## A negative Enhancement for Join Trees

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

1. Lemma. The endofunctor $$\mathcal{R}$$ is a $$1$$-homomorphism from $$\mathbf{D}$$ to $$\mathbf{D}$$ and $$\pi \colon {\operatorname{id}}\Rightarrow \mathcal{R}$$ is a $$2$$-homomorphism.

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.)