# HG changeset patch
# User Benedikt Fluhr <http://bfluhr.com>
# Date 1503318826 -7200
#      Mon Aug 21 14:33:46 2017 +0200
# Node ID c94ee4e0161216e1bfde11d4f70c73badfa37056
# Parent  d8d0d68546ae37de5189622cff1ba10c6397f326
Added File forgotten with last Commit

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+In the section
+[Equivalence to descending Precosheaf][]
+we discussed the $\overline{\E}$-category of
+precosheaves on $\overline{\R}_{-\infty}$
+with smoothing functor $\overline{\mathcal{S}}$
+and the $1$-homomorphism $\pi^2_*$ from
+the $D$-category of precosheaves on $\overline{D}$.
+Together with $\mathcal{C}$ and $\tilde{\mathcal{C}}$ we obtain
+the functor $\pi^2_* \circ \mathcal{C}$ with the positive
+persistence-enhancement $\pi^2_* \circ \tilde{\mathcal{C}}$.
+And this is pretty much the precosheaf-theoretical version of the join tree
+introduced by @bubenik2014[example 1.2.3].
+We name $\pi^2_* \circ \mathcal{C}$ the *join precosheaf*.
+With $\pi^2_*$ being a $1$-homomorphism we conclude
+from corollary @1HomLowerBd that the interleaving distances
+of join precosheaves provide lower bounds to the corresponding distances
+of Reeb precosheaves.
+Now we will see that this join precosheaf is indeed closely
+related to the other versions of join trees we have seen.
+We start with a self-contained description of a complete
+persistence-enhancement for $\pi^2_* \circ \mathcal{C}$
+extending $\pi^2_* \circ \tilde{\mathcal{C}}$.
+To this end let $f \colon X \rightarrow \R$
+and $g \colon Y \rightarrow \R$ be continuous functions
+and let $(a, b), (c, d) \in \overline{\mathbb{E}}$.
+We set
+$\widetilde{(\pi^2_* \circ \mathcal{C})}((f, (a, b))) :=
+(s^{-b} \circ \pi^2 \circ \Delta \circ f)_* \Lambda$.
+Now let 
+$\varphi \colon (f, (a, b)) \rightarrow (g, (c, d))$
+be a homomorphism in $\pm \mathbf{F}$
+and let $U \subseteq \overline{\R}_{-\infty}$ be an open subset,
+then we have
+$\varphi((s^{-b} \circ \pi^2 \circ \Delta \circ f)^{-1} (U)) =
+\varphi((s^{-b} \circ f)^{-1} (U)) \subseteq
+\varphi((s^{-d} \circ f)^{-1} (U))$
+and thus the restriction
+$\varphi |_{(s^{-b} \circ f)^{-1} (U)} \colon
+(s^{-b} \circ f)^{-1} (U) \rightarrow (s^{-d} \circ f)^{-1} (U)$.
+We set
+$\widetilde{(\pi^2_* \circ \mathcal{C})}(\varphi)_U :=
+\Lambda \big( \varphi |_{(s^{-b} \circ f)^{-1} (U)} \big)$.
+
+* **Lemma.**
+We have
+$\widetilde{(\pi^2_* \circ \mathcal{C})} |_{\mathbf{F}} =
+\pi^2_* \circ \tilde{C}$.
+
+So the positive persistence-enhancement we get from
+$\widetilde{(\pi^2_* \circ \mathcal{C})}$ coincides with
+$\pi^2_* \circ \tilde{C}$.
+Now we examine the negative persistence-enhancement
+$\widetilde{(\pi^2_* \circ \mathcal{C})} |_{(-\mathbf{F})}$
+and how it relates to
+$\pi^2_* \circ \mathcal{C} \circ \tilde{\mathcal{E}}$.
+
+* **Lemma.**
+The homomorphism
+$(\pi^2_* \circ \mathcal{C} \circ \kappa)_f$
+is an isomorphism from
+$(\pi^2_* \circ \mathcal{C})(f)$ to
+$(\pi^2_* \circ \mathcal{C} \circ \mathcal{E})(f)$.
+
+The proof of this lemma is similar to the proof of lemma @etaCEIso.
+
+* **Corollary.**
+The interleavings of
+$(\pi^2_* \circ \mathcal{C})(f)$ and
+$(\pi^2_* \circ \mathcal{C})(g)$
+are in bijection to those of
+$(\pi^2_* \circ \mathcal{C} \circ \mathcal{E})(f)$ and
+$(\pi^2_* \circ \mathcal{C} \circ \mathcal{E})(g)$.
+
+This corollary already captures what we care about
+most of the time, but we might also like to know
+whether the interleavings that we get from interleavings in
+$-\mathbf{F}$ are preserved by this bijection.
+An affirmative answer is provided by remark @combine2Hom and the following
+
+* **Lemma.**
+The functors
+$\overline{\mathcal{S}}$,
+$\widetilde{(\pi^2_* \circ \mathcal{C})} |_{(-\mathbf{F})}$,
+and
+$\pi^2_* \circ \mathcal{C} \circ \tilde{\mathcal{E}}$
+combine to a negative persistence-enhancement for
+$\pi^2_* \circ \mathcal{C} \circ \kappa$.
+
+The proof of this lemma is similar to the proof of lemma @shiftDescJoinTree.