## Positive Persistence-Enhancements

In the previous section we defined strict $$D$$-categories, interleavings of objects in $$D$$-categories, and showed that interleavings of precosheaves are an instance of this. Then we derived some properties about the interleaving distances, such as the triangle inequality and this compensates for the triangle inequality having been left out in the section before. However we did not prove, that the interleaving distances of Reeb precosheaves provide lower bounds to the corresponding distances of spaces. And it is the aim of this section to make up for this deficit. In the previous section we already observerd that $$1$$-homomorphisms of $$D$$-categories yield inequalities for the corresponding interleaving distances. To use this result we will take two steps. First we will embed the category of $${\mathbb{R}}$$-spaces into a $$D$$-category $$\mathbf{F}$$, so that the interleaving distances on the image of this embedding are equal to the corresponding distances on $${\mathbb{R}}$$-spaces. Second we factor $$\mathcal{C}$$ into this embedding and a $$1$$-homomorphism $$\tilde{\mathcal{C}}$$ from $$\mathbf{F}$$ to the category of precosheaves on $$\overline{D}$$.

• Definition. Here we define the category $$\mathbf{F}$$.

The class of objects of $$\mathbf{F}$$ is the class of all pairs $$(f, \mathbf{a})$$, where $$f \colon X \rightarrow {\mathbb{R}}$$ is a continuous real-valued function and $$\mathbf{a} \in D$$.

Next we specify the homomorphsims of $$\mathbf{F}$$. To this end let $$f \colon X \rightarrow {\mathbb{R}}$$ and $$g \colon Y \rightarrow {\mathbb{R}}$$ be continuous functions and $$(a, b), (c, d) \in D$$. Then a homomorphism from $$(f, (a, b))$$ to $$(g, (c, d))$$ is a continuous map $$\varphi \colon X \rightarrow Y$$ such that $$b - d \leq f(p) - g(\varphi(p)) \leq a - c$$ for all $$p \in X$$.

For $$b = d = \infty$$ we read the left hand side of the previous inequality as $$-\infty$$ and for $$a = c = -\infty$$ we read the right hand side to be $$\infty$$.

So far we have the category $$\mathbf{F}$$ and the full embedding $(\_, \mathbf{o}) \colon \begin{cases} \begin{split} f & \mapsto (f, \mathbf{o}) \\ \varphi & \mapsto \varphi . \end{split} \end{cases}$ Next we provide a smoothing functor for $$\mathbf{F}$$.

• Definition. Let $$f \colon X \rightarrow {\mathbb{R}}$$ be a continuous function and let $$\mathbf{a}, \mathbf{b}, \mathbf{c} \in D$$ with $$\mathbf{b} \preceq \mathbf{c}$$. Then we set $$\mathcal{T}(\mathbf{b})((f, \mathbf{a})) := (f, \mathbf{a} + \mathbf{b})$$ and $$\mathcal{T}(\mathbf{b} \preceq \mathbf{c})_{(f, \mathbf{a})} := {\operatorname{id}}_X$$.

The previous definition augments $$\mathbf{F}$$ with the structure of a strict $$D$$-category. Now we compare the distances of functions to the interleaving distances of $$\mathbf{F}$$. To this end let $$f \colon X \rightarrow {\mathbb{R}}$$ and $$g \colon Y \rightarrow {\mathbb{R}}$$ be continuous functions.

• Lemma. Let $$r \in {\mathbb{R}}$$, $$\varepsilon \geq 0$$, and let $$\varphi \colon X \rightarrow Y$$ be a homeomorphism. Then we have $$-\varepsilon \leq r + f(p) - g(\varphi(p)) \leq \varepsilon$$ if and only if $$\varphi$$ and $$\varphi^{-1}$$ form an $$(-r + \varepsilon, r + \varepsilon)$$-interleaving of $$(f, \mathbf{o})$$ and $$(g, \mathbf{o})$$.

• Corollary. We have $$M(f, g) = M_{\mathcal{T}} ((f, \mathbf{o}), (g, \mathbf{o}))$$ and $$\mu(f, g) = \mu_{\mathcal{T}} ((f, \mathbf{o}), (g, \mathbf{o}))$$.

