### Interleaving Precosheaves on D

In the interlude on precosheaves we described how continuous and Galois maps between intersection-based spaces yield functors between the corresponding categories of precosheaves. We start by describing some self-maps on $$\overline{D}$$ to yield us endofunctors on the category of precosheaves on $$\overline{D}$$.

• Definition. For $$a \in {\mathbb{R}}$$ we define $$s^a \colon \overline{{\mathbb{R}}} \rightarrow \overline{{\mathbb{R}}}, t \mapsto t + a$$
and we set $$s^{\pm \infty} \colon \overline{{\mathbb{R}}} \rightarrow \overline{{\mathbb{R}}}, t \mapsto \pm \infty$$.

We note that $$s^a$$ is continuous and Galois on $$\overline{{\mathbb{R}}}_{\infty}$$ for $$- \infty \leq a < \infty$$ and on $$\overline{{\mathbb{R}}}_{- \infty}$$ for $$- \infty < a \leq \infty$$. Thus the following definition yields a continuous and Galois self-map on $${\overline{{\mathbb{E}}}}$$ for all $$(a, b) \in D$$.

• Definition. For $$(a, b) \in D$$ we set $$S^{(a, b)} := (s^a \circ \pi^1) \times (s^b \circ \pi^2)$$.

More explicitly we have $$S^{(a, b)} (x, y) = (s^a (x), s^b (y))$$ for all $$(a, b) \in D$$ and $$x, y \in \overline{{\mathbb{R}}}$$. For all $$\mathbf{a} \in D$$ we have $$S^{\mathbf{a}} \big(\overline{D}\big) \subseteq \overline{D}$$, hence $$S^{\mathbf{a}}$$ also defines a continuous and Galois self-map on $$\overline{D}$$. Now let $$F$$ be a set-valued precosheaf on $$\overline{D}$$. Using the definitions from the interlude on precosheaves we get the precosheaf $$S^{\mathbf{a}}_p F$$ for any $$\mathbf{a} \in D$$. Now for $$\mathbf{o} \preceq \mathbf{a}$$ and any distinguished open subset $$U \subseteq \overline{D}$$ we have $$U \subseteq (S^{\mathbf{a}})^{+1} (U)$$, hence the precosheaf $$F$$ itself yields a map from $$F(U)$$ to $$S^{\mathbf{a}}_p F (U)$$. Now $$(S^{\mathbf{a}})^{+1}$$ is monotone and thus these maps describe a homomorphism from $$F$$ to $$S^{\mathbf{a}}_p F$$.

• Definition. We denote the previously described homomorphism by
$$\Sigma^{\mathbf{a}}_F \colon F \rightarrow S^{\mathbf{a}}_p F$$.

We note that $$\Sigma^{\mathbf{a}}$$ is a natural transformation from the identity functor $${\operatorname{id}}$$ to $$S^{\mathbf{a}}_p$$. With these definitions in place we can define interleavings of precosheaves. To this end let $$F$$ and $$G$$ be precosheaves on $$\overline{D}$$.

• Definition. For $$(\mathbf{a}, \mathbf{b}) \in \mathcal{D}$$ an $$(\mathbf{a}, \mathbf{b})$$-interleaving of $$F$$ and $$G$$ is a pair of homomorphisms $$\varphi \colon F \rightarrow S^{\mathbf{a}}_p G$$ and $$\psi \colon G \rightarrow S^{\mathbf{b}}_p F$$ such that $\xymatrix{ F \ar@/^/[rr]^{\Sigma^{\mathbf{a} + \mathbf{b}}_F} \ar[dr]_{\varphi} & & S^{\mathbf{a} + \mathbf{b}}_p F \\ & S^{\mathbf{a}}_p G \ar[ru]_{S^{\mathbf{a}}_p \psi} }$ and $\xymatrix{ G \ar@/^/[rr]^{\Sigma^{\mathbf{a} + \mathbf{b}}_G} \ar[dr]_{\psi} & & S^{\mathbf{a} + \mathbf{b}}_p G \\ & S^{\mathbf{b}}_p F \ar[ru]_{S^{\mathbf{b}}_p \varphi} }$ commute.

We say $$F$$ and $$G$$ are $$(\mathbf{a}, \mathbf{b})$$-interleaved if there is an $$(\mathbf{a}, \mathbf{b})$$-interleaving of $$F$$ and $$G$$.

Now we use the weightings on $$\mathcal{D}$$ we defined previously to describe two interleaving distances for precosheaves on $$\overline{D}$$.

• Definition. Let $$\mathcal{I}$$ be the set of all $$(\mathbf{a}, \mathbf{b}) \in \mathcal{D}$$ such that there is an $$(\mathbf{a}, \mathbf{b})$$-interleaving of $$F$$ and $$G$$.
Then we set $$M (F, G) := \inf (\epsilon \circ \gamma \circ \delta) (\mathcal{I})$$ and $$\mu (F, G) := \inf (\epsilon \circ \delta) (\mathcal{I})$$.

We name $$M (F, G)$$ the absolute interleaving distance of $$F$$ and $$G$$ and $$\mu (F, G)$$ the relative interleaving distance of $$F$$ and $$G$$.

Now let $$f \colon X \rightarrow {\mathbb{R}}$$ and $$g \colon Y \rightarrow {\mathbb{R}}$$ be continuous functions. At this point the next steps would be to proof the triangle inequalities for $$M$$ and $$\mu$$ and to show, that the interleaving distances $$M(\mathcal{C} f, \mathcal{C} g)$$ and $$\mu(\mathcal{C} f, \mathcal{C} g)$$ really provide lower bounds to $$M(f, g)$$ respectively $$\mu(f, g)$$ as we did for join trees. Instead we will derive these results from more general statements however.

Nevertheless we can now repeat another question that we pledged to address in a more precise way. In the beginning we introduced the interleaving distances of join trees $$\mathcal{R} \mathcal{E} f$$ and $$\mathcal{R} \mathcal{E} g$$. Above we argued that the functors $$\mathcal{R}$$ and $$\mathcal{C}$$ are closely related, so it seems very reasonable to compare the interleavings distances of $$\mathcal{R} \mathcal{E} f$$ and $$\mathcal{R} \mathcal{E} g$$ to those of the precosheaves $$\mathcal{C} \mathcal{E} f$$ and $$\mathcal{C} \mathcal{E} g$$. Later we will show the following

1. Theorem. If $$X$$ and $$Y$$ are smooth and compact manifolds, then $M_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) = M (\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g)$ and $\mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) = \mu (\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g) .$