@@ 404,7 404,7 @@ With these definitions in place a reform
 whether $f$ and $g$ are isomorphic as objects of the
 category of $\overline{\R}$-spaces.
 
-### Further Prior Art
+### Further References
 
 Join trees have been harnessed for a similar purpose by @saikia14a.
 Further methods to compare Reeb graphs have been proposed by

          
@@ 414,3 414,7 @@ The two distances from [@bauer2014] and 
 equivalent in some sense by @bauer2015.
 Moreover the work presented here
 borrows from [@bubenik2014] on several aspects.
+There is much more prior art and the papers we mentioned merely represent
+our primary references.
+Most of the papers mentioned by @deSilva2016 and @bubenik2014
+in their introductions are antecedents to our work as well.

          

@@ 1,4 1,4 @@ 
-# Positive Persistence Enhancements
+# Positive Persistence-Enhancements
 
 In the previous section we defined strict $D$-categories,
 interleavings of objects in $D$-categories,

          
@@ 7,7 7,7 @@ are an instance of this.
 Then we derived some properties about the interleaving distances,
 such as the triangle inequality and this compensates
 for the triangle inequality having been left out in the section before.
-However we did not prove that the interleaving distances of
+However we did not prove, that the interleaving distances of
 Reeb precosheaves provide lower bounds to the corresponding
 distances of spaces.
 And it is the aim of this section to make up for this deficit.

          
@@ 95,7 95,7 @@ The first equation follows in conjunctio
 see [the section on properties of interleavings](#properties-of-interleavings),
 and the second equation follows in conjunction with corollary @abInterlDist.
 
-Now we have an embedding of the category of $\R$-spaces into $\mathbf{F}$
+Now we have an embedding of the category of $\R$-spaces into $\mathbf{F}$,
 that preserves the distances.
 In example @DCatPrecosh from the
 [previous section](#interleavings-in-d-categories)

          
@@ 110,7 110,7 @@ to the category of set-valued precosheav
 such that $\mathcal{C} = \tilde{\mathcal{C}}((\_, \mathbf{o}))$.
 Applying the same procedure to an arbitrary functor $F$
 on the category of $\R$-spaces
-is what we name a *positive persistence enhancement of $F$*.
+is what we name a *positive persistence-enhancement of $F$*.
 Now let $F$ be a functor from the category of $\R$-spaces
 to some category $\mathbf{C}$
 

          

@@ 1,9 1,10 @@ 
-# Complete Persistence Enhancements
+# Complete Persistence-Enhancements
 
 In the previous section we defined positive persistence enhancements
-of functor on $\R$-spaces and provided one for $\mathcal{C}$,
+of functors on $\R$-spaces and provided one for $\mathcal{C}$,
 thereby finally establishing that the interleaving distances of
-the Reeb precosheaves provide lower bounds to the distances on functions.
+the Reeb precosheaves provide lower bounds
+to the correspoding distances on functions.
 The next aim is to connect this to the interleaving distances of
 [join-trees](#join-trees) and to proof theorem @interEq.
 Unfortunately there is a bit of a problem with our use of infinity.

          
@@ 36,8 37,8 @@ Now $\Ec$ contains $D$ as a submonoid.
 Moreover $-D$ is a commutative submonoid of $\Ec$ as well
 and negation yields an order-reversing monoid isomorphism from
 $D$ to $-D$.
-Similarly to our definition of $D$-categories we now define
-$-D$ and $\Ec$-categories.
+Similarly to $D$-categories we now define
+$-D$- and $\Ec$-categories.
 
 * **Definition.**
 A *strict $\Ec$-category* respectively *$-D$-category*

          
@@ 94,8 95,7 @@ of $A$ and $B$.
 (@dualDCat) *Remark.*
 By passing from $\mathbf{C}$ to $\mathbf{C}^{\op}$
 we can turn any $-D$-category into a $D$-category.
-In this sense each result from the section on
-[properties of interleavings](#properties-of-interleavings)
+In this sense each result from the previous section
 has an analogous statement for $-D$-categories.
 
