Relation to the Join Precosheaf

In the section Equivalence to descending Precosheaf we discussed the \(\overline{{\mathbb{E}}}\)-category of precosheaves on \(\overline{{\mathbb{R}}}_{-\infty}\) 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 Bubenik, de Silva, and Scott (2014 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 15 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 \(f \colon X \rightarrow {\mathbb{R}}\) and \(g \colon Y \rightarrow {\mathbb{R}}\) 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 \(U \subseteq \overline{{\mathbb{R}}}_{-\infty}\) 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 19.

  • 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 17 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 26.