### A second type of Interleavings

We reuse the notation and the definitions from the previous subsection. In the present subsection we define the structure of a $$-D$$-category on $$\mathbf{C}$$ with the smoothing functor named $$\mathcal{S}'$$. Now for a smoothing functor we need two things, first we need the endofunctors $$\mathcal{S}'(\mathbf{a})$$ associated to any $$\mathbf{a} \in -D$$ and second we need the natural transformations $$\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})$$ associated to any $$\mathbf{a}, \mathbf{b} \in D$$ with $$\mathbf{a} \preceq \mathbf{b}$$. Now in order to get there, we will do something seemingly unnecessary. We will define the endofunctors $$\mathcal{S}'(\mathbf{a})$$ not just on $$\mathbf{D}$$ but on the whole category of set-valued precosheaves on $$\overline{D}$$. Then we will show, that $$\mathbf{C}$$ is invariant under these endofunctors. In this sense the endofunctors $$\mathcal{S}'(\mathbf{a})$$ restrict to endofunctors on $$\mathbf{C}$$. Then we define the natural transformations.

So let us start with the endofunctors setting $$\mathcal{S}'((a, b)) := S^{(-b, -b)}_*$$ for all $$(a, b) \in -D$$. Now in order to see that $$\mathbf{C}$$ is invariant under these endofunctors we will show that $$\pi^2_*$$, $$\pi^2_p$$, $$\eta^{\pi^2}$$, and $$\overline{\mathcal{S}}$$ are compatible with these endofunctors in the same way they would by compatible if $$\pi^2_*$$ and $$\pi^2_p$$ were $$1$$-homomorphisms and $$\eta^{\pi^2}$$ was a $$2$$-homomorphism. To this end we convince ourselves that the diagram $\xymatrix@+=3pc{ \mathcal{Q} \ar[d]_{\big(S^{(-b, -b)}\big)^{-1}} \ar[r]^{(\pi^2)^{+1}} & \mathcal{U} \ar[d]|-{(s^{-b})^{-1}} \ar[r]^{(\pi^2)^{-1}} & \mathcal{Q} \ar[d]^{\big(S^{(-b, -b)}\big)^{-1}} \\ \mathcal{Q} \ar[r]_{(\pi^2)^{+1}} & \mathcal{U} \ar[r]_{(\pi^2)^{-1}} & \mathcal{Q} }$ commutes for all $$b < \infty$$.

• Lemma. The functor $$\pi^2_*$$ commutes with $$\mathcal{S}'(\mathbf{a})$$ and $$\overline{\mathcal{S}}(\mathbf{a})$$ for all $$\mathbf{a} \in -D$$.

• Proof. This follows from the commutativity of the right square in the above diagram.

• Lemma. The functor $$\pi^2_p$$ commutes with $$\overline{\mathcal{S}}(\mathbf{a})$$ and $$\mathcal{S}'(\mathbf{a})$$ for all $$\mathbf{a} \in -D$$.

• Proof. This follows from the commutativity of the left square in the above diagram.

• Lemma. The natural transformation $$\eta^{\pi^2}$$ commutes with $$\mathcal{S}'(\mathbf{a})$$ for all $$\mathbf{a} \in -D$$.

• Proof. This follows from the commutativity of the outer square in the above diagram.

The previous three lemmata have the following

• Corollary. The subcategory $$\mathbf{C}$$ is invariant under $$\mathcal{S}'(\mathbf{a})$$ for all $$\mathbf{a} \in -D$$.

So far we defined the endofunctors for the smoothing functor $$\mathcal{S}'$$ on $$\mathbf{C}$$. The next step is to define the natural transformations $$\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})$$ for any $$\mathbf{a}, \mathbf{b} \in D$$ with $$\mathbf{a} \preceq \mathbf{b}$$. Now we already convinced ourselves that $$\pi^2_*$$, $$\pi^2_p$$, and $$\eta^{\pi^2}$$ are compatible with the endofunctors of $$\mathcal{S}'$$ and $$\overline{\mathcal{S}}$$. So that is already half the part of $$\pi^2_*$$ and $$\pi^2_p$$ being $$1$$-homomorphisms and $$\eta^{\pi^2}$$ being a $$2$$-homomorphism for this hypothetical smoothing functor. So it would be nice if we could define these natural transformations in such a way that $$\pi^2_*$$, $$\pi^2_p$$, and $$\eta^{\pi^2}$$ were actually $$1$$- respectively $$2$$-homomorphisms. Now suppose we were already there, then we would have the commutative diagram $\xymatrix@C+=5pc{ \mathcal{S}'(\mathbf{a}) \ar@{=>}[r]^{\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})} \ar@{=>}[d]_{\eta^{\pi^2} \circ \mathcal{S}(\mathbf{a})} & \mathcal{S}'(\mathbf{b}) \ar@{=>}[d]^{\eta^{\pi^2} \circ \mathcal{S}(\mathbf{b})} \\ \pi^2_p \circ \overline{\mathcal{S}}(\mathbf{a}) \circ \pi^2_* \ar@{=>}[r]_{\pi^2_p \circ \overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ \pi^2_*} & \pi^2_p \circ \overline{\mathcal{S}}(\mathbf{b}) \circ \pi^2_* }$ for all $$\mathbf{a}, \mathbf{b} \in D$$ with $$\mathbf{a} \preceq \mathbf{b}$$. And this determines $$\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})$$, since $$\eta^{\pi^2}$$ is a natural isomorphism on $$\mathbf{C}$$. Also it does the job.

So now that $$\pi^2_*$$ is a $$1$$-homomorphism with respect to this second smoothing functor $$\mathcal{S}'$$ on $$\mathbf{C}$$ and $$\overline{\mathcal{S}}$$, the interleavings of $$\mathcal{C} \mathcal{E} f$$ and $$\mathcal{C} \mathcal{E} g$$ with respect to $$\mathcal{S}'$$ are in bijection with those of $$\pi^2_* \mathcal{C} \mathcal{E} f$$ and $$\pi^2_* \mathcal{C} \mathcal{E} g$$. Now these are all interleavings in $$-D$$-categories. But with $$\overline{\mathcal{S}}$$ we gave the category of set-valued precosheaves on $$\overline{{\mathbb{R}}}_{-\infty}$$ the structure of an $${\overline{{\mathbb{E}}}}$$-category. So by lemma 18, from the section on complete persistence-enhancements, the interleavings of $$\mathcal{C} \mathcal{E} f$$ and $$\mathcal{C} \mathcal{E} g$$, with respect to the $$-D$$-category structure given by $$\overline{\mathcal{S}}$$, are in canonical bijection with those given by the $$\overline{\mathcal{S}}$$-induced structure of a $$D$$-category. So in conjunction with corollary 20 from the previous subsection we have the following

1. Propostion. The interleavings of $$\mathcal{C} \mathcal{E} f$$ and $$\mathcal{C} \mathcal{E} g$$ with respect to $$\mathcal{S}$$ and with respect to $$\mathcal{S}'$$ are in bijection.