 Now let us assume $\mathbf{C}$ is an $\Ec$-category

          

@@ 162,14 162,15 @@ associated to any $\mathbf{a} \in -D$
 and second we need the natural transformations
 $\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})$ associated to any
 $\mathbf{a}, \mathbf{b} \in D$ with $\mathbf{a} \preceq \mathbf{b}$.
-Now in order to get there we will do something seemingly unnecessary.
+Now in order to get there, we will do something seemingly unnecessary.
 We will define the endofunctors $\mathcal{S}'(\mathbf{a})$
 not just on $\mathbf{D}$ but on the whole category of
 set-valued precosheaves on $\overline{D}$.
-Then we will show $\mathbf{C}$ is invariant under these
-endofunctors and sothat these endofunctors are also
-endofunctors for $\mathbf{C}$
-and then we define the natural transformations.
+Then we will show, that $\mathbf{C}$ is invariant under these
+endofunctors.
+In this sense the endofunctors $\mathcal{S}'(\mathbf{a})$
+restrict to endofunctors on $\mathbf{C}$.
+Then we define the natural transformations.
 
 So let us start with the endofunctors setting
 $\mathcal{S}'((a, b)) := S^{(-b, -b)}_*$ for all $(a, b) \in -D$.

          
@@ 214,8 215,6 @@ for all $\mathbf{a} \in -D$.
 This follows from the commutativity of the right square in the above
 diagram.
 
-Now we consider the functor $\pi^2_p$.
-
 * **Lemma.**
 The functor $\pi^2_p$ commutes with
 $\overline{\mathcal{S}}(\mathbf{a})$ and $\mathcal{S}'(\mathbf{a})$

          
@@ 238,8 237,8 @@ The previous three lemmata have the foll
 The subcategory $\mathbf{C}$ is invariant under
 $\mathcal{S}'(\mathbf{a})$ for all $\mathbf{a} \in -D$.
 
-So now we have the endomorphisms for $\mathcal{S}'$ as a smoothing
-functor on $\mathbf{C}$.
+So far we defined the endofunctors for the smoothing functor
+$\mathcal{S}'$ on $\mathbf{C}$.
 The next step is to define the natural transformations
 $\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})$ for any
 $\mathbf{a}, \mathbf{b} \in D$ with $\mathbf{a} \preceq \mathbf{b}$.

          
@@ 292,12 291,12 @@ we gave the category of set-valued preco
 $\overline{\R}_{-\infty}$
 the structure of an $\Ec$-category.
 So by lemma @pmDEquiv, from the section on
-[complete persistence enhancements][],
+[complete persistence-enhancements][],
 the interleavings of
-$\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$
+$\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$,
 with respect to the $-D$-category structure given by
-$\overline{\mathcal{S}}$ are in canonical bijection with those
-with respect to the $D$-category structure given by $\overline{\mathcal{S}}$.
+$\overline{\mathcal{S}}$, are in canonical bijection with those
+given by the $\overline{\mathcal{S}}$-induced structure of a $D$-category.
 So in conjunction with the last corollary from the previous subsection
 we have the following
 

          

@@ 1,6 1,6 @@ 
 # A negative Enhancement of Join Trees
 
-To provide a negative persistence enhancement
+To provide a negative persistence-enhancement
 for $\mathcal{R} \circ \mathcal{E}$
 we first provide one for $\mathcal{E}$.
 To this end let $\mathbf{D}$ be the [full subcategory][] of

          
@@ 21,21 21,27 @@ then we set
 And for any homomorphism $\varphi$
 in the category of $(-\infty, \infty]$-spaces we set
 $\mathcal{S}((a, b))(\varphi) := \varphi$.
-And this defines the endofunctors
+This defines the endofunctors
 $\{\mathcal{S}(\mathbf{a}) ~|~ \mathbf{a} \in -D\}$ on all
 $(-\infty, \infty]$-spaces.
+
 Now let $f \colon X \rightarrow \R$
 be an $\R$-space,
 let $r \in \R$, and let $(a, b), (c, d) \in -D$
 with $(a, b) \preceq (c, d)$.
 Then we set
-$\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f) \colon
- \epi f \rightarrow \epi f,
- (p, t) \mapsto (p, t - b + d)$.
+$$
+\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f) \colon
+\epi f \rightarrow \epi f,
+(p, t) \mapsto (p, t - b + d) .
+$$
 Moreover we set
-$\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{R} \mathcal{E} f) :=
- \mathcal{R}(\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f))$.
+$$
+\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{R} \mathcal{E} f) :=
+\mathcal{R}(\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f)) .
+$$
 And this concludes our definition of $\mathcal{S}$.
+
 Now for $(a, b) \in D^{\perp}$,
 the $(a, b)$-interleavings of two join trees are precisely the
 $(a, b)$-interleavings with respect to $\mathcal{S}$.

