# HG changeset patch
# User Benedikt Fluhr <http://bfluhr.com>
# Date 1503318248 -7200
#      Mon Aug 21 14:24:08 2017 +0200
# Node ID d8d0d68546ae37de5189622cff1ba10c6397f326
# Parent  343ef1eebcbee2293ae7872fdce0643ae1fb2882
Discussion of the Join Precosheaf

diff --git a/EqualityOfInterlDist.md b/EqualityOfInterlDist.md
--- a/EqualityOfInterlDist.md
+++ b/EqualityOfInterlDist.md
@@ -65,7 +65,7 @@
 To this end let $\mathbf{a}, \mathbf{b} \in -D$ with
 $\mathbf{a} \preceq \mathbf{b}$.
 
-* **Lemma.**
+(@shiftDescJoinTree) **Lemma.**
 We have
 $(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ
   \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_f =
diff --git a/complPersistenceEnhancements.md b/complPersistenceEnhancements.md
--- a/complPersistenceEnhancements.md
+++ b/complPersistenceEnhancements.md
@@ -49,7 +49,7 @@
 
 * *Remark.*
 If $\mathbf{C}$ is a strict $-D$-category with smoothing functor $\mathcal{S}$,
-then the opposite category $\mathcal{C}^{\op}$ is a strict
+then the opposite category $\mathbf{C}^{\op}$ is a strict
 $D$-category with smoothing functor $\mathcal{S}(- (\_))$.
 
 Now we define interleavings in $-D$-categories.
diff --git a/links.md b/links.md
--- a/links.md
+++ b/links.md
@@ -26,3 +26,4 @@
 [Godement product]: https://ncatlab.org/nlab/show/Godement+product
 [strict 2-category]: https://ncatlab.org/nlab/show/strict+2-category
 [galois]: https://en.wikipedia.org/wiki/Galois_connection#.28Monotone.29_Galois_connection
+[category of arrows]: https://unapologetic.wordpress.com/2007/05/23/arrow-categories/
diff --git a/posPersistenceEnhancements.md b/posPersistenceEnhancements.md
--- a/posPersistenceEnhancements.md
+++ b/posPersistenceEnhancements.md
@@ -108,12 +108,12 @@
 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 for $F$*.
 Now let $F$ be a functor from the category of $\R$-spaces
 to some category $\mathbf{C}$
 
 * **Definition** (Positive Persistence-Enhancement)**.**
-A *postive persistence-enhancement of $F$*
+A *postive persistence-enhancement for $F$*
 is the structure of a strict $D$-category on
 $\mathbf{C}$ together with a $1$-homomorphism $\tilde{F}$ from
 $\mathbf{F}$ to $\mathbf{C}$ such that $F = \tilde{F}((\_, \mathbf{o}))$.
@@ -131,7 +131,7 @@
 and the previous corollary.
 
 * *Example.*
-To conclude this section we provide a positive persistence-enhancements for
+As a first example we provide a positive persistence-enhancements for
 the Reeb precosheaf $\mathcal{C}$.
 In the previous section we already defined the structure of a
 strict $D$-category on the category of precosheaves.
@@ -141,7 +141,7 @@
 $\Delta^{\mathbf{a}} \colon \R \rightarrow \Ec,
  t \mapsto (t, t) - \mathbf{a}$ 
 and
-$\overline{C}((f, \mathbf{a})) :=
+$\overline{\mathcal{C}}((f, \mathbf{a})) :=
  (\Delta^{\mathbf{a}} \circ f)_* \Lambda$ for any $\mathbf{a} \in D$.
 Now let
 $\varphi \colon (f, \mathbf{a}) \rightarrow (g, \mathbf{b})$
@@ -229,3 +229,32 @@
     $\R$-space and $r \in \R$
     we set
 -->
+
+Now let $F$ and $G$ be functors from the category of $\R$-spaces
+to some $D$-category $\mathbf{C}$ with smoothing functor $\mathcal{S}$
+and let $\eta \colon F \rightarrow G$
+be a natural transformation.
+We consider $\eta$ a functor from the category of $\R$-spaces
+to the [category of arrows][] in $\mathbf{C}$.
+Moreover we consider the [category of arrows][] in $\mathbf{C}$
+a $D$-category with smoothing functor $\mathcal{S}$.
+(We just apply the smoothing functor to the homomorphisms.)
+Then a (positive) persistence-enhancement of
+$\eta$ with smoothing functor $\mathcal{S}$
+is already determined by the corresponding enhancements for $F$ and $G$.
+Now suppose $\tilde{F}$ and $\tilde{G}$ are arbitrary persistence-enhancements
+for $F$ and $G$ both with smoothing functor $\mathcal{S}$.
+
+* **Definition.**
+We say *$\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$*
+if the map
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a natural transformation from $\tilde{F}$ to $\tilde{G}$.
+
+(@combine2Hom) *Remark.*
+If $\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$,
+then
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a $2$-homomorphism from $\tilde{F}$ to $\tilde{G}$.
diff --git a/someEquivalences.md b/someEquivalences.md
--- a/someEquivalences.md
+++ b/someEquivalences.md
@@ -142,7 +142,7 @@
 
 Now let $g \colon Y \rightarrow \R$ be another continuous function.
 
-* **Corollary.**
+(@etaCEIsoCor) **Corollary.**
 The interleavings of
 $\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$
 are in bijection to those of
@@ -295,7 +295,7 @@
 with respect to the $-D$-category structure given by
 $\overline{\mathcal{S}}$, 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
+So in conjunction with corollary @etaCEIsoCor from the previous subsection
 we have the following
 
 (@bijD) **Propostion.**
diff --git a/struct.yaml b/struct.yaml
--- a/struct.yaml
+++ b/struct.yaml
@@ -41,6 +41,10 @@
 - title: Equality of Interleaving Distances
   files:
   - EqualityOfInterlDist.md
+- title: Relation to the Join Precosheaf
+  name: join-precosheaf
+  files:
+  - joinPrecosheaf.md
 - title: Appendix
   sections:
   - title: Constructible Spaces over the Reals