresponding author

###### 2000 Mathematics Subject Classification:

35B10, 35J61, 47A75.

Giovana Alves

Centro de Ciências Exatas Naturais e Tecnológicas

Universidade Estadual da Região Tocantina do Maranhão

Imperatriz, Maranhão, 65900-000, Brazil.

<EMAIL_ADDRESS>
Fábio Natali∗

Departamento de Matemática

Universidade Estadual de Maringá

Maringá, Paraná, CEP 87020-900, Brazil.

<EMAIL_ADDRESS>
##  1\. Introduction

Let $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic solution of the
equation

$-\varphi^{\prime\prime}+\varphi-\varphi^{k}=0,$ (1.1)

where $k>1$ is a real number and $\varphi=\varphi(x)$ is a function depending
on $x\in\mathbb{R}$. We easily verify that , the function defined by

$\mathcal{E}(\varphi,\xi)=\frac{\xi^{2}}{2}-\frac{\varphi^{2}}{2}+\frac{\varphi^{k+1}}{k+1},$
(1.2)

is a first integral for the two dimensional system associated to equation
(1.1), where $\xi=\varphi^{\prime}$. In other words, this means that the pair
$(\varphi,\xi)$ satisfies $\mathcal{E}(\varphi,\xi)=B$ for convenient values
$B\in\mathbb{R}$. When $k>1$ and $\frac{1-k}{2(k+1)}<B<0$, we obtain periodic
orbits that spinning around the center point $(1,0)$. When $k>1$ is particular
an odd number, we also obtain periodic solutions but for all $B>0$. In this
case, the periodic solutions change their sign, and all initial conditions
associated with the solution are located outside the separatrix curve. (a
curve that converges to the saddle point $(0,0)$ in the associated phase
portrait).
The period map $L=L(B)$ of the periodic solution $\varphi$ depends on $B$ and
it can be expressed by

$L=\displaystyle
2\int_{b_{1}}^{b_{2}}\frac{dh}{\sqrt{-\frac{2h^{k+1}}{k+1}+h^{2}+2B}},$ (1.3)

where $b_{1}=\displaystyle\min_{x\in[0,L]}\varphi(x)$ and
$b_{2}=\displaystyle\max_{x\in[0,L]}\varphi(x)$. This expression is not very
handy to obtain the smoothness of the period function $L$ in terms of $B$
since $b_{1}$ and $b_{2}$ are zeroes of the function
$G(\varphi)=-\frac{2\varphi^{k+2}}{k+2}+\varphi^{2}+2B$. However, from
standard theory of ODE, the solution $\varphi$ of $(\ref{ode1})$ depends
smoothly on the initial conditions $\varphi(t_{0})=\varphi_{0}$ and
$\varphi^{\prime}(t_{0})=\varphi_{1}$, $t_{0}\in\mathbb{R}$ and in particular,
$\varphi$ and $L$ also depend smoothly on $B$ in a convenient set of
parameters (see [9, Chapter 1, Theorem 3.3]).
With the smoothness of the period map in hands, it is important to know the
behaviour of this function in terms of the energy levels $B$ or even in terms
of the initial data. In fact, questions related to the behaviour of the period
function have attracted a considerable attention in the last decades and the
study of the monotonicity of the period map is important for several reasons.
For instance, if the period map is monotonic, solutions with large periods
correspond to initial conditions that are far away from the critical points.
On the other hand, if the period map is not monotonic, then the periodic
solutions may exhibit bifurcations points in the associated phase portrait and
the uniqueness of solutions may be not valid.
Next, we present some contributors. Loud [10] and Urabe [17] have proved the
isochronicity111Just to make clear the comprehension of the reader, a center
point is said to be isochronous when all periodic solutions turning around the
center point have the same period of the period function for specific families
of planar vector fields. Using tools of ODE, Schaaf [16] has established a
general set of sufficient conditions for the monotonicity of the period
function for a class of Hamiltonian systems of the form

$\left\\{\begin{array}[]{llll}\varphi^{\prime}=-f(\psi),\\\
\psi^{\prime}=g(\varphi).\end{array}\right.$ (1.4)

By assuming suitable assumptions on the smooth functions $f$ and $g$, it is
possible to prove that the period function is strictly increasing for periodic
solutions turning around the center point $(0,0)$ (see also Rothe [15] for an
extension of the results in [16]).
Cima, Gasull and Mañosas [5] (see also Coppel and Gavrilov [6]) studied the
same system in $(\ref{hamiltgen})$ and they characterized the limiting
behaviour of the period $L$ at infinity when the origin is a global center. In
adddition, they apply this result to prove that there are no nonlinear
polynomial isochronous centers in this family, showing the monotonicity of the
period in terms of the energy levels $B$. Rothe [14] and Waldvogel [19] showed
that all Lotka-Volterra systems of the form

$\varphi’=\varphi(a-b\psi),\quad\psi’=-\psi(c-d\varphi),$

for $a$ and $b$ in a suitable set of conditions, have monotonic period
functions. Chow and Wang [4] gave some characterization of the first and
second derivatives for the period function in terms of $B$ for the equation

$\varphi^{\prime\prime}+g(\varphi)=0,$

where $g$ satisfies a suitable set of conditions. As an application, they
determine the monotonicity of the period function when $g(s)=e^{s}-1$. Using a
different approach, Chicone [2] also obtained part of the results in [4] and
gave a general condition for the monotonicity of period functions for a class
of planar Hamiltonian systems. In addition, [2] was the inspiration for the
study in Yagasaki [18] to prove the monotonicity of the period function for
the equation $(\ref{ode1})$. In this work, the authors also considered $k>1$ a
real number and the periodic solutions turning around the center point
$(1,0)$. A revisited proof of [18] has been established by Benguria, Depassier
and Loss in [1].
In our approach, we use a different method to establish the monotonicity of
the period map for positive solutions as determined in [18] and we realize
that when $k>1$ is in particular an odd number, we can use the same method to
determine the same behaviour for the periodic solutions of the equation
$(\ref{ode1})$ that change their sign. As far as we know, this last question
never been treated in the current literature and our intention is to give a
new perspective to solve this kind of problems since our approach can be used
in other models.
Without further ado, we will describe our methods. Indeed, let us consider the
linearized operator around $\varphi$ given by

$\mathcal{L}(y)=-y^{\prime\prime}+y-k\varphi^{k-1}y,$ (1.5)

defined in $L_{per}^{2}([0,L])$ with domain $H_{per}^{2}([0,L])$. Deriving
equation $(\ref{ode1})$ with respect to $x$, we obtain that $\varphi^{\prime}$
satisfies $\mathcal{L}\varphi^{\prime}=0$, so that $\varphi^{\prime}$ is a
periodic element which solves the equation

$-y^{\prime\prime}+y-k\varphi^{k-1}y=0.$ (1.6)

Since $(\ref{fundset})$ is a second order linear equation, there exists
another solution $\bar{y}$ for the equation $\mathcal{L}\bar{y}=0$ and
$\\{\varphi^{\prime},\bar{y}\\}$ is a fundamental set of solutions for the
equation $(\ref{fundset})$. Using tools of the Floquet theory (see [11]), we
see that function $\bar{y}$ can be periodic or not and it is related with
$\varphi^{\prime}$ through the equality (see [11])

$\bar{y}(x+L)=\bar{y}(x)+\theta\varphi^{\prime}(x),$ (1.7)

for all $x\in\mathbb{R}$. Constant $\theta$ is a real parameter and it
determines if $\bar{y}$ is periodic or not in the sense that $\bar{y}$ is
periodic if and only if $\theta=0$.
Our strategy to prove the monotonicity of $L$ in terms of $B$ is then to show
that $\bar{y}$ is always a non-periodic function by proving that the kernel of
$\mathcal{L}$ in $(\ref{operator})$ in $(\ref{operator1})$ defined in
$L_{per}^{2}([0,L])$ with domain $H_{per}^{2}([0,L])$ is always simple (see
Lemma 3.7 and Lemma 3.5). After that, we need to use the smoothness of the
period function with respect to $B$ to prove that $\frac{\partial L}{\partial
B}=-\theta$ (see Lemma 3.2). Thus, since $\bar{y}$ is always non-periodic, we
can conclude that $\theta\neq 0$, and the monotonicity of the period map is
determined. Our main result can be summarized as follows:

######  Theorem 1.1.

Let $L=L(B)$ be the period function associated to the periodic solution
$\varphi$ of the equation $(\ref{ode1})$.
i) If $k>1$ is odd and $\varphi$ is a solution that changes its sign, then
$\frac{\partial L}{\partial B}<0$.
ii) If $k>1$ and $\varphi$ is a positive non-constant periodic solution, then
$\frac{\partial L}{\partial B}>0$.

Our paper is organized as follows. In Section 2 we present a brief explanation
concerning the existence of positive and solutions that change their sign.
Section 3 is devoted to prove Theorem 1.1.

Notation. Here, we introduce the basic notation concerning the periodic
Sobolev spaces that will be useful in our paper. For a more complete
introduction to these spaces we refer the reader to [8]. In fact, by
$L^{2}_{per}([0,L])$, we denote the space of all square integrable functions
which are $L$-periodic. For $m\geq 0$ integer, the Sobolev space
$H^{m}_{per}([0,L])$ is the set of all periodic functions such that

$\|f\|_{H^{m}_{per}}^{2}:=||f||_{L_{per}^{2}}^{2}+\sum_{j=1}^{m}||f^{(j)}||_{L_{per}^{2}}^{2}<\infty,$

where $f^{(j)}$, $j=1,\cdots,m$, indicates the $j$-th derivative of $f$ in the
sense of periodic distributions (see [8] for further details). The space
$H^{m}_{per}([0,L])$ is a Hilbert space with natural inner product denoted by
$(\cdot,\cdot)_{H_{per}^{m}}$. When $m=0$, the space $H^{m}_{per}([0,L])$ is
isometrically isomorphic to the space $L^{2}_{per}([0,L])$. The norm and inner
product in $L^{2}_{per}([0,L])$ will be denoted by $\|\cdot\|_{L_{per}^{2}}$
and $(\cdot,\cdot)_{L_{per}^{2}}$, respectively.

##  2\. Existence of periodic solutions via planar analysis.

Our purpose in this section is to present some facts concerning the existence
of periodic solutions for the nonlinear ODE given by

$-\varphi^{\prime\prime}+\varphi-\varphi^{k}=0,$ (2.1)

where $k>1$ is a real number.

