@@ 1,5 1,3 @@ 
-# Equality of Interleaving Distances
-
 Finally we get to proving theorem @interEq.
 We reuse the notation and the definitions from the previous two subsections.
 We aim to show that the Reeb precosheaf $\mathcal{C}$

          

@@ 1,5 1,3 @@ 
-# Interleavings in *D*-Categories
-
 Up to this point we have seen two notions of an interleaving,
 the first for join trees and the second for precosheaves.
 In order to show theorem @interEq we will use several more

          

@@ 1,5 1,3 @@ 
-## Graphs over the Reals
-
 * **Definition.**
 Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
 non-negative integer $n$.

          

@@ 1,5 1,3 @@ 
-# Reeb Graphs
-
 In this section we introduce Reeb graphs.
 
 * **Definition** (Reeb Graph)**.**

          

@@ 1,5 1,3 @@ 
-## A Skeleton for the Epigraph {#skeleton-epigraph}
-
 Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
 non-negative integer $n$
 and let $X$ be an $S$-skeleton for a bounded $\R$-space.

          

@@ 1,5 1,3 @@ 
-# Complete Persistence-Enhancements
-
 In the previous section we defined positive persistence enhancements
 of functors on $\R$-spaces and provided one for $\mathcal{C}$,
 thereby finally establishing that the interleaving distances of

          

@@ 1,5 1,3 @@ 
-## Constructible Spaces over the Reals {#constructible-spaces}
-
 * **Definition.**
 Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
 non-negative integer $n$

          

@@ 1,5 1,3 @@ 
-# Interleaving Reeb Graphs
-
 In this section we define the interleaving distance of Reeb graphs
 due to @deSilva2016.
 Strictly speaking it is somewhat misleading to name this the interleaving

          

@@ 1,5 1,3 @@ 
-# Introduction
-
 The topic of the present text is the interleaving distance
 of join trees by @morozov2013.
 Just like this paper we focus on topological data analysis (TDA),

          

@@ 1,5 1,3 @@ 
-# Join Trees
-
 In this section we define join trees and their interleaving distance
 due to @morozov2013.
 We start with an auxiliary

          

@@ 1,5 1,3 @@ 
-# Monoidal Posets for 1D-Interleavings
-
 Before we defined [interleavings of join trees](#join-trees) we introduced the
 poset $D^{\perp}$ and the two weightings
 $\epsilon'$ and $\epsilon''$ on $D^{\perp}$.

          

@@ 1,5 1,3 @@ 
-# A negative Enhancement of Join Trees
-
 To provide a negative persistence-enhancement
 for $\mathcal{R} \circ \mathcal{E}$
 we first provide one for $\mathcal{E}$.

          

@@ 1,5 1,3 @@ 
-# Positive Persistence-Enhancements
-
 In the previous section we defined strict $D$-categories,
 interleavings of objects in $D$-categories,
 and showed that interleavings of precosheaves

          

@@ 1,5 1,3 @@ 
-# Interlude on Precosheaves {#precosheaves}
-
 In this section we develop the theory of precosheaves to the extend needed
 for the interleaving distance of Reeb graphs by @deSilva2016
 and subsequent sections.

          

@@ 1,5 1,3 @@ 
-# Some Equivalences
-
 In this section we move one step closer to proving theorem @interEq.
 We consider interleavings of precosheaves in the image
 of the functor $\mathcal{C} \mathcal{E}$ and transform those into