Now let \(\mathbf{C}\) be a strict \(D\)-category with smoothing functor \(\mathcal{S}\) and let \(A\) and \(B\) be objects of \(\mathbf{C}\). We aim to provide some useful properties about interleavings in \(\mathbf{C}\).

  1. Corollary. Let \(\mathcal{I}\) be the set of all \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\) such that \(A\) and \(B\) are \((\mathbf{a}, \mathbf{b})\)-interleaved. Further let \(\mathcal{I}' := \mathcal{I} \cap \triangledown\) and \(\mathcal{I}'' := \mathcal{I} \cap \blacktriangledown\). Then we have \(\mu_{\mathcal{S}} (A, B) = \inf \epsilon (\mathcal{I}')\) and \(M_{\mathcal{S}} (A, B) = \inf (\epsilon \circ \gamma)(\mathcal{I}') = \inf \epsilon (\mathcal{I}'')\).

We will now use this corollary to give more concise descriptions of the interleaving distances. The reason we didn’t define the interleaving distances with these more concise descriptions in the first place is that we believe, having those additional interleavings around, can help with the computation of the interleaving distances. Now suppose \((a, b; c, d) \in \triangledown\), then \((a, b; c, d) = (-d, b; -b, d)\). Thus we have the bijection \[ \Phi \colon \triangledown \rightarrow D^{\perp}, (a, b; c, d) \mapsto (d, b) \] with inverse \[ \Psi \colon D^{\perp} \rightarrow \triangledown, (a, b) \mapsto (-a, b; -b, a) . \]

With this definition we get a corollary to the previous corollary.

  1. Corollary. Let \(\mathcal{J}\) be the set of all \((a, b) \in D^{\perp}\) such that \(A\) and \(B\) are \((a, b)\)-interleaved, then \(\mu_{\mathcal{S}} (A, B) = \inf \epsilon' (\mathcal{J})\) and \(M_{\mathcal{S}} (A, B) = \inf \epsilon'' (\mathcal{J})\).

We further simplify the absolute interleaving distance of \(A\) and \(B\).

Now an \((\varepsilon, \varepsilon)\)-interleaving of \(A\) and \(B\) is just a \((-\varepsilon, \varepsilon; -\varepsilon, \varepsilon)\)-interleaving. Further we have \(\epsilon ((-\varepsilon, \varepsilon; -\varepsilon, \varepsilon)) = \varepsilon\). And as noted earlier \(\epsilon |_{\blacktriangledown}\) is a bijection. Thus we have yet another corollary to corollary 11.

  1. Corollary. Let \(\mathcal{V}\) be the set of all \(\varepsilon \geq 0\) such that \(A\) and \(B\) are \(\varepsilon\)-interleaved. Then \(M_{\mathcal{S}} (A, B) = \inf \mathcal{V}\).

Next we proof a type of triangle inequality for both interleaving distances. To this end let \(C\) be another object of \(\mathbf{C}\).

  1. Corollary (Triangle Inequality). We have \[ M_{\mathcal{S}} (A, C) \leq M_{\mathcal{S}} (A, B) + M_{\mathcal{S}} (B, C) \] and \[ \mu_{\mathcal{S}} (A, C) \leq \mu_{\mathcal{S}} (A, B) + \mu_{\mathcal{S}} (B, C) . \]