Equation (2.1) is conservative with

$\mathcal{E}(\varphi,\xi)=\frac{\xi^{2}}{2}-\frac{\varphi^{2}}{2}+\frac{\varphi^{k+1}}{k+1},$
(2.2)

as a first integral, where $\xi=\varphi^{\prime}$, and thus its solutions are
contained on the level curves of the energy.
According to the classical ODE theory (see [3], [9] and [12] for further
details), $\varphi$ is a periodic solution of the equation $(\ref{ode})$ if
and only if $(\varphi,\varphi^{\prime})$ is a periodic orbit of the planar
differential system

$\left\\{\begin{array}[]{lllll}\varphi^{\prime}=\xi,\\\ \\\
\xi^{\prime}=\varphi-\varphi^{k}.\end{array}\right.$ (2.3)

The periodic orbits for the equation $(\ref{planarODE})$ can be determined by
considering the energy levels of the function $\mathcal{E}$ defined in
$(\ref{energyODE})$. This means that the pair $(\varphi,\xi)$ satisfies the
equation $\mathcal{E}(\varphi,\xi)=B$. If $k>1$ and
$B\in\left(\frac{1-k}{2(k+1)},0\right)$, we obtain periodic orbits that turn
round at the equilibrium points $(1,0)$ and these orbits are called positive
since they are associated with positive periodic solutions of the equation
$(\ref{ode})$. In our specific case, $(\ref{planarODE})$ admits at least two
critical points, being a saddle point at the origin and a center point at
$(1,0)$. According to the standard ODE theory, the periodic orbits emanate
from the center points to the separatrix curve which is represented by a
smooth solution $\widetilde{\varphi}:\mathbb{R}\rightarrow\mathbb{R}$ of
$(\ref{ode})$ satisfying
$\lim_{x\rightarrow\pm\infty}\widetilde{\varphi}^{(n)}(x)=0$ for all
$n\in\mathbb{\mathbb{N}}$. In particular, when $k$ is odd, we see that the
presence of two symmetric center points $(\pm 1,0)$ allows to conclude that
the periodic orbits which turn around these points can be positive and
negative222The terminology positive (negative) periodic orbit means that the
associated periodic solution for the equation $(\ref{ode})$ is positive
(negative).. For every orbit with initial condition outside the separatrix, we
obtain periodic solutions for the equation $(\ref{ode})$ that change their
sign 333Here, we can also define an appropriate terminology for this kind of
periodic solution as periodic orbit that changes its sign.. Indeed, if $B>0$
we also have periodic orbits and the corresponding periodic solutions
$\varphi$ with the zero mean property, that is, periodic solutions satisfying
$\int_{0}^{L}\varphi(x)dx=0$.

Figure 2.1. Periodic orbits for the equation $(\ref{ode})$. Left: positive and
periodic orbits for the case $k>1$ that spinning around the center point
$(1,0)$. Right: Positive and negative periodic orbits for the case $k>1$ and
odd that spinning around the center points $(1,0)$ and $(-1,0)$, respectively.
Right: periodic orbits that change their sign when the initial conditions are
located outside the the separatrix.

Independently of the type of periodic solutions on which we are working, the
period $L=L(B)$ of the solution $\varphi$ can be expressed (formally) as
follows:

$L=\displaystyle
2\int_{b_{1}}^{b_{2}}\frac{dh}{\sqrt{-\frac{2h^{k+1}}{k+1}+h^{2}+2B}},$ (2.4)

where $b_{1}=\displaystyle\min_{x\in[0,L]}\varphi(x)$ and
$b_{2}=\displaystyle\max_{x\in[0,L]}\varphi(x)$.
On the other hand, the energy levels of the first integral $\mathcal{E}$ in
$(\ref{energyODE})$ parametrize the unbounded set of periodic orbits
$\\{\Gamma_{B}\\}_{B}$ which emanate from the separatrix curve. Thus, we can
conclude that the set of smooth periodic solutions of $(\ref{ode})$ can be
expressed by a smooth family $\varphi=\varphi_{B}$ which is parametrized by
the value $B$. Moreover, for the case $B\in\left(\frac{1-k}{2(k+1)},0\right)$,
we see that if $B\rightarrow\frac{1-k}{2(k+1)}$ thus $L\rightarrow 2\pi$,
where $2\pi$ is the period of the equilibrium solution $(1,0)$ of the system
$(\ref{planarODE})$ (see [12, Section 2]). On the other hand, if $B\rightarrow
0$ we have $L\rightarrow+\infty$ and both convergences suggest that the period
map is increasing in the interval $\left(\frac{1-k}{2(k+1)},0\right)$. On the
other hand, when $B\in(0,+\infty)$, we see that if $B\rightarrow 0$, we have
$L\rightarrow+\infty$, and if $B\rightarrow+\infty$, we obtain $L\rightarrow
0$. In this case, the suggestion is that the period map is decreasing.

##  3\. The monotonicity of the period map - proof of Theorem 1.1.

We need to recall some basic facts concerning Floquet’s theory (see [7] and
[11]). Let $Q$ be a smooth $L$-periodic function. Consider $\mathcal{P}$ the
Hill operator defined in $L_{per}^{2}([0,L])$, with domain
$D(\mathcal{P})=H_{per}^{2}([0,L])$, given by

$\mathcal{P}=-\partial_{x}^{2}+Q(x).$ (3.1)

The spectrum of $\mathcal{P}$ is formed by an unbounded sequence of real
eigenvalues

$\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\lambda_{4}\leq\cdots\;\leq\lambda_{2n-1}\leq\lambda_{2n}\;\cdots,$
(3.2)

where equality means that $\lambda_{2n-1}=\lambda_{2n}$ is a double
eigenvalue. Moreover, according with the Oscillation Theorem (see [11]), the
spectrum is characterized by the number of zeros of the eigenfunctions as: if
$p$ is an eigenfunction associated to either $\lambda_{2n-1}$ or
$\lambda_{2n}$, then $p$ has exactly $2n$ zeros in the half-open interval
$[0,L)$. In particular, the even eigenfunction associated to the first
eigenvalue $\lambda_{0}$ has no zeros in $[0,L]$.

Let $p$ be a nontrivial $L$-periodic solution of the equation

$\mathcal{P}(p)=-p^{\prime\prime}+Q(x)p=0.$ (3.3)

Consider $\bar{y}$ the other solution of (3.3) linearly independent of $p$.
There exists a constant $\theta$ (depending on $\bar{y}$ and $p$) such that
(see [11, page 5])

$\bar{y}(x+L)=\bar{y}(x)+\theta p(x).$ (3.4)

Consequently, $\theta=0$ is a necessary and sufficient condition to all
solutions of (3.3) to be $L$-periodic. This criterion is very useful to
establish if the kernel of $\mathcal{P}$ is one-dimensional or not.
Next, let $\varphi=\varphi_{B}$ be any periodic solution of (2.1). Consider
$\mathcal{L}=\mathcal{L}_{B}:H_{per}^{2}([0,L])\subset
L_{per}^{2}([0,L])\rightarrow L_{per}^{2}([0,L])$ the linearized operator
arising from the linearization of (2.1) at $\varphi$, that is,

$\mathcal{L}(y)=-y^{\prime\prime}+y-k\varphi^{k-1}y.$ (3.5)

Clearly $\mathcal{P}=\mathcal{L}$, when $Q(x)y(x)=y(x)-k\varphi^{k-1}(x)y(x)$.
By taking the derivative with respect to $x$ is (2.1), we have that
$p:=\varphi^{\prime}$ belongs to the kernel of the operator $\mathcal{L}$. In
addition, from our construction, $\varphi^{\prime}$ has exactly two zeros in
the half-open interval $[0,L)$ (see Figure 2.1), which implies that zero is
the second or the third eigenvalue of $\mathcal{L}$ (see Oscillation’s Theorem
in [11])

Next, let $\bar{y}=\bar{y}_{B}$ be the unique even solution of the initial-
value problem

$\left\\{\begin{array}[]{l}-\bar{y}^{\prime\prime}+\bar{y}-k\varphi^{k-1}\bar{y}=0,\\\
\bar{y}(0)=-\frac{1}{\varphi^{\prime\prime}(0)},\\\
\bar{y}^{\prime}(0)=0.\end{array}\right.$ (3.6)

Since $\varphi^{\prime}$ is an $L$-periodic solution for the equation in (3.6)
and the corresponding Wronskian of $\varphi^{\prime}$ and $\bar{y}$ is 1,
there is a constant $\theta=\theta_{\bar{y}}$ such that

$\bar{y}(x+L)=\bar{y}(x)+\theta\varphi^{\prime}(x).$ (3.7)

By taking the derivative in this last expression and evaluating at $x=0$, we
obtain

$\theta=\frac{\bar{y}^{\prime}(L)}{\varphi^{\prime\prime}(0)}.$ (3.8)

The next result gives that it is possible to decide the exact position of the
zero eigenvalue by knowing the precise sign of $\theta$ in $(\ref{theta1})$.

######  Lemma 3.1.

i) If $\theta>0$, operator $\mathcal{L}$, defined in $L_{per}^{2}([0,L])$,
with domain $H_{per}^{2}([0,L])$, has exactly two negative eigenvalue which
are simple, a simple eigenvalue at zero and the rest of the spectrum is
positive and bounded away from zero.
ii) If $\theta<0$, the same operator $\mathcal{L}$ has exactly one negative
eigenvalue which is simple, a simple eigenvalue at zero and the rest of the
spectrum is positive and bounded away from zero.

###### Proof.

See Theorem 3.1 in [13]. ∎

Now we can give a relation between $\frac{\partial L}{\partial B}$ and
$\theta$.

######  Lemma 3.2.

We have that $\frac{\partial L}{\partial B}=-\theta$, where $\theta$ is the
constant in (3.8).

###### Proof.

Consider $\bar{y}$ and $\varphi^{\prime}$ as above. Since $\varphi$ is even
and periodic one has $\varphi^{\prime}(0)=\varphi^{\prime}(L)=0$. Thus, the
smoothness of $\varphi$ in terms of the parameter $B$ enables us to take the
derivative of $\varphi^{\prime}(L)=0$ with respect to $B$ to obtain

$\varphi^{\prime\prime}(L)\frac{\partial L}{\partial
B}+\frac{\partial\varphi^{\prime}(L)}{\partial B}=0.$ (3.9)

Next, we turn back to equation $(\ref{ode})$ and multiply it by
$\varphi^{\prime}$ to deduce, after integration, the quadrature form

$\frac{\varphi^{\prime
2}(x)}{2}-\frac{\varphi(x)^{2}}{2}+\frac{\varphi(x)^{k+1}}{k+1}-B=0,\ \ \ \
\mbox{for all}\ x\in[0,L].$ (3.10)

Deriving equation $(\ref{eq12344})$ with respect to $B$ and taking $x=0$ in
the final result, we obtain from $(\ref{ode})$ that
$\frac{\partial\varphi(0)}{\partial B}=\frac{1}{\varphi^{\prime\prime}(0)}$.
In addition, since $\varphi$ is even one has that
$\frac{\partial\varphi}{\partial B}$ is also even and thus
$\frac{\partial\varphi^{\prime}(0)}{\partial B}=0$. On the other hand,
deriving equation $(\ref{ode})$ with respect to $B$, we obtain that
$\frac{\partial\varphi}{\partial B}$ satisfies the initial-value problem

$\left\\{\begin{array}[]{lllll}\displaystyle-\frac{\partial\varphi}{\partial
B}^{\prime\prime}+\frac{\partial\varphi}{\partial
B}-k\varphi^{k-1}\frac{\partial\varphi}{\partial B}=0,\\\ \\\
\displaystyle\frac{\partial\varphi(0)}{\partial
B}=\frac{1}{\varphi^{\prime\prime}(0)},\\\ \\\
\displaystyle\frac{\partial\varphi^{\prime}(0)}{\partial
B}=0.\end{array}\right.$ (3.11)

The existence and uniqueness theorem for ordinary differential equations
applied to the problem $(\ref{y})$ enables us to deduce that
$\bar{y}=-\frac{\partial\varphi}{\partial B}$. Therefore, we can combine
$(\ref{theta})$ with $(\ref{eq12343})$ to obtain that $\frac{\partial
L}{\partial B}=-\theta$. ∎

Next lemma is basic for our purposes.

