While it may be quite hard to compute the two distances \(M\) and \(\mu\) defined above we may try to compute lower bounds. Topological data analysis is a vast field and several solutions exist in this direction. Here we focus on two of them, namely the interleaving distance of join trees introduced by Morozov, Beketayev, and Weber (2013) and the interleaving distance of Reeb graphs by de Silva, Munch, and Patel (2016).

Join trees and Reeb graphs themselves are very similar. The join
tree associated to a function \(f\) can be seen as a real-valued
function itself whose fibers encode the connected components of
the corresponding sublevel sets of \(f\) and the Reeb graph can be
seen as a function whose fibers encode the connected components of
the level sets. Join trees are much easier to work with however.
While Morozov, Beketayev, and Weber
(2013) defined their
interleaving distance in an ad hoc manner, de Silva, Munch, and
Patel (2016) introduce
set-valued (pre)cosheaves first and then they define their
interleaving distance. (Actually de Silva, Munch, and Patel
(2016) introduce a second
interleaving distance, that could be seen as a more ad hoc
approach to the problem, but in the author’s opinion^{[1]} their first notion of an interleaving distance using
the theory of precosheaves is more convenient for our
considerations.)

Here is the main motivation behind this document. For a continuous
function \(f\) Morozov, Beketayev, and Weber
(2013) define the join tree
of \(f\) as the
Reeb
graph of the
epigraph
associated to \(f\). Later de Silva, Munch, and Patel
(2016) introduced the
*interleaving distance of Reeb graphs* and
therefore there are now two different notions of an interleaving
distance on join trees. For two continuous functions
\(f \colon X \rightarrow {\mathbb{R}}\) and
\(g \colon Y \rightarrow {\mathbb{R}}\) we have the interleaving
distance of join trees associated to \(f\) and \(g\) as defined by
Morozov, Beketayev, and Weber
(2013) and we have the
interleaving distance of the Reeb graphs associated to the
epigraph of \(f\) respectively \(g\) as defined by de Silva,
Munch, and Patel (2016). We
aim to show that the two distances are the same when \(X\) and
\(Y\) are compact smooth manifolds.

It is not really essential and more of a personal preference that from this point onward we work with functions with values in the extended real line \(\overline{{\mathbb{R}}} := [-\infty, \infty]\) and consider real-valued functions a subclass.

When working with different notions of an interleaving distance
and in particular when comparing them, the use of some basic
category
theory seems very natural and simplifies several of our
arguments. More specifically we will use
functors
and
natural
transformations on several occasions. Moreover we augment
the class of \(\overline{{\mathbb{R}}}\)-valued continuous
functions with the structure of a category, the *category
of \(\overline{{\mathbb{R}}}\)-spaces* for short.

**Definition.**For two continuous functions \(f \colon X \rightarrow \overline{{\mathbb{R}}}\) and \(g \colon Y \rightarrow \overline{{\mathbb{R}}}\), a*homomorphism \(\varphi\) from \(f\) to \(g\)*, also denoted by \(\varphi \colon f \rightarrow g\), is a continuous map \(\varphi \colon X \rightarrow Y\) such that the diagram \[ \xymatrix{ X \ar@/^/[rr]^{\varphi} \ar[dr]_{f} & & Y \ar[dl]^{g} \\ & \overline{{\mathbb{R}}} } \] commutes.We define the composition of homomorphisms in the

*category of \(\overline{{\mathbb{R}}}\)-spaces*by the composition of maps.The

*category of \({\mathbb{R}}\)-spaces*is the full subcategory of all real-valued continuous functions.*Remark.*With these definitions in place a reformulation of question 1 is, whether \(f\) and \(g\) are isomorphic as objects of the category of \(\overline{{\mathbb{R}}}\)-spaces.