          
@@ 55,7 61,7 @@ similarly for $(g, (c, d))$ of course,
 and
 $\tilde{\mathcal{E}}(\varphi) \colon \epi f \rightarrow \epi g,
  (p, t) \mapsto (\varphi(p), t - b + d)$.
-And this defines the negative persistence-enhancement.
+This defines our negative persistence-enhancement of $\mathcal{E}$.
 (Again a bit of a cheat.)
 
 We note that we have the following

          

@@ 17,6 17,7 @@ To proof the previous lemma it suffices 
 $\mathcal{C} (r + \mathcal{E} f)$ and
 $\mathcal{C} (r + \mathcal{R} \mathcal{E} f)$
 are an object of $\mathbf{C}$ for all $r \in (-\infty, \infty]$.
+
 We consider the case $r = 0$ first.
 By lemma @etaCEIso the precosheaf $\mathcal{C} \mathcal{E} f$
 is an object of $\mathbf{C}$.

          
@@ 34,13 35,14 @@ is an isomorphism of precosheaves.
 
 Now $\mathbf{C}$ is closed under isomorphisms and thus also
 $\mathcal{C} \mathcal{R} \mathcal{E} f$ lies in $\mathbf{C}$.
-Now we consider the case of $r$ not necessarily being $0$.
+
+We continue with the case of $r$ not necessarily being $0$.
 To this end let $g \colon Y \rightarrow (-\infty, \infty]$
 be a continuous function,
 possibly an object of $\mathbf{D}$.
 We note that for any
 $r \in (-\infty, \infty]$ we have
-$r + g = \mathcal{S}((\infty, -r)) (g)$
+$r + g = \mathcal{S}((\infty, -r)) (g)$,
 by definition of $\mathcal{S}$.
 We now recall that we also defined the endofunctors
 associated to the smoothing functor

          
@@ 76,27 78,37 @@ We have
 First we set $(a', a) := \mathbf{a}$ and $(b', b) := \mathbf{b}$.
 Now let $r \in \R$.
 If we unravel the definitions we obtain
-$(\overline{\mathcal{S}}(\mathbf{a}) \pi^2_* \mathcal{C} \mathcal{E} f)(
-  [-\infty, r)
- ) =
- \Lambda(\epi f \cap X \times [-\infty, r + a))$
+$$
+(\overline{\mathcal{S}}(\mathbf{a}) \pi^2_* \mathcal{C} \mathcal{E} f)(
+ [-\infty, r)
+) =
+\Lambda(\epi f \cap X \times [-\infty, r + a))
+$$
 and
-$(\overline{\mathcal{S}}(\mathbf{b}) \pi^2_* \mathcal{C} \mathcal{E} f)(
-  [-\infty, r)
- ) =
- \Lambda(\epi f \cap X \times [-\infty, r + b))$.
+$$
+(\overline{\mathcal{S}}(\mathbf{b}) \pi^2_* \mathcal{C} \mathcal{E} f)(
+ [-\infty, r)
+) =
+\Lambda(\epi f \cap X \times [-\infty, r + b)) .
+$$
 Let $i$ be the inclusion of $\epi f \cap X \times [-\infty, r + a)$
 into $\epi f \cap X \times [-\infty, r + b)$
 and let
-$\tau \colon \epi f \cap X \times [-\infty, r + a) \rightarrow
- \epi f \cap X \times [-\infty, r + b),
- (p, t) \mapsto (p, t - a + b)$,
+$$
+\tau \colon \epi f \cap X \times [-\infty, r + a) \rightarrow
+\epi f \cap X \times [-\infty, r + b),
+(p, t) \mapsto (p, t - a + b) ,
+$$
 then
-$(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ
-  \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(i)$
+$$
+(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ
+ \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(i)
+$$
 and
-$(\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b})
-  \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(\tau)$.
+$$
+(\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b})
+ \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(\tau) .
+$$
 Now let $(p, t) \in \epi f \cap X \times [-\infty, r + a)$,
 then $\{p\} \times [t, t - a + b]$ is contained in
 $\epi f \cap X \times [-\infty, r + b)$.