######  Lemma 3.3.

Let $\mathcal{L}$ be the linearized operator defined in $(\ref{operator})$.
Thus, $1\in\rm{range}(\mathcal{L})$.

###### Proof.

First, we see that (2.1) is invariant under translations. This means that if
$\varphi$ is a solution of $(\ref{ode})$, thus $\psi_{r}=\varphi(\cdot-r)$ is
also a solution for all $r\in\mathbb{R}$. In particular, if $r=L/4$ we obtain
that $\psi:=\psi_{L/4}=\varphi(\cdot-L/4)$ is also a periodic solution for the
equation $(\ref{ode})$. Therefore, $\varphi^{\prime}$ is an odd element of
$\ker(\mathcal{L})$, so that $\psi^{\prime}$ results to be even and it is an
element of the kernel of
$\widetilde{\mathcal{L}}=-\partial_{x}^{2}+1-k\psi^{k-1}$
($\widetilde{\mathcal{L}}$ is the operator obtained from $\mathcal{L}$ by
considering the linearization at $\psi$ instead of $\varphi$). Consider then,
$\chi=\bar{y}(\cdot-L/4)$ the corresponding element of the formal equation
$\mathcal{L}(\bar{y})=0$ associated to the translation solution $\psi$. Since
$\chi$ results to be $L-$periodic and odd, we have

$\begin{array}[]{lllll}\displaystyle\int_{0}^{L}\chi(x)dx&=&\displaystyle\frac{1}{2}\int_{0}^{L}\chi(x)dx+\frac{1}{2}\int_{L}^{2L}\chi(x)dx\\\
\\\
&=&\displaystyle\frac{1}{2}\int_{0}^{2L}\chi(x)dx=\displaystyle\frac{1}{2}\int_{-L}^{L}\chi(x)dx=0.\end{array}$

that is, $\chi$ has the zero mean property.
Let us define the following function:

$h(x)=\left(\int_{0}^{x}\chi(s)ds\right)\psi^{\prime}(x)-\left(\int_{0}^{x}\psi^{\prime}(s)ds\right)\chi(x).$
(3.12)

Using the method of variation of parameters, we see that $h$ is a particular
solution (not necessarily periodic) of the equation

$-u^{\prime\prime}+u-k\psi^{k-1}u=1,$ (3.13)

in the sense that $Y=c_{1}\psi^{\prime}+c_{2}\chi+h$ is a general solution
associated with the equation $(\ref{gensol})$, where $c_{1}$ and $c_{2}$ are
real constants. We need to prove that $h$ is periodic. In fact, it is clear
$h(0)=0$ and since $\int_{0}^{L}\chi(x)dx=0$, one has from $(\ref{varform})$
and the fact that $\psi^{\prime}$ is periodic that $h(L)=0$. Deriving
$(\ref{varform})$ with respect to $x$, we obtain after some computations
$h^{\prime}(x)=\left(\int_{0}^{x}\chi(s)ds\right)\psi^{\prime\prime}(x)-\left(\int_{0}^{x}\psi^{\prime}(s)ds\right)\chi^{\prime}(x)$,
so that $h^{\prime}(0)=0$. Since $\psi$ is both odd and periodic, it follows
that $\psi^{\prime}$ and $\psi^{\prime\prime}$ are also periodic, with
$\psi^{\prime\prime}$ being an odd function. We have

$\begin{array}[]{lll}h^{\prime}(L)&=&\displaystyle\left(\int_{0}^{L}\chi(s)ds\right)\psi^{\prime\prime}(L)-\left(\int_{0}^{L}\psi^{\prime}(s)ds\right)\chi^{\prime}(L)\\\
\\\
&=&\displaystyle\left(\int_{0}^{L}\chi(s)ds\right)\psi^{\prime\prime}(0)-(\psi(L)-\psi(0))\chi^{\prime}(L)=0\end{array}$

so that, $h$ is periodic. Equality $(\ref{gensol})$ is then satisfied for the
periodic function $h$, that is,

$-h^{\prime\prime}(x)+h(x)-k\psi(x)^{k-1}h(x)=1,$

for all $x\in\mathbb{R}$. In particular, for $x=t+L/4$ and since
$\varphi(t)=\psi(t+L/4)$, we have

$-h^{\prime\prime}(t+L/4)+h(t+L/4)-k\varphi(t)^{k-1}h(x+L/4)=1.$

Defining $\tilde{h}(t)=h(t+L/4)$, we obtain that $\tilde{h}$ is $L-$periodic
and it satisfies the equation
$-\tilde{h}^{\prime\prime}(t)+\tilde{h}(t)-k\varphi(t)^{k-1}\tilde{h}(t)=1$,
so that $\mathcal{L}(\tilde{h})=1$ as requested in lemma. ∎

######  Lemma 3.4.

We have that $\\{\varphi^{k-1},\varphi^{k}\\}\subset{\rm range}(\mathcal{L})$.

###### Proof.

By $(\ref{ode})$, we obtain

$\mathcal{L}(\varphi)=-\varphi^{\prime\prime}+\varphi-k\varphi^{k}=-\varphi^{\prime\prime}+\varphi-k\varphi^{k}+\varphi^{k}-\varphi^{k}=(1-k)\varphi^{k}.$

Thus, $\varphi^{k}\in{\rm range}(\mathcal{L})$.
To prove that $\varphi^{k-1}\in{\rm range}(\mathcal{L})$, we need to use the
fact that $\mathcal{L}(1)=1-k\varphi^{k-1}$. By Lemma 3.3 one has $1\in{\rm
range}(\mathcal{L})$, so that $\varphi^{k-1}\in{\rm range}(\mathcal{L})$. ∎

Next lemma establishes item i) of Theorem 1.1. In what follows
$n(\mathcal{A})$ and $z(\mathcal{A})$ are respectively the number of negative
eigenvalues (counting multiplicities) and the dimension of the kernel of a
certain linear operator $\mathcal{A}$.

######  Lemma 3.5.

Let $k>1$ be a positive odd integer. If $\varphi$ is the zero mean periodic
solution of $(\ref{ode})$, thus $n(\mathcal{L})=2$ and $z(\mathcal{L})=1$. In
particular, we have that $\frac{\partial L}{\partial B}<0$.

###### Proof.

First, we see that $\varphi^{\prime}$ is an odd eigenfunction of $\mathcal{L}$
associated to the eigenvalue $0$ having two zeroes in the interval $[0,L)$.
From the Oscillation theorem (see [11]), we obtain that $0$ needs to be the
second or the third eigenvalue in the sequence of real numbers in
$(\ref{seqeigen})$.
On the other hand, since $\varphi$ is even and $\int_{0}^{L}\varphi(x)dx=0$,
one has $\psi=\varphi(\cdot-L/4)$ is odd and it has the zero mean property.
Consider again $\widetilde{\mathcal{L}}=-\partial_{x}^{2}+1-k\psi^{k-1}$ the
translated operator obtained from $\mathcal{L}$ by considering the
linearization at $\psi$ instead of $\varphi$ and let
$\widetilde{\mathcal{L}}_{odd}$ be the restriction of
$\widetilde{\mathcal{L}}$ in the odd sector of $L_{per}^{2}([0,L])$. Notice
that such restriction is possible because $\psi^{k-1}$ is even since $k-1$ is
an even number. Since $k+1>2$ is also even, we have
$(\widetilde{\mathcal{L}}_{odd}(\psi),\psi)_{L_{per}^{2}}=(1-k)\int_{0}^{L}\psi(x)^{k+1}dx<0$,
so that by Courant’s min-max characterization of eigenvalues, we obtain
$n(\widetilde{\mathcal{L}}_{odd})\geq 1$. The fact
$n(\widetilde{\mathcal{L}})=n(\widetilde{\mathcal{L}}_{odd})+n(\widetilde{\mathcal{L}}_{even})$
and Krein-Rutman’s Theorem enable us to conclude that the first eigenvalue of
$\widetilde{\mathcal{L}}$ is simple and it is associated to a positive
(negative) eigenfunction which needs to be even. Thus, we obtain since $0$ is
the second or third eigenvalue of $\widetilde{\mathcal{L}}$ that
$n(\widetilde{\mathcal{L}})=n(\mathcal{L})=2$ as requested.
We prove that $z(\mathcal{L})=1$. Indeed, since
$n(\widetilde{\mathcal{L}}_{odd})=1$, we see that the corresponding
eigenfunction $p$ associated to the first eigenvalue of
$\widetilde{\mathcal{L}}_{odd}$ is odd and consequently, $q=p(\cdot-L/4)$ is
an even function that changes its sign. Again, by Krein-Rutman’s theorem we
have that the first eigenfunction $\lambda_{1}$ of $\mathcal{L}$ is simple and
it is associated to a positive (negative) even periodic function, so that $0$
can not be an eigenvalue associated to $\widetilde{\mathcal{L}}_{odd}$. Since
$z(\widetilde{\mathcal{L}})=z(\widetilde{\mathcal{L}}_{odd})+z(\widetilde{\mathcal{L}}_{even})$,
we obtain from the fact $\psi^{\prime}$ is even that
$z(\widetilde{\mathcal{L}})=z(\widetilde{\mathcal{L}}_{even})=1$. Therefore,
using the translation transformation $f=g(\cdot-L/4)$, we obtain
$z(\mathcal{L})=z(\mathcal{L}_{odd})=1$ as requested. ∎

We prove the second part of Theorem 1.1. To do so, it is necessary to prove
the following basic result.

######  Lemma 3.6.

Let $L>0$ and $k>1$ be fixed. There exists $\varphi\in H_{per}^{1}([0,L])$
solution of the following constrained minimization problem

$\nu=\inf\left\\{D(u);\ u\in H_{per}^{1}([0,L]),\
\int_{0}^{L}u(x)^{k+1}dx=1\right\\},$ (3.14)

where $D:H_{per}^{1}([0,L])\rightarrow\mathbb{R}$ is the functional defined as

$D(u)=\frac{1}{2}\int_{0}^{L}u^{\prime}(x)^{2}+u(x)^{2}dx.$ (3.15)

In addition, $\varphi$ is smooth and it satisfies equation $(\ref{ode})$.
Conversely, if $\varphi$ is a positive non-constant solution of $(\ref{ode})$,
then $\varphi$ satisfies the minimization problem $(\ref{minP1})$.

