Up to this point we have seen two notions of an interleaving, the first for join trees and the second for precosheaves. In order to show theorem 9 we will use several more and to not repeatedly define new notions with slight modifications we define a common generalization. The idea is to formalize the additional structure, needed on a certain category \(\mathbf{C}\), to define the notion of an interleaving between any two objects in \(\mathbf{C}\).
Definition (Strict \(D\)-Categories). A strict \(D\)-category is a category \(\mathbf{C}\) with a strict monoidal functor \(\mathcal{S}\) from \(D\) to the category of endofunctors on \(\mathbf{C}\). We refer to \(\mathcal{S}\) as the smoothing functor of \(\mathbf{C}\).
Now let \(\mathbf{C}\) be a strict \(D\)-category with smoothing functor \(\mathcal{S}\) and let \(A\) and \(B\) be two objects in \(\mathbf{C}\). It turns out that the smoothing functor \(\mathcal{S}\) is all we need in order to define the notion of an interleaving between \(A\) and \(B\).
Definition (Interleavings). For \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\) an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(A\) and \(B\) is a pair of homomorphisms \(\varphi \colon A \rightarrow \mathcal{S}(\mathbf{a})(B)\) and \(\psi \colon B \rightarrow \mathcal{S}(\mathbf{b})(A)\) such that the diagrams \[ \xymatrix{ A \ar@/^/[rr]^{\mathcal{S}(\mathbf{o} \preceq \mathbf{a} + \mathbf{b})_A \quad} \ar[dr]|-{\varphi} & & \mathcal{S}(\mathbf{a} + \mathbf{b})(A) \\ & \mathcal{S}(\mathbf{a})(B) \ar[ru]|-{\mathcal{S}(\mathbf{a})(\psi)} } \] and \[ \xymatrix{ B \ar@/^/[rr]^{\mathcal{S}(\mathbf{o} \preceq \mathbf{a} + \mathbf{b})_B \quad} \ar[dr]|-{\psi} & & \mathcal{S}(\mathbf{a} + \mathbf{b})(B) \\ & \mathcal{S}(\mathbf{b})(A) \ar[ru]|-{\mathcal{S}(\mathbf{b})(\varphi)} } \] commute.
We say \(A\) and \(B\) are \((\mathbf{a}, \mathbf{b})\)-interleaved if there is an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(A\) and \(B\).
The two interleaving distances of \(A\) and \(B\) are defined similarly to those of two precosheaves, we spell out the definitions nevertheless.
Definition (Interleaving Distances). 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 \(A\) and \(B\).
Then we set \(M_{\mathcal{S}} (A, B) := \inf (\epsilon \circ \gamma \circ \delta) (\mathcal{I})\) and \(\mu_{\mathcal{S}} (A, B) := \inf (\epsilon \circ \delta) (\mathcal{I})\).
In the following example we show that interleavings of precosheaves and the interleaving distances of precosheaves on \(\overline{D}\) are an instance of the notions we defined here.
Example. Let \(\mathcal{B}\) be the intersection-base of \(\overline{D}\), then we may identify the monoidal poset of monotone self-maps \({\operatorname{End}(\mathcal{B})}\) with the category of endofunctors on \(\mathcal{B}\). Now let \(\mathbf{C}\) be the category of set-valued precosheaves on \(\overline{D}\). We define the precomposition functor \[ \tilde{\mathcal{S}} \colon {\operatorname{End}(\mathcal{B})} \rightarrow {\operatorname{End}(\mathbf{C})}, \begin{cases} \begin{split} T & \mapsto (F \mapsto F \circ T) \\ \eta & \mapsto (F \mapsto F \circ \eta) . \end{split} \end{cases} \] We note that \(\tilde{\mathcal{S}}\) is a strict monoidal functor. Moreover the map \[ \big(S^{(\_)}\big)^{+1} \colon D \rightarrow {\operatorname{End}(\mathcal{B})}, \mathbf{a} \mapsto (S^{\mathbf{a}})^{+1} \] is a homomorphism of monoidal posets, hence the functor \(\mathcal{S} := \tilde{\mathcal{S}} \circ \big(S^{(\_)}\big)^{+1}\) is strict monoidal as well. Now we observe that \(\mathcal{S}(\mathbf{a}) = S^{\mathbf{a}}_p\) for \(\mathbf{a} \in D\). And if \(\mathbf{o} \preceq \mathbf{a}\), then \(\mathcal{S}(\mathbf{o} \preceq \mathbf{a}) = \Sigma^{\mathbf{a}}\).