• Proof. The first equation follows in conjunction with corollary 13, see the section on properties of interleavings, and the second equation follows in conjunction with corollary 12.

Now we have an embedding of the category of $${\mathbb{R}}$$-spaces into $$\mathbf{F}$$, that preserves the distances. In example 10 from the previous section we augmented the category of set-valued precosheaves on $$\overline{D}$$ with the structure of a $$D$$-category and we observed that the interleavings of precosheaves are precisely the interleavings with respect to this structure. Our next and final step towards a lower bound is to provide a $$1$$-homomorphism $$\tilde{\mathcal{C}}$$ from $$\mathbf{F}$$ to the category of set-valued precosheaves on $$\overline{D}$$ such that $$\mathcal{C} = \tilde{\mathcal{C}}((\_, \mathbf{o}))$$. Applying the same procedure to an arbitrary functor $$F$$ on the category of $${\mathbb{R}}$$-spaces is what we name a positive persistence-enhancement for $$F$$. Now let $$F$$ be a functor from the category of $${\mathbb{R}}$$-spaces to some category $$\mathbf{C}$$

• Definition (Positive Persistence-Enhancement). 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 $$F = \tilde{F}((\_, \mathbf{o}))$$.

1. Propostion. If there is a positive persistence-enhancement of $$F$$ with smoothing functor $$\mathcal{S}$$ on $$\mathbf{C}$$, then we have $$M_{\mathcal{S}}(F(f), F(g)) \leq M(f, g)$$ and $$\mu_{\mathcal{S}}(F(f), F(g)) \leq \mu(f, g)$$.

• Proof. This follows from corollary 15, see the section on homomorphisms in $$D$$-categories, and the previous corollary.

• Example. 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. So the only thing left to define for a positive persistence-enhancement is the functor $$\tilde{\mathcal{C}}$$. To this end we set $$\Delta^{\mathbf{a}} \colon {\mathbb{R}}\rightarrow {\overline{{\mathbb{E}}}}, t \mapsto (t, t) - \mathbf{a}$$ and $$\overline{\mathcal{C}}((f, \mathbf{a})) := (\Delta^{\mathbf{a}} \circ f)_* \Lambda$$ for any $$\mathbf{a} \in D$$. Now let $$\varphi \colon (f, \mathbf{a}) \rightarrow (g, \mathbf{b})$$ be a homomorphism in $$\mathbf{F}$$ for some $$\mathbf{a}, \mathbf{b} \in D$$ and let $$U \subseteq {\overline{{\mathbb{E}}}}$$ be a distinguished open subset, then we have $$\varphi((\Delta^{\mathbf{a}} \circ f)^{-1}(U)) \subseteq (\Delta^{\mathbf{b}} \circ g)^{-1}(U)$$ and thus the restriction $$\varphi |_{(\Delta^{\mathbf{a}} \circ f)^{-1}(U)} \colon (\Delta^{\mathbf{a}} \circ f)^{-1}(U) \rightarrow (\Delta^{\mathbf{b}} \circ g)^{-1}(U)$$. We set $$\overline{\mathcal{C}}(\varphi)_U := \Lambda\big(\varphi |_{(\Delta^{\mathbf{a}} \circ f)^{-1}(U)}\big)$$. This defines the functor $$\overline{\mathcal{C}}$$ from $$\mathbf{F}$$ to the category of set-valued precosheaves on $${\overline{{\mathbb{E}}}}$$. Now let $$i \colon \overline{D} \subseteq {\overline{{\mathbb{E}}}}$$ denote the inclusion, then the composition $$\tilde{\mathcal{C}} := i_p \circ \overline{\mathcal{C}}$$ defines a $$1$$-homomorphism of strict $$D$$-categories with $$\mathcal{C} = \tilde{\mathcal{C}}((\_, \mathbf{o}))$$.

Now let $$F$$ and $$G$$ be functors from the category of $${\mathbb{R}}$$-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 $${\mathbb{R}}$$-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 $$\tilde{G}$$.

1. 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 $$\tilde{G}$$.