###### Proof.

Since $D$ is smooth and $D(u)\geq 0$ for all $u\in H_{per}^{1}([0,L])$, there
exists $(u_{n})_{n\in\mathbb{N}}$ a minimizing sequence associated to the
problem $(\ref{minP1})$, that is, there exists
$(u_{n})_{n\in\mathbb{N}}\subset H_{per}^{1}([0,L])$ satisfying
$\int_{0}^{L}u_{n}(x)^{k+1}dx=1$ for all $n\in\mathbb{N}$, and

$D(u_{n})\rightarrow\nu,$ (3.16)

as $n\rightarrow+\infty$. The convergence in $(\ref{conv1})$ allows us to
conclude that $(u_{n})_{n\in\mathbb{N}}$ is a bounded sequence in
$H_{per}^{1}([0,L])$. Since $H_{per}^{1}([0,L])$ is a Hilbert space, there
exists $\varphi\in H_{per}^{1}([0,L])$ such that

$u_{n}\rightharpoonup\varphi.$ (3.17)

On the other hand, the compact embedding $H_{per}^{1}([0,L])\hookrightarrow
L_{per}^{k+1}([0,L])$ gives us

$u_{n}\rightarrow\varphi\ \mbox{in}\ L_{per}^{k+1}([0,L]).$ (3.18)

Consider the function $q:\mathbb{R}\rightarrow\mathbb{R}$ given by
$q(s)=s^{k+1}$. The mean value theorem gives us
$s^{k+1}-t^{k+1}=(k+1)w^{k}(s-t)$, where $w$ is point in the open interval
$(t,s)$. Since $|w|\leq|s|+|t|$, we obtain

$|u_{n}(x)^{k+1}-\varphi(x)^{k+1}|\leq
2^{k+1}(k+1)(|u_{n}(x)|^{k}+|\varphi(x)|^{k})|u_{n}(x)-\varphi(x)|,$ (3.19)

for all $x\in[0,L]$. The Hölder inequality and $(\ref{est1})$ allow us to
conclude

$\begin{array}[]{llll}\displaystyle\left|\int_{0}^{L}u_{n}(x)^{k+1}-\varphi(x)^{k+1}dx\right|&\leq&\displaystyle\int_{0}^{L}|u_{n}(x)^{k+1}-\varphi(x)^{k+1}|dx\\\
\\\
&\leq&C_{0}\displaystyle||u_{n}-\varphi||_{L_{per}^{k+1}}(||u_{n}||_{L_{per}^{k+1}}^{k}+||\varphi||_{L_{per}^{k+1}}^{k}),\end{array}$
(3.20)

where $C_{0}=2^{k+1}(k+1)$. By the convergence in $(\ref{conv3})$, we obtain
from $(\ref{conv4})$ that $\int_{0}^{L}\varphi(x)^{k+1}dx=1$ and thus,
$\nu\leq D(\varphi)$. On the other hand, the weak lower semi-continuity of $D$
and the convergence in $(\ref{conv2})$ give us that
$D(\varphi)\leq\lim\inf_{n\rightarrow+\infty}D(u_{n})=\nu$. Thus
$D(\varphi)=\nu$ and the infimum is attained at $\varphi$.
An application of the Lagrange multiplier theorem guarantees the existence of
$C_{1}$ such that

$-\varphi^{\prime\prime}+\varphi=C_{1}\varphi^{k+1}.$ (3.21)

The function $\varphi$ is nontrivial because
$\int_{0}^{L}\varphi(x)^{k+1}dx=1$. Using a standard rescaling argument, we
can deduce that the Lagrange multiplier $C_{1}$ can be chosen as $1$, and thus
we deduce that $\varphi$ solves equation $(\ref{ode})$. In addition, a
bootstrap argument applied to the equality in $(\ref{ode})$ also gives that
$\varphi$ is smooth. Next, multiplying equation $(\ref{ode})$ by $\varphi$ and
integrating the result over $[0,L]$, we conclude that $\nu=\frac{1}{2}$.
Conversely, suppose that $\varphi$ is a positive non-constant periodic
solution of $(\ref{ode})$. According to the arguments presented in Section 2,
we can deduce that the period of $\varphi$ satisfies $L>2\pi$. Additionally,
since $\varphi>0$ and non-constant, we can assume, without loss of generality,
that $\int_{0}^{L}\varphi(x)^{k+1}dx=1$. Multiplying the equation
$(\ref{ode})$ by $\varphi$ and integrating the result over $[0,L]$, we obtain
$D(\varphi)=\frac{1}{2}$. Since every solution $\phi$ of the minimization
problem $(\ref{minP1})$ satisfies $D(\phi)=\nu=\frac{1}{2}$, we obtain that
the positive non-constant periodic solution $\varphi$ also satisfies the
minimization $(\ref{minP1})$ as requested in lemma. ∎

The proof of the second part of Theorem 1.1 is now presented.

######  Lemma 3.7.

Let $\varphi$ be a positive non-constant periodic solution associated with the
equation $(\ref{ode})$. We have that $n(\mathcal{L})=z(\mathcal{L})=1$ and in
particular $\frac{\partial L}{\partial B}>0$.

###### Proof.

Let $k>1$ be fixed. For $L>2\pi$, we have by Lemma 3.6 that the positive non-
constant solution $\varphi$ of the equation $(\ref{ode})$ satisfies the
minimization problem $(\ref{minP1})$.
Next, thanks to the Lemma 3.6, we obtain that $\varphi$ is a minimizer of the
functional $G:H_{per}^{1}([0,L])\rightarrow\mathbb{R}$ given by
$G(u)=D(u)-\frac{1}{k+1}\int_{0}^{L}u(x)^{k+1}dx$ subject to the same
constraint $\int_{0}^{L}u(x)^{k+1}dx=1$. Since
$\mathcal{L}=-\partial_{x}^{2}+1-k\varphi^{k-1}$ is the Hessian operator for
$G(u)$ at the point $\varphi$, we obtain that $n(\mathcal{L})\leq 1$. On the
other hand, solution $\varphi$ is positive and thus

$(\mathcal{L}(\varphi),\varphi)_{L_{per}^{2}}=(1-k)\int_{0}^{L}\varphi(x)^{k+1}dx<0.$

We deduce by Courant’s min-max characterization of eigenvalues that
$n(\mathcal{L})\geq 1$ and both results allow to conclude that
$n(\mathcal{L})=1$. Let us assume that $z(\mathcal{L})=2$ for some
$B_{0}\in\left(\frac{1-k}{2(k+1)},0\right)$. By Oscillation’s Theorem in [11],
it follows that the periodic solution $\bar{y}$ of the Cauchy problem
$(\ref{y})$ has exactly two zeroes in the interval $[0,L)$ and by periodicity,
also in the interval $\left[-L/2,L/2\right)$. Since
$\varphi^{\prime}\in\ker(\mathcal{L})$ is odd, we see that the periodic
function $\bar{y}\in\ker(\mathcal{L})$ is even and it has exactly two
symmetric zeroes in the interval $\left[-L/2,L/2\right)$. Hence, there exists
$x_{0}\in\left(-L/2,L/2\right)$ such that $\bar{y}(\pm x_{0})=0$. Without loss
of generality, we can still suppose that

$\bar{y}(x)>0,\;x\in(-x_{0},x_{0})\quad\text{and}\quad\bar{y}(x)<0,\;x\in\left[-L/2,-x_{0}\right)\cup\left(x_{0},L/2\right).$
(3.22)

Furthermore, by Lemma 3.4 we have
$\varphi^{k-1},\varphi^{k}\in\ker(\mathcal{L})^{\bot}={\rm
range}(\mathcal{L})$, so that

$(\bar{y},\varphi^{k-1})_{L^{2}_{per}}=0\quad\text{and}\quad(\bar{y},\varphi^{k})_{L^{2}_{per}}=0.$
(3.23)

Since $\varphi>0$, we obtain that
$\varphi(x)^{k-1}(\varphi(x)-\varphi(x_{0}))$ is positive over
$(-x_{0},x_{0})$ and negative over
$\left[-L/2,x_{0}\right)\cup\left(x_{0},L/2\right)$ and it has the same
behaviour as $\bar{y}$ in (3.22). Thus,
$(\varphi^{k-1}(\varphi-\varphi(x_{0})),\bar{y})_{L^{2}_{per}}\neq 0$ which
leads a contradiction with (3.23). Consequently, we have
$\ker(\mathcal{L})=[\varphi^{\prime}]$. ∎

## Acknowledgments

F. Natali is partially supported by CNPq/Brazil (grant 303907/2021-5) and
CAPES MathAmSud (grant 88881.520205/2020-01).

## Data availability

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## Conflict of interest

The authors declare that they have no conflict of interest.

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# Metric spaces and homotopy types

J.F. Jardine Supported by NSERC. Department of Mathematics
University of Western Ontario
London, Ontario, Canada

###### Abstract

By analogy with methods of Spivak, there is a realization functor which takes
a persistence diagram $Y$ in simplicial sets to an extended pseudo-metric
space (or ep-metric space) $\operatorname{Re}(Y)$. The functor
$\operatorname{Re}$ has a right adjoint, called the singular functor, which
takes an ep-metric space $Z$ to a persistence diagram $S(Z)$. We give an
explicit description of $\operatorname{Re}(Y)$, and show that it depends only
on the $1$-skeleton $\operatorname{sk}_{1}Y$ of $Y$. If $X$ is a totally
ordered ep-metric space, then there is an isomorphism
$\operatorname{Re}(V_{\ast}(X))\cong X$, between the realization of the
Vietoris-Rips diagram $V_{\ast}(X)$ and the ep-metric space $X$. The
persistence diagrams $V_{\ast}(X)$ and $S(X)$ are sectionwise equivalent for
all such $X$.

## Introduction

An extended pseudo-metric space, here called an ep-metric space, is a set $X$
together with a function $d:X\times X\to[0,\infty]$ such that the following
conditions hold:

  * 1)

$d(x,x)=0$,

  * 2)

$d(x,y)=d(y,x)$,

  * 3)

$d(x,z)\leq d(x,y)+d(y,z)$.

There is no condition that $d(x,y)=0$ implies $x$ and $y$ coincide — this is
where the adjective “pseudo” comes from, and the gadget is “extended” because
we are allowing an infinite distance.

A metric space is an ep-metric space for which $d(x,y)=0$ implies $x=y$, and
all distances $d(x,y)$ are finite.

The traditional objects of study in topological data analysis are finite
metric spaces $X$, and the most common analysis starts by creating a family of
simplicial complexes $V_{s}(X)$, the Vietoris-Rips complexes for $X$, which
are parameterized by a distance variable $s$.

To construct the complex $V_{s}(X)$, it is harmless at the outset is to list
the elements of $X$, or give $X$ a total ordering — one can always do this
without damaging the homotopy type. Then $V_{s}(X)$ is a simplicial complex
(and a simplicial set), with simplices given by strings

$x_{0}\leq x_{1}\leq\dots\leq x_{n}$

of elements of $X$ such that $d(x_{i},x_{j})\leq s$ for all $i,j$. If $s\leq
t$ then there is an inclusion $V_{s}X\subset V_{t}(X)$, and varying the
distance parameter $s$ gives a diagram (functor) $V_{\ast}(X):[0,\infty]\to
s\mathbf{Set}$, taking values in simplicial sets.

Following Spivak [4] (sort of), one can take an arbitrary diagram
$Y:[0,\infty]\to s\mathbf{Set}$, and produce an ep-metric space
$\operatorname{Re}(Y)$, called its realization. This realization functor has a
right adjoint $S$, called the singular functor, which takes an ep-metric space
$Z$ and produces a diagram $S(Z):[0,\infty]\to s\mathbf{Set}$ in simplicial
sets.

One needs good cocompleteness properties to construct the realization functor
$\operatorname{Re}$. Ordinary metric spaces are not well behaved in this
regard, but it is shown in the first section (Lemma 3) that the category of
ep-metric spaces has all of the colimits one could want. Then
$\operatorname{Re}(Y)$ can be constructed as a colimit of finite metric spaces
$U^{n}_{s}$, one for each simplex $\Delta^{n}\to Y_{s}$ of some section of
$Y$.

The metric space $U^{n}_{s}$ is the set $\\{0,1,\dots,n\\}$, equipped with a
metric $d$, where $d(i,j)=s$ for $i\neq j$. A morphism $U^{n}_{s}\to Z$ of ep-
metric spaces is a list $(x_{0},x_{1},\dots,x_{n})$ of elements of $Z$ such
that $d(x_{i},x_{j})\leq s$ for all $i,j$. Such lists have nothing to with
orderings on $Z$, and could have repeats.

