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