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.