With a bit of categorical homotopy theory, one shows (Proposition 7) that
$\operatorname{Re}(Y)$ is the set of vertices of the simplicial set
$Y_{\infty}$ (evaluation of $Y$ at $\infty$), equipped with a metric that is
imposed by the proof of Lemma 3.

One wants to know about the homotopy properties of the counit map $\eta:Y\to
S(\operatorname{Re}(Y))$, especially when $Y$ is an old friend such as the
Vietoris-Rips system $V_{\ast}(X)$. But $\operatorname{Re}(V_{\ast}(X))$ is
the original metric space $X$ (Example 13), the object $S(X)$ is the diagram
$[0,\infty]\to s\mathbf{Set}$ with $(S(X)_{t})_{n}=\hom(U^{n}_{t},X)$, and the
counit $\eta:V_{t}(X)\to S_{t}(X)$ in simplicial sets takes an $n$-simplex
$\sigma:\Delta^{n}\to V_{t}(X)$ to the list
$(\sigma(0),\sigma(1),\dots,\sigma(n))$ of its vertices.

We show in Section 3 (Theorem 16, the main result of this paper) that the map
$\eta:V_{t}(X)\to S_{t}(X)$ is a weak equivalence for all distance parameter
values $t$. The proof proceeds in two main steps, and involves technical
results from the theory of simplicial approximation. The steps are the
following:

1) We show (Lemma 14) that the map $\eta$ induces a weak equivalence
$\eta_{\ast}:BNV_{t}(X)\to BNS_{t}(X)$, where $\eta_{\ast}:NV_{t}(X)\to
NS_{t}(X)$ is the induced comparison of posets of non-degenerate simplices.
Here, $V_{t}(X)$ is a simplicial complex, so that $BNV_{t}(X)$ is a copy of
the subdivision $\operatorname{sd}(V_{t}(X))$, and is therefore weakly
equivalent to $V_{t}(X)$.

2) There is a canonical map $\pi:\operatorname{sd}S_{t}(X)\to BNS_{t}(X)$, and
the second step in the proof of Theorem 16 is to show (Lemma 15) that this map
$\pi$ is a weak equivalence.

It follows that the map $\eta$ induces a weak equivalence
$\operatorname{sd}(V_{t}(X))\to\operatorname{sd}(S_{t}(X))$, and Theorem 16 is
a consequence.

The fact that the space $S_{t}(X)$ is weakly equivalent to $V_{t}(X)$ for each
$t$ means that we have yet another system of spaces $S_{\ast}(X)$ that models
persistent homotopy invariants for a data set $X$.

One should bear in mind, however, that $S_{t}(X)$ is an infinite complex. To
see this, observe that if $x_{0}$ and $x_{1}$ are distinct points in $X$ with
$d(x_{0},x_{1})\leq t$, then all of the lists

$(x_{0},x_{1},x_{0},x_{1},\dots,x_{0},x_{1})$

define non-degenerate simplices of $S_{t}(X)$.

###### Contents

  1. 1 ep-metric spaces
  2. 2 Metric space realizations
    1. 2.1 The realization functor
    2. 2.2 Partial realizations
  3. 3 The singular functor

##  1 ep-metric spaces

An extended pseudo-metric space [2] (or an uber metric space [4]) is a set
$Y$, together with a function $d:Y\times Y\to[0,\infty]$, such that the
following conditions hold:

  * a)

$d(x,x)=0$,

  * b)

$d(x,y)=d(y,x)$,

  * c)

$d(x,z)\leq d(x,y)+d(y,z)$.

Following [3], I use the term ep-metric spaces for these objects, which will
be denoted by $(Y,d)$ in cases where clarity is required for the metric.

Every metric space $(X,d)$ is an ep-metric space, by composing the distance
function $d:X\times X\to[0,\infty)$ with the inclusion
$[0,\infty)\subset[0,\infty]$.

A morphism between ep-metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ is a function
$f:X\to Y$ such that

$d_{Y}(f(x),f(y))\leq d_{X}(x,y).$

These morphisms are sometimes said to be non-expanding [2].

I shall use the notation $ep-\mathbf{Met}$ to denote the category of ep-metric
spaces and their morphisms.

######  Example 1 (Quotient ep-metric spaces).

Suppose that $(X,d)$ is an ep-metric space and that $p:X\to Y$ is a surjective
function.

For $x,y\in Y$, set

$D(x,y)=\inf_{P}\ \sum_{i}\ d(x_{i},y_{i}),$ (1)

where each $P$ consists of pairs of points $(x_{i},y_{i})$ with $x=x_{0}$ and
$y_{k}=y$, such that $p(y_{i})=p(x_{i+1})$.

Certainly $D(x,x)=0$ and $D(x,y)=D(y,x)$. One thinks of each $P$ in the
definition of $D(x,y)$ as a “polygonal path” from $x$ to $y$. Polygonal paths
concatenate, so that $D(x,z)\leq D(x,y)+D(y,z)$, and $D$ gives the set $Y$ an
ep-metric space structure. This is the quotient ep-metric space structure on
$Y$.

If $x,y$ are elements of $X$, the pair $(x,y)$ is a polygonal path from $x$ to
$y$, so that $D(p(x),p(y))\leq d(x,y)$. It follows that the function $p$
defines a morphism $p:(X,d)\to(Y,D)$ of ep-metric spaces.

######  Example 2 (Dividing by zero).

Suppose that $(X,d)$ is an ep-metric space. There is an equivalence relation
on $X$, with $x\sim y$ if and only if $d(x,y)=0$. Write $p:X\to X/\sim\ =:Y$
for the corresponding quotient map.

Given a polygonal path $P=\\{(x_{i},y_{i})\\}$ from $x$ to $y$ in $X$ as
above, $d(y_{i},x_{i+1})=0$, so the sum corresponding to $P$ in (1) can be
rewritten as

$d(x,y_{0})+d(y_{0},x_{1})+d(x_{1},y_{1})+\dots+d(x_{k},y).$

It follows that $d(x,y)\leq D(p(x),p(y))$, whereas $D(p(x),p(y))\leq d(x,y)$
by construction.

Thus, if $D(p(x),p(y))=0$, then $d(x,y)=0$ so that $p(x)=p(y)$.

######  Lemma 3.

The category $ep-\mathbf{Met}$ of ep-metric spaces is cocomplete.

###### Proof.

The empty set is the initial object for this category,

Suppose that $(X_{i},d_{i}),i\in I$, is a list of ep-metric spaces. Form the
set theoretic disjoint union $X=\sqcup_{i}\ X_{i}$, and define a function

$d:X\times X\to[0,\infty]$

by setting $d(x,y)=d_{i}(x,y)$ if $x,y$ belong to the same summand $X_{i}$ and
$d(x,y)=\infty$ otherwise. Any collection of morphisms $f_{i}:X_{i}\to Y$ in
$ep-\mathbf{Met}$ defines a unique function $f=(f_{i}):X\to A$, and this
function is a morphism of $ep-\mathbf{Met}$ since

$d(f(x),f(y))=d(f_{i}(x),f_{j}(y))\leq\infty=d(x,y)$

if $x\in X_{i}$ and $y\in X_{j}$ with $i\neq j$.

Suppose given a pair of morphisms

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{X}$

in $ep-\mathbf{Met}$, and form the set theoretic coequalizer $\pi:X\to C$. The
function $p$ is the canonical map onto a set of equivalence classes of $X$,
which classes are defined by the relations $f(a)\sim g(a)$ for $a\in A$. We
give $C$ the quotient ep-metric space structure, as in Example 1.

Suppose that $\alpha:(X,d_{X})\to(Z,d_{Z})$ is an morphism of ep-metric spaces
such that $\alpha\cdot f=\alpha\cdot g$. Write $\alpha_{\ast}:C\to Z$ for the
unique function such that $\alpha_{\ast}\cdot p=\alpha$.

Suppose given a polygonal path $P=\\{(x_{i},y_{i})\\}$ from $x$ to $y$ in $X$.
Then $\alpha(y_{i})=\alpha(x_{i+1})$, so that

$d_{Y}(\alpha(x),\alpha(y))\leq\sum_{i}\
d_{Y}(\alpha(x_{i}),\alpha(y_{i}))\leq\sum_{i}\ d_{X}(x_{i},y_{i}).$

This is true for every polygonal path from $x$ to $y$ in $X$, so that

$d_{Y}(\alpha_{\ast}p(x),\alpha_{\ast}p(y))\leq d_{C}(p(x),p(y)).$

It follows that $\alpha_{\ast}:(C,d_{C})\to(Z,d_{Z})$ is a morphism of ep-
metric spaces. ∎

######  Example 4 (“Bad” filtered colimit).

If one starts with a diagram of metric spaces, the colimit $C$ that is
produced by Lemma 3 is an ep-metric space, and it may be that $d(x,y)=0$ in
the coequalizer $C$ for some elements $x,y$ with $x\neq y$.

In particular, suppose that
$X_{s}=\\{(\frac{1}{s\sqrt{2}},0),(0,\frac{1}{s\sqrt{2}})\\}\subset\mathbb{R}^{2}$
for $0<s<\infty$. Write $p_{s}=(\frac{1}{s\sqrt{2}},0)$ and
$q_{s}=(0,\frac{1}{s\sqrt{2}})$ in $X_{s}$. Then $d(p_{s},q_{s})=\frac{1}{s}$.
For $s\leq t$ there is an ep-metric space map $X_{s}\to X_{t}$ which is
defined by $p_{s}\mapsto p_{t}$ and $q_{s}\mapsto q_{t}$.

The filtered colimit $\varinjlim_{s}\ X_{s}$ has two distinct points, namely
$p_{\infty}$ and $q_{\infty}$, and $d(p_{\infty},q_{\infty})\leq
d(p_{s},q_{s})=\frac{1}{s}$ for all $s>0$. It follows that
$d(p_{\infty},q_{\infty})=0$, whereas $p_{\infty}\neq q_{\infty}$.

######  Lemma 5.

Suppose that $X$ is an ep-metric space. Then there is an isomorphism of ep-
metric spaces

$\psi:\varinjlim_{F}\ F\xrightarrow{\cong}X,$

where $F$ varies over the finite subsets of $X$, with their induced ep-metric
space structures.

###### Proof.

The collection of finite subsets of $X$ is filtered, and the set $X$ is a
filtered colimit of its finite subsets, so the function defining the ep-metric
space map $\psi$ is a bijection. Write $d_{\infty}$ for the metric on the
filtered colimit.

If $x,y\in X$ and $d(x,y)=s\leq\infty$ in $X$, then there is a finite subset
$F$ with $x,y\in F$ such that $d(x,y)=s$ in $F$. The list $(x,y)$ is a
polygonal path from $x$ to $y$ in $F$, so that $d_{\infty}(x,y)\leq d(x,y)$.
It follows that $d(x,y)=d_{\infty}(x,y)$, and so $\psi$ is an isomorphism. ∎

An ep-metric space $(X,d)$ has an associated system of posets
$P_{\ast}(X):[0,\infty]\to s\mathbf{Set}$, where $P_{s}(X)$ is the collection
of finite subsets $F$ of $X$ such that $d(x,y)\leq s$ for any two members
$x,y$ of $X$.

