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 \[\mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) = \mu (\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g) .\]