Equivalence to descending Precosheaf

We start by describing the structure of an \({\overline{{\mathbb{E}}}}\)-category on the the category of set-valued precosheaves on \(\overline{{\mathbb{R}}}_{-\infty}\). We proceed similar to example 10. To this end let \(\mathcal{U}\) be the topology or intersection-base of \(\overline{{\mathbb{R}}}_{-\infty}\) (here they are the same), \(\mathcal{Q}\) the intersection-base of \(\overline{D}\), and let \[ \overline{s} \colon {\overline{{\mathbb{E}}}}\mapsto {\operatorname{End}(\mathcal{U})}, (a, b) \mapsto \begin{cases} (s^b)^{+1} & b > -\infty \\ (s^{-b})^{-1} & b < \infty . \end{cases} \] We note that this definition is not over-determined and that \(\overline{s}\) is monotone. Now post-composition with the precomposition functor \(\tilde{\mathcal{S}}\) yields the smoothing functor \(\overline{\mathcal{S}} := \tilde{\mathcal{S}} \circ \overline{s}\). We note that \(\overline{\mathcal{S}}((a, b)) = s^b_p\) for \(b > -\infty\) and \(\overline{\mathcal{S}}((a, b)) = s^{-b}_*\) for \(b < \infty\). Now we have the continuous map \(\pi^2\) from \(\overline{D}\) to \(\overline{{\mathbb{R}}}_{-\infty}\) and we aim to show that \(\pi^2_*\) and \(\pi^2_p\) are \(1\)-homomorphisms between the corresponding \(D\)-categories and that \(\eta^{\pi^2}\) is a \(2\)-homomorphism. For \((a, b) \in D\) we have \(\pi^2 \circ S^{(a, b)} = s^b \circ \pi^2\) and with this we convince ourselves that the diagram \[ \xymatrix@+=3pc{ \mathcal{Q} \ar[d]_{\big(S^{(a, b)}\big)^{+1}} \ar[r]^{(\pi^2)^{+1}} & \mathcal{U} \ar[d]|-{(s^b)^{+1}} \ar[r]^{(\pi^2)^{-1}} & \mathcal{Q} \ar[d]^{\big(S^{(a, b)}\big)^{+1}} \\ \mathcal{Q} \ar[r]_{(\pi^2)^{+1}} & \mathcal{U} \ar[r]_{(\pi^2)^{-1}} & \mathcal{Q} } \] commutes for all \((a, b) \in D\). (We note that lemma 5 is the commutativity of the left square.)

  • Lemma. The functor \(\pi^2_*\) is a \(1\)-homomorphism of \(D\)-categories.

  • Proof. This follows from the commutativity of the right square in the above diagram and the monotonicity of post-composition by \((\pi^2)^{-1}\).

  • Lemma. The functor \(\pi^2_p\) is a \(1\)-homomorphism of \(D\)-categories.

  • Proof. This follows from the commutativity of the left square in the above diagram and the monotonicity of post-composition by \((\pi^2)^{+1}\).

  • Lemma. The natural transformation \(\eta^{\pi^2}\) is a \(2\)-homomorphism of \(D\)-categories.

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

The previous three lemmata have the following

  • Corollary. The full subcategory of set-valued precosheaves \(F\) on \(\overline{D}\), with \(\eta^{\pi^2}_F\) an isomorphism, is a sub-\(D\)-category in the sense that it is invariant under \(\mathcal{S}(\mathbf{a})\) for all \(\mathbf{a} \in D\).

Now the functor \(\pi^2_*\) is full and faithful on the category of precosheaves \(F\), with \(\eta^{\pi^2}_F\) an isomorphism. Since this is also a \(D\)-category and \(\pi^2_*\) is a \(1\)-homomorphism on this \(D\)-category, \(\pi^2_*\) yields bijections of interleavings.

Now let \(f \colon X \rightarrow {\mathbb{R}}\) a continuous function.

  1. Lemma. The homomorphism \((\eta^{\pi^2} \circ \mathcal{C} \circ \mathcal{E})_f\) from \(\mathcal{C} \mathcal{E} f\) to \(\pi^2_p \pi^2_* \mathcal{C} \mathcal{E} f\) is an isomorphism.

  • Proof. For \(a < r < b\) we set \(U := (a, \infty] \times [-\infty, b)\), then we have \((\mathcal{C} \mathcal{E} f)(U) = \Lambda({\operatorname{epi}}f \cap X \times (a, b))\) and \((\pi^2_p \pi^2_* \mathcal{C} \mathcal{E} f)(U) = \Lambda({\operatorname{epi}}f \cap X \times [-\infty, b))\). Now let \(i\) be the inclusion of \({\operatorname{epi}}f \cap X \times [r, b)\) into \({\operatorname{epi}}f \cap X \times (a, b)\) and let \(j\) be the inclusion of \({\operatorname{epi}}f \cap X \times (a, b)\) into \({\operatorname{epi}}f \cap X \times [-\infty, b)\). First we prove that \(\Lambda(i)\) is a bijection. Let \((x, y) \in {\operatorname{epi}}f \cap X \times (a, r)\), then \(c \colon [0, 1] \rightarrow {\operatorname{epi}}f \cap X \times (a, b), t \mapsto (x, t(r-y) + y)\) defines a path from \((x, y)\) to \((x, r)\), hence \(\Lambda(i)\) is surjective. Now let \(R \colon {\operatorname{epi}}f \cap X \times (a, b) \rightarrow {\operatorname{epi}}f \cap X \times [r, b), (x, y) \mapsto (x, \max \{r, y\})\), then \(R \circ i = {\operatorname{id}}\) and thus \(\Lambda(R) \circ \Lambda(i) = {\operatorname{id}}\), hence \(\Lambda(i)\) is injective. By a similar argument we obtain that \(\Lambda(j \circ i) = \Lambda(j) \circ \Lambda(i)\) is bijective. Now that both \(\Lambda(i)\) and \(\Lambda(j) \circ \Lambda(i)\) are bijective, we also have that \(\Lambda(j)\) is bijective. But \(\Lambda(j)\) is just \((\eta^{\pi^2} \circ \mathcal{C} \circ \mathcal{E})_{f ~