This construction defines a system of abstract simplicial complexes
$V_{\ast}(X)$, which can be constructed entirely within simplicial sets when
$X$ has a total ordering. In that case, the $n$-simplices of the simplicial
set $V_{s}(X)$ are the strings $x_{0}\leq x_{1}\leq\dots\leq x_{n}$ such that
$d(x_{i},x_{j})\leq s$. The diagram $V_{\ast}(X):[0,\infty]\to s\mathbf{Set}$
is the Vietoris-Rips system. The spaces $V_{s}(X)$ are independent up to weak
equivalence of the ordering on $X$, because there is a canonical weak
equivalence (a “last vertex map”) $\gamma:BP_{s}(X)\to V_{s}(X)$ of systems,
while the spaces $BP_{s}(X)$ are defined independently from the ordering. In
classical terms, the nerve $BP_{s}(X)$ of the poset $P_{s}(X)$ (non-degenerate
simplices of the Vietoris-Rips complex $V_{s}(X)$) is the barycentric
subdivision of $V_{s}(X)$.

######  Example 6 (Excision for path components).

Suppose that $X$ and $Y$ are finite subsets of an ep-metric space $Z$, with
the induced ep-metric space structures. Consider the inclusions of finite ep-
metric spaces

$\textstyle{X\cap
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\cup
Y}$

inside $Z$. Write $X\cup_{m}Y$ for the corresponding pushout in the category
of ep-metric spaces. The unique map

$X\cup_{m}Y\to X\cup Y$

of ep-metric spaces is the identity on the underlying point set. Write $d_{m}$
for the metric on $X\cup_{m}Y$. Then $d_{m}(x,y)$ is the minimum of sums

$\sum\ d(x_{i},x_{i+1}),$ (2)

indexed over paths

$P:\ x=x_{0},x_{1},\dots.x_{n}=y,$

such that for each $i$ the points $x_{i},x_{i+1}$ are either both in $X$ or
both in $Y$.

All sums in (2) are finite, and $d_{m}(x,y)$ is realized by a particular path
$P$ since $X$ and $Y$ are finite. Note that $d(x,y)\leq d_{m}(x,y)$, by
construction, and that $d(x,y)=d_{m}(x,y)$ if $x,y$ are both in either $X$ or
$Y$.

There are induced simplicial set maps

$V_{s}(X)\cup V_{s}(Y)\to V_{s}(X\cup_{m}Y)\to V_{s}(X\cup Y),$

all of which are the identity on vertices. There is a $1$-simplex
$\sigma=\\{x,y\\}$ of $V_{s}(X\cup_{m}Y)$ if and only if there is a path

$P:\ x=x_{0},x_{1},\dots,x_{n}=y$

consisting of $1$-simplices in either $X$ or $Y$, such that

$\sum d(x_{i},x_{i+1})\leq s.$

Then all $d(x_{i},x_{i+1})\leq s$, so that $x$ and $y$ are in the same path
component of $V_{s}(X)\cup V_{s}(Y)$. It follows that there is an induced
isomorphism

$\pi_{0}(V_{s}(X)\cup V_{s}(Y))\cong\pi_{0}V_{s}(X\cup_{m}Y).$ (3)

The isomorphisms (3) induce isomorphisms

$\pi_{0}(V_{s}(X)\cup V_{s}(Y))\cong\pi_{0}V_{s}(X\cup_{m}Y).$ (4)

for arbitrary subsets $X$ and $Y$ of an ep-metric space $Z$, by an application
of Lemma 5.

##  2 Metric space realizations

Write $U^{n}_{s}$ for the collection of axis points
$x_{i}=\frac{s}{\sqrt{2}}e_{i}$, where

$e_{i}=(0,\dots,\overset{i+1}{1},\dots,0)\in\mathbb{R}^{n+1}.$

for $0\leq i\leq n$. Observe that $d(x_{i},x_{j})=s$ in $\mathbb{R}^{n+1}$ for
$i\neq j$. Another way of looking at it: $U^{n}_{s}$ is the set
$\mathbf{n}=\\{0,1,\dots,n\\}$ with $d(i,j)=s$ for $i\neq j$.

An ep-metric space morphism $f:U^{n}_{s}\to Y$ consists of points $f(x_{i})$,
$0\leq i\leq n$, such that $d_{Y}(f(x_{i}),f(x_{j}))\leq s$ for all $i,j$.

Write $s\mathbf{Set}^{[0,\infty]}$ for the category of diagrams (functors)
$X:[0,\infty]\to s\mathbf{Set}$ and their natural transformation, which take
values in simplicial sets and are defined on the poset $[0,\infty]$. I usually
write $s\mapsto X_{s}$ for such a diagram $X$. In particular, $X_{\infty}$ is
the value that the diagram $X$ takes at the terminal object of $[0,\infty]$.

Suppose that $K$ is a simplicial set. The representable diagram $L_{s}K$
satisfies the universal property

$\hom(L_{s}K,X)\cong\hom(K,X_{s}).$

One shows that

$(L_{s}K)_{t}=\begin{cases}\emptyset&\text{if $t<s$,}\\\ K&\text{if $t\geq
s$.}\end{cases}$

The set of maps $L_{s}\Delta^{n}\to X$ can be identified with the set of
$n$-simplices of the simplicial set $X_{s}$.

A morphism $L_{t}\Delta^{m}\to L_{s}\Delta^{n}$ consists of a relation $s\leq
t$ and a simplicial map $\theta:\Delta^{m}\to\Delta^{n}$. In the presence of
such a morphism, the function $\theta:\mathbf{m}\to\mathbf{n}$ defines an ep-
metric space morphism $U^{m}_{t}\to U^{n}_{s}$, since
$s=d(\theta(i),\theta(j))\leq d(i,j)=t$.

###  2.1 The realization functor

Suppose that $X:[0,\infty]\to s\mathbf{Set}$ is a diagram. The category
$\mathbf{\Delta}/X$ of simplices of $X$ has maps $L_{s}\Delta^{n}\to X$ as
objects and commutative diagrams

---
$\textstyle{L_{t}\Delta^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\tau}$$\textstyle{L_{s}\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{X}$

as morphisms.

Equivalently, a simplex of $X$ is a simplicial set map $\Delta^{n}\to X_{s}$,
and a morphism of simplices is a diagram

$\textstyle{\Delta^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\tau}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{X_{t}}$$\textstyle{X_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(5)

Every simplex $\Delta^{n}\to X_{s}$ determines a simplex

$\Delta^{n}\to X_{s}\to X_{\infty},$

and we have a functor $r:\mathbf{\Delta}/X\to\mathbf{\Delta}/X_{\infty}$,
where $\mathbf{\Delta}/X_{\infty}$ is the simplex category of the simplicial
set $X_{\infty}$.

There is an inclusion $i:\mathbf{\Delta}/X_{\infty}\to\mathbf{\Delta}/X$, and
the composite $r\cdot i$ is the identity. The maps

$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{r(\sigma)}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{X_{\infty}}$$\textstyle{X_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(6)

define a natural transformation $h:i\cdot r\to 1$.

There is a functor

$\mathbf{\Delta}/X\to s\mathbf{Set}$ (7)

which takes a morphism (5) to the map $\theta:\Delta^{m}\to\Delta^{n}$.

The translation category $E_{X}$ for the functor (7) is a simplicial category
that has objects consisting of pairs $(\sigma,x)$ where $\sigma:\Delta^{n}\to
X_{s}$ and $x\in\Delta^{n}$ (of a fixed dimension). A morphism
$(\tau,y)\to(\sigma,x)$ of $E_{X}$ is a morphism $\theta:\tau\to\sigma$ as in
(5) such that $\theta(y)=x$. The path component simplicial set $\pi_{0}E_{X}$
of the category $E_{X}$ is isomorphic to the colimit

$\varinjlim_{L_{s}\Delta^{n}\to X}\ \Delta^{n}.$

There is a correponding translation category $E_{X_{\infty}}$ for the functor
which takes the simplex $\Delta^{n}\to X_{\infty}$ to the simplicial set
$\Delta^{n}$, and there is an induced functor $i_{\ast}:E_{X_{\infty}}\subset
E_{X}$. The functor $r:\mathbf{\Delta}/X\to\mathbf{\Delta}/X_{\infty}$ induces
a functor $r_{\ast}:E_{X}\to E_{X_{\infty}}$. The composite $r_{\ast}\cdot
i_{\ast}$ is the identity on $E_{X_{\infty}}$, and the map (6) defines a
natural transformation $i_{\ast}\cdot r_{\ast}\to 1$ of functors
$E_{X_{\infty}}\to E_{X_{\infty}}$.

The translation categories $E_{X}$ and $E_{X_{\infty}}$ are therefore homotopy
equivalent, and thus have isomorphic simplicial sets of path components. It
follows that there are isomorphisms of simplicial sets

$X_{\infty}\xleftarrow{\cong}\varinjlim_{\Delta^{n}\to X_{\infty}}\
\Delta^{n}\xrightarrow{\cong}\varinjlim_{L_{s}\Delta^{n}\to X}\ \Delta^{n}$
(8)

Suppose that $X:[0,\infty]\to s\mathbf{Set}$ is a diagram, and set

$\operatorname{Re}(X)=\varinjlim_{L_{s}\Delta^{n}\to X}\ U^{n}_{s}$

in the category of ep-metric spaces.

It follows from the identifications of (8) that $\operatorname{Re}(X)$ is the
set of vertices of $X_{\infty}$, equipped with an ep-metric space structure.

If $x$ and $y$ are two such vertices, and are the boundary of a $1$-simplex

$\Delta^{1}\to X_{s}\to X_{\infty}$

then $x$ and $y$ are in the image of a map $U^{1}_{s}\to\operatorname{Re}(X)$,
so that $d(x,y)\leq s$. If there is a sequence of $1$-simplices
$\omega_{i}:\Delta^{1}\to X_{s_{i}}$ that define a polygonal path

$P:x=x_{0}\leftrightarrows x_{1}\leftrightarrows\dots\leftrightarrows x_{k}=y$

of $1$-simplices in $X_{\infty}$, then $d(x,y)\leq\sum_{i}\ s_{i}$ by
definition. Formally, we set

$d(x,y)=\inf_{P}\ \\{\sum_{i}\ s_{i}\\}.$ (9)

provided such polygonal paths exist. Otherwise, we set $d(x,y)=\infty$.

The resulting metric $d$ is the metric which is imposed on the set of vertices
of $X_{\infty}$ by the requirement that

$\operatorname{Re}(X)=\varinjlim_{L_{s}\Delta^{n}\to X}\ U^{n}_{s}$

in the category of ep-metric spaces — see Lemma 3. We have shown the
following:

######  Proposition 7.

Suppose that $X:[0,\infty]\to s\mathbf{Set}$ is a diagram. Then the ep-metric
space $\operatorname{Re}(X)$ has underlying set given by the set of vertices
of $X_{\infty}$, with metric defined within path components by (9). Elements
$x$ and $y$ that are in distinct path components have $d(x,y)=\infty$.

######  Example 8 (Realization of Vietoris-Rips systems).

Suppose that $X$ is a finite ep-metric space, and that $X$ is totally ordered.

