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 \(\overline{{\mathbb{R}}}\)-spaces of the form \(r + \mathcal{E} f\) or \(r + \mathcal{R} \mathcal{E} f\) for some bounded constructible \({\mathbb{R}}\)-space \(f\) and some \(r \in (-\infty, \infty]\). (We don’t need constructibility for the results of this section, but we will need it later and we don’t intend to introduce even more notation.) Now we define the smoothing functor \(\mathcal{S}\) on \(\mathbf{D}\). The easiest part of the definition are the endofunctors. Moreover these endofunctors exist on the whole category of \((-\infty, \infty]\)-spaces and not just \(\mathbf{D}\). Now let \(f \colon X \rightarrow (-\infty, \infty]\) be a continuous function, then we set \(\mathcal{S}((a, b))(f) := f - b\) for all \((a, b) \in -D\). And for any homomorphism \(\varphi\) in the category of \((-\infty, \infty]\)-spaces we set \(\mathcal{S}((a, b))(\varphi) := \varphi\). This defines the endofunctors \(\{\mathcal{S}(\mathbf{a}) ~|~ \mathbf{a} \in -D\}\) on all \((-\infty, \infty]\)-spaces.

Now let \(f \colon X \rightarrow {\mathbb{R}}\) be an \({\mathbb{R}}\)-space, let \(r \in {\mathbb{R}}\), and let \((a, b), (c, d) \in -D\) with \((a, b) \preceq (c, d)\). Then we set \[ \mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f) \colon {\operatorname{epi}}f \rightarrow {\operatorname{epi}}f, (p, t) \mapsto (p, t - b + d) . \] Moreover we set \[ \mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{R} \mathcal{E} f) := \mathcal{R}(\mathcal{S}((a, b) \preceq (c, d))(r + \mathcal{E} f)) . \] 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 \(\mathcal{S}\). In particular the interleaving distances coincide by corollary 12. Admittedly it is a bit of a cheat or kind of trivial, since we specifically defined this subcategory \(\mathbf{D}\) so that this was going to work out.

Next we define the functor \(\tilde{\mathcal{E}}\). To this end let \(f \colon X \rightarrow {\mathbb{R}}\) and \(g \colon Y \rightarrow {\mathbb{R}}\) be continuous functions, let \((a, b), (c, d) \in -D\), and let \(\varphi \colon X \rightarrow Y\) be a homomorphism from \((f, (a, b))\) to \((g, (c, d))\) in \(-\mathbf{F}\). Then we set \(\tilde{\mathcal{E}}(f, (a, b)) := -b + \mathcal{E} f\), similarly for \((g, (c, d))\) of course, and \(\tilde{\mathcal{E}}(\varphi) \colon {\operatorname{epi}}f \rightarrow {\operatorname{epi}}g, (p, t) \mapsto (\varphi(p), t - b + d)\). This defines our negative persistence-enhancement of \(\mathcal{E}\). (Again a bit of a cheat.)

We note that we have the following

In particular \(\mathcal{S}\) and \(\mathcal{R} \circ \tilde{\mathcal{E}}\) form a negative persistence-enhancement of join trees. (This does not yield any new results about join trees, we just meant to show, they fit into this framework.)