Interleavings in D-Categories

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.

  1. 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 \(\mathcal{S}(\mathbf{o} \preceq \mathbf{a}) = \Sigma^{\mathbf{a}}\).