The realization $\operatorname{Re}(V_{\ast}(X))$ has $X$ as its underlying
set, and $V_{\infty}(X)=\Delta^{X}$ is a finite simplex, which is connected,
so that there is a finite polygonal path in $X$ between any two points $x,y\in
X$. We have a relation

$d(x,y)\leq\sum_{i}\ s_{i}$

in $X$ for any polygonal path $P$ which is defined by $1$-simplices
$\omega_{i}\in V_{s_{i}}(X)$. This means that $d(x,y)$ in $X$ coincides with
the distance between $x$ and $y$ in the metric space $\operatorname{Re}(X)$.
It follows that the identity on the set $X$ induces an isomorphism of ep-
metric spaces

$\phi:\operatorname{Re}(V_{\ast}(X))\xrightarrow{\cong}X.$

This map $\phi$ is an isomorphism of ep-metric spaces, by Lemma 5 and the
previous paragraphs.

######  Example 9 (Degree Rips systems).

Continue with a finite totally ordered ep-metric space $X$ as in Example 8,
let $k$ be a positive integer, and consider the degree Rips system
$L_{\ast,k}(X)$. We choose $k$ such that the system of complexes
$L_{\ast,k}(X)$ is non-empty, ie. such that $k\leq|X|$. Then
$L_{t,k}(X)=V_{t}(X)$ for $t$ sufficiently large, and $X$ is the underlying
set of $\operatorname{Re}(L_{\ast,k}(X))$.

The maps

$\operatorname{Re}(L_{\ast,k}(X))\to\operatorname{Re}(V_{\ast}(X))\to X$

are isomorphisms of metric spaces, by a cofinality argument.

Here is a special case:

######  Lemma 10.

Suppose that $K$ is a simplicial complex and that $s>0$. Then
$\operatorname{Re}(L_{s}K)=\operatorname{Re}(L_{s}\operatorname{sk}_{1}(K))$,
and $\operatorname{Re}(L_{s}K)$ is the set of vertices $K_{0}$ with a metric
$d$ defined by

$d(x,y)=\begin{cases}\infty&\text{if $[x]\neq[y]$ in $\pi_{0}(K)$,}\\\
\min_{P}\ s\cdot k&\text{if $[x]=[y]$.}\end{cases}$

where $P$ varies through the polygonal paths

$P:x=x_{0}\leftrightarrows x_{1}\leftrightarrows\dots\leftrightarrows x_{k}=y$

of $1$-simplices between $x$ and $y$.

###### Proof.

Write $\operatorname{Re}(K)=\operatorname{Re}(L_{s}K)$. The simplicial set $K$
is a colimit of its simplices, and so there is an isomorphism

$\varinjlim_{\Delta^{n}\to K}\ L_{s}\Delta^{n}\xrightarrow{\cong}L_{s}K.$

It follows that there is an isomorphism

$\varinjlim_{\Delta^{n}\to K}\
U^{n}_{s}\xrightarrow{\cong}\operatorname{Re}(L_{s}K).$

Suppose that $n\geq 2$. Then $\partial\Delta^{n}$ and $\Delta^{n}$ have the
same vertices, and any two vertices $x,y$ are on a common face
$\Delta^{n-1}\subset\Delta^{n}$. It follows that $d(x,y)=s$ in
$\operatorname{Re}(\partial\Delta^{n})$ and $\operatorname{Re}(\Delta^{n})$,
and the induced map

$\operatorname{Re}(\partial\Delta^{n})\to\operatorname{Re}(\Delta^{n})$

is an isomorphism for $n\geq 2$.

The displayed metric $d$ on the vertices of $K$ defines a metric space
$\operatorname{Re}(K)$, with maps
$\sigma_{\ast}:U^{n}_{s}\to\operatorname{Re}(K)$ for all simplices
$\sigma:\Delta^{n}\to K$, which maps are natural with respect to the
simplicial structure of $K$.

Any family of metric space morphisms $f_{\sigma}:U^{n}_{s}\to Y$ determines a
unique function $f:K_{0}\to Y$. Also, $d(f(x),f(y))\leq s$ if $x,y$ are in a
common simplex $\Delta^{1}\to K$. If $P$ is a polygonal path between $x$ and
$y$ as above, then $d(f(x),f(y))\leq k\cdot s$. This is true for all such
polygonal paths, so $d(f(x),f(y))\leq d(x,y)$.

If $x$ and $y$ are in distinct components of $K$, then $d(f(x),f(y))\leq
d(x,y)=\infty$. ∎

The following result says that the realization $\operatorname{Re}(X)$ of a
diagram $X:[0,\infty]\to s\mathbf{Set}$ depends only on the associated diagram
of graphs $\operatorname{sk}_{1}(X)$.

######  Lemma 11.

Suppose that $X:[0,\infty]\to s\mathbf{Set}$ is a diagram. Then the inclusion
$\operatorname{sk}_{1}X\subset X$ induces an isomorphism

$\operatorname{Re}(\operatorname{sk}_{1}X)\xrightarrow{\cong}\operatorname{Re}(X).$

###### Proof.

The diagram of $1$-skeleta $\operatorname{sk}_{1}X$ is a colimit

$\varinjlim_{L_{s}\Delta^{n}\to X}\ L_{s}\operatorname{sk}_{1}\Delta^{n},$

since the functor $\operatorname{sk}_{1}$ preserves colimits. There are
commutative diagrams

$\textstyle{\operatorname{Re}(L_{s}\operatorname{sk}_{1}\Delta^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Re}(\operatorname{sk}_{1}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Re}(L_{s}\Delta^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Re}(X)}$

that are natural in the simplices of $X$, and it follows that the induced map
$\operatorname{Re}(\operatorname{sk}_{1}X)\to\operatorname{Re}(X)$ is an
isomorphism, as required. ∎

###  2.2 Partial realizations

Suppose again that $X:[0,\infty]\to s\mathbf{Set}$ is a diagram in simplicial
sets. We construct partial realizations by writing

$\operatorname{Re}(X)_{s}=\varinjlim_{L_{t}\Delta^{n}\to X,\ t\leq s}\
U^{n}_{t}.$

This is the colimit of a functor taking values in ep-metric spaces, which is
defined on the full subcategory $\mathbf{\Delta}/X_{\leq s}$ of
$\mathbf{\Delta}/X$ having objects $L_{t}\Delta^{n}\to X$ with $t\leq s$. A
map $L_{t}\Delta^{n}\to X$ can be identified with a simplex $\Delta^{n}\to
X_{t}$, and the relation $t\leq s$ defines a simplex $\Delta^{n}\to X_{t}\to
X_{s}$, so that we have a functor

$r:\Delta/X_{\leq s}\to\mathbf{\Delta}/X_{s},$

along with an inclusion $i:\mathbf{\Delta}/X_{s}\subset\mathbf{\Delta}/X_{\leq
s}$. The composite $r\cdot i$ is the identity, and the composite $i\cdot r$ is
homotopic to the identity, just as before.

There is a functor $\mathbf{\Delta}/X_{\leq s}\to s\mathbf{Set}$ which takes a
simplex $\Delta^{n}\to X_{t}$ to the simplicial set $\Delta^{n}$. By
manipulating path components of homotopy colimits, one finds isomorphisms

$X_{s}\xleftarrow{\cong}\varinjlim_{\Delta^{n}\to X_{s}}\
\Delta^{n}\xrightarrow{\cong}\varinjlim_{L_{t}\Delta^{n}\to X,\ t\leq s}\
\Delta^{n}.$

that are analogous to the isomorphisms of (9). It follows, as in Theorem 7,
that the set underlying the metric space $\operatorname{Re}(X)_{s}$ is the set
of vertices of the simplicial set $X_{s}$.

The metric $d$ on $(X_{s})_{0}$ is defined as before: $d(x,y)=\infty$ if $x$
and $y$ not in the same path component of $X_{s}$. Otherwise

$d(x,y)=\inf_{P}\ \\{\sum t_{i}\\},$ (10)

indexed over all polygonal paths

$P:x=x_{0}\leftrightarrows x_{1}\leftrightarrows\dots\leftrightarrows x_{k}=y$

that are defined by $1$-simplices $\omega:\Delta^{1}\to X_{t_{i}}$ with
$t_{i}\leq s$.

We then have the following analogue of Proposition 7:

######  Proposition 12.

Suppose that $X:[0,\infty]\to s\mathbf{Set}$ is a functor. Then the ep-metric
space $\operatorname{Re}(X)_{s}$ has underlying set given by the set of
vertices of $X_{s}$, with metric defined within path components by (10).
Elements $x,y$ in distinct path components have $d(x,y)=\infty$.

The map $X_{s}\to X_{\infty}$ defines a map
$\operatorname{Re}(X)_{s}\to\operatorname{Re}(X)$ and
$\operatorname{Re}(X)_{\infty}=\operatorname{Re}(X)$. There is an isomorphism
of ep-metric spaces

$\varinjlim_{s}\
\operatorname{Re}(X)_{s}\xrightarrow{\cong}\operatorname{Re}(X),$ (11)

since the element $\infty$ is terminal in $[0,\infty]$ and
$\operatorname{Re}(X)_{\infty}=\operatorname{Re}(X)$.

######  Example 13 (Partial metrics for Vietoris-Rips complexes).

Suppose that $X$ is a finite totally ordered ep-metric space. Consider the
associated functor $V_{\ast}(X):[0,\infty]\to s\mathbf{Set}$.

The associated ep-metric space $\operatorname{Re}(X)_{s}$ has underlying set
$X$. We have $d(x,y)=\infty$ if $x,y$ are in distinct path components of
$V_{s}(X)$. Otherwise

$d(x,y)=\inf_{P}\ \\{\sum\ d(x_{i},x_{i+1})\\},$

indexed over all polygonal paths

$P:x=x_{0},x_{1},\dots,x_{n}=y,$

with $d(x_{i},x_{i+1})\leq s$.

If $d(x,y)=t\leq s$ in $X$ then $d(x,y)=t$ in $\operatorname{Re}(X)_{s}$.
Otherwise, the distance between $x$ and $y$ in the same path component of
$\operatorname{Re}(X)_{s}$ is more interesting — it is achieved by a
particular path $P$ since $X$ is finite, and $d(x,y)$ is a type of weighted
path length.

We see in Example 8 that there is an isomorphism of ep-metric spaces
$\phi:\operatorname{Re}(V_{\ast}(X))\xrightarrow{\cong}X$. It follows that
there is an ep-metric space map $\phi_{s}:\operatorname{Re}(X)_{s}\to X$ which
is the identity on the underlying point set $X$, and compresses distances.

##  3 The singular functor

The right adjoint $S$ of the realization functor $\operatorname{Re}$ is
defined for an ep-metric space $Y$ by

$S(Y)_{s,n}=\hom(U_{s}^{n},Y),$

where $\hom(U_{s}^{n},Y)$ is the collection of ep-metric space morphisms
$U^{n}_{s}\to Y$. Equivalently, $S(Y)_{s,n}$ is the set of families of points
$(x_{0},x_{1},\dots,x_{n})$ in $Y$ such that $d(x_{i},x_{j})\leq s$.

A simplex $(x_{0},x_{1},\dots,x_{n})$ is alternatively a function
$\mathbf{n}\to Y$ (a “bag of words”), with a distance restriction. There is no
requirement that the elements $x_{i}$ are distinct. This simplex is non-
degenerate if and only if $x_{i}\neq x_{i+1}$ for $0\leq i\leq n-1$.

Suppose that an ep-metric space $X$ is totally ordered, as in Example 8 above.
Then, in view of the discussion of Example 8, the canonical map
$\eta:V_{\ast}(X)\to S\operatorname{Re}(V_{\ast}(X))$ consists of functions
$\eta:V_{t}(X)\to S_{t}(X)$ which send simplices $\sigma:x_{0}\leq
x_{1}\leq\dots\leq x_{n}$ with $d(x_{i},x_{j})\leq t$ to the list of points
$(x_{0},x_{1},\dots,x_{n})$.

If $\sigma$ is non-degenerate, so that the vertices $x_{i}$ are distinct, then
$\eta(\sigma)$ is a non-degenerate simplex of $S_{t}(X)$.

The poset $NZ$ of non-degenerate simplices of a simplicial set $Z$ has
$\sigma\leq\tau$ if there is a subcomplex inclusion
$\langle\sigma\rangle\subset\langle\tau\rangle$, where $\langle\sigma\rangle$
is the subcomplex of $Z$ which is generated by the simplex $\sigma$.
Equivalently, $\sigma\leq\tau$ if there is an ordinal number map $\theta$ such
that $\theta^{\ast}(\tau)=\sigma$.

The map $\eta$ induces a morphism $\eta_{\ast}:NV_{t}(X)\to NS_{t}(X)$ of
posets of non-degenerate simplices.

######  Lemma 14.

Suppose that $X$ is a totally ordered ep-metric space. Then the induced
simplicial set map

$\eta_{\ast}:BNV_{t}(X)\to BNS_{t}(X)$

of associated nerves is a weak equivalence.

###### Proof.

Given a non-degenerate simplex $\sigma\in S_{t}(X)$, write $L(\sigma)$ for its
list of distinct elements.

Suppose that $\langle\tau\rangle\subset\langle\sigma\rangle$, where $\tau$ and
$\sigma$ are non-degenerate simplices of $S_{t}(X)$. Then $\tau=s\cdot
d(\sigma)$ for an (iterated) face map $d$ and degeneracy $s$. Then

$L(\tau)=L(s\cdot d(\sigma))=L(d(\sigma))\subset L(\sigma).$

It follows that the assignment $\sigma\mapsto L(\sigma)$ defines a poset
morphism

$L:NS_{t}(X)\to NV_{t}(X).$

The composite

$NV_{t}(X)\xrightarrow{\eta}NS_{t}(X)\xrightarrow{L}NV_{t}(X)$

is the identity on $NV_{t}(X)$.

Consider the composite poset morphism

$NS_{t}(X)\xrightarrow{L}NV_{t}(X)\xrightarrow{\eta}NS_{t}(X).$ (12)

Given a non-degenerate simplex $\tau=(y_{0},\dots y_{r})$ of $S_{t}(X)$, write
$L(\tau)=(s_{0},\dots s_{k})$ for the list of distinct elements of $\tau$, in
the order specified by the total order for $X$. Then the list

$V(\tau)=(y_{0},\dots,y_{r},s_{0},\dots,s_{k})$

is a simplex of $S_{t}(X)$, since each $s_{j}$ is some $y_{i_{j}}$, and there
are relations

$\langle\tau\rangle\leq\langle V(\tau)\rangle\geq\langle L(\tau)\rangle$

as subcomplexes of $S_{t}(X)$.

The simplex $V(\tau)$ has the form $V(\tau)=s(V_{\ast}(\tau))$ for a unique
iterated degeneracy $s$ and a unique non-degenerate simplex $V_{\ast}(\tau)$
(see Lemma 18), and $\langle V(\tau)\rangle=\langle V_{\ast}(\tau)\rangle$.

Suppose that $\gamma$ is non-degenerate in $S_{t}(X)$ and that
$\gamma\in\langle\tau\rangle$. Then $\gamma=d(\tau)$ for some face map $d$,
and $\gamma=(x_{0},\dots,x_{k})$ is a sublist of $\tau=(y_{0},\dots,y_{r})$.
The ordered list $L(\gamma)$ of distinct elements of $\gamma$ is a sublist of
$L(\tau)$, and $V(\gamma)$ is a sublist of $V(\tau)$. There is a diagram of
relations

$\textstyle{\langle\tau\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
V_{\ast}(\tau)\rangle}$$\textstyle{\langle
L(\tau)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle\gamma\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
V_{\ast}(\gamma)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
L(\gamma)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

It follows that the composite (12) is homotopic to the identity on the poset
$NS_{t}(X)$, and the Lemma follows. ∎

The subdivision $\operatorname{sd}(Z)$ of a simplicial set $Z$ is defined by

$\operatorname{sd}(Z)=\varinjlim_{\Delta^{n}\to Z}\ BN\Delta^{n}.$

The poset morphisms $N\Delta^{n}\to NZ$ that are induced by simplices
$\Delta^{n}\to Z$ together induce a map

$\pi:\operatorname{sd}(Z)\to BNZ.$

It is known [1] (and not difficult to prove) that the map $\pi$ is a bijection
for simplicial sets $Z$ that are polyhedral.

A polyhedral simplicial set is a subobject of the nerve of a poset. All
oriented simplicial complexes are polyhedral in this sense. Examples include
the Vietoris-Rips systems $V_{s}(X)$ associated to a totally ordered ep-metric
space $X$, since

$V_{s}(X)\subset V_{\infty}(X)=BX,$

where $BX$ is the nerve of the totally ordered poset $X$.

######  Lemma 15.

Suppose that $X$ is an ep-metric space. Then the map

$\pi:\operatorname{sd}(S_{t}(X))\to BNS_{t}(X)$

is a weak equivalence.

###### Proof.

We show that all subcomplexes $\langle\sigma\rangle$ which are generated by
non-degenerate simplices $\sigma$ of $S_{t}(X)$ are contractible. Then Lemma
4.2 of [1] implies that the map $\pi$ is a weak equivalence.

A non-degenerate simplex $\sigma$ has the form
$\sigma=(x_{0},x_{1},\dots,x_{k})$ with $x_{i}\neq x_{i+1}$. The simplices
$\tau$ of $\langle\sigma\rangle$ have the form

$\tau=\theta^{\ast}\sigma=(x_{\theta(0)},\dots,x_{\theta(k)}),$

where $\theta:\mathbf{k}\to\mathbf{n}$ is an ordinal number morphism.

For each such $\theta$, the list

$(x_{0},x_{\theta(0)},\dots,x_{\theta(k)})$

defines a simplex $\tau_{\ast}$ of $\langle\sigma\rangle$, since $\tau_{\ast}$
is a face of the simplex

$s_{0}(\sigma)=(x_{0},x_{0},\dots,x_{k}).$

The simplices $\tau_{\ast}$ define functors

$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\theta(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\theta(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{\theta(m)}}$

or homotopies, that consist of simplices of $\langle\sigma\rangle$ that patch
together to give a contracting homotopy
$\langle\sigma\rangle\times\Delta^{1}\to\langle\sigma\rangle$. ∎

######  Theorem 16.

Suppose that $X$ is a totally ordered ep-metric space. Then there is a diagram
of weak equivalences

$\textstyle{BNV_{t}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\ast}}$$\textstyle{\operatorname{sd}(V_{t}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\cong}$$\scriptstyle{\gamma}$$\scriptstyle{\eta_{\ast}}$$\textstyle{V_{t}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{BNS_{t}(X)}$$\textstyle{\operatorname{sd}(S_{t}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\gamma}$$\textstyle{S_{t}(X)}$

In particular, the map $\eta:V_{t}(X)\to S_{t}(X)$ is a weak equivalence.

This diagram is natural in $t$.

###### Proof.

The map $\eta_{\ast}:BNV_{t}(X)\to BNS_{t}(X)$ is a weak equivalence by Lemma
14. The map $\pi:\operatorname{sd}(S_{t}(X))\to BNS_{t}(X)$ is a weak
equivalence by Lemma 15. The instances of the maps $\gamma$ are weak
equivalences [1]. It follows that the maps
$\eta_{\ast}:\operatorname{sd}(V_{t}(X))\to\operatorname{sd}(S_{t}(X))$ and
$\eta:V_{t}(X)\to S_{t}(X)$ are weak equivalences. ∎

######  Remark 17.

The total ordering on the ep-metric space $X$ in Theorem 16 is intimately
involved in the definition of the Vietoris-Rips system $V_{\ast}(X)$, the
morphism

$\eta:V_{t}(X)\to S_{t}(\operatorname{Re}(V_{\ast}(X)))=S_{t}(X),$

and all induced maps $\eta_{\ast}$.

The counit $\eta:Z\to S(\operatorname{Re}(Z))$ is not a sectionwise weak
equivalence in general. One can show that $S_{\infty}(\operatorname{Re}(Z))$
is the nerve of the trivial groupoid on the vertex set of $Z_{\infty}$
(Proposition 7), and is therefore contractible, whereas the space $Z_{\infty}$
may not be contractible. For example, if $K$ is a simplicial set, then there
is an identification $K=(L_{s}K)_{\infty}$.

It may be that the map

$\eta:BP_{t}(X)\to S_{t}(\operatorname{Re}(BP_{\ast}(X)))$

is a weak equivalence for arbitrary ep-metric spaces $X$, but this has not
been proved. Such a result would give a non-oriented version of Theorem 16.

The following is a classical result, which is included here for the sake of
completeness. This result is usually neither expressed nor proved in the form
displayed here.

######  Lemma 18.

Suppose that $\sigma$ is an $n$-simplex of a simplicial set $X$. Then there is
a unique iterated degeneracy and a non-degenerate simplex $x$ such that
$\sigma=s(x)$.

An iterated degeneracy is a surjective ordinal number map
$s:\mathbf{n}\to\mathbf{k}$. Such a map induces a function $s:X_{k}\to X_{n}$
for a simplicial set $X$. Lemma 18 says that $\sigma=s(x)$ for some iterated
degeneracy $s$ and a non-degenerate simplex $x$, and that this representation
is unique.

###### Proof of Lemma 18.

Suppose that $\sigma=s(x)=s^{\prime}(x^{\prime})$ where $s,s^{\prime}$ are
iterated degeneracies and $x,x^{\prime}$ are non-degenerate.

The $x=d(\sigma)$ for some face map $d$ such that $d\cdot s=1$, and so
$d(s^{\prime}(x^{\prime}))=s^{\prime\prime}(d^{\prime\prime}(x))$ for some
iterated degeneracy $s^{\prime\prime}$ and face map $d^{\prime\prime}$. But
$x$ is non-degenerate, so that $s^{\prime\prime}=1$ and
$x=d^{\prime\prime}(x^{\prime})$. Similarly, $x^{\prime}=\tilde{d}(x)$ for
some face map $\tilde{d}$. But then $x$ and $x^{\prime}$ have the same
dimension, and $d=1$, so that $x=x^{\prime}$.

If $s\neq s^{\prime}$ there is a face map $d$ such that $d\cdot s=1$ but
$d\cdot s^{\prime}\neq 1$. Then $\sigma=s(x)=s^{\prime}(x)$ for $s\neq
s^{\prime}$ and $x$ non-degenerate, then

$d(\sigma)=x=d(s^{\prime}(x))=s^{\prime\prime}(d^{\prime\prime}(x))$

for some degeneracy $s^{\prime\prime}$ and face map $d^{\prime\prime}$, at
least one of which is non-trivial.

But $x$ is non-degenerate, so that $s^{\prime\prime}=1$ and
$x=d^{\prime\prime}(x)$ only if $d^{\prime\prime}=1$. This contradicts the
assumption that $s\neq s^{\prime}$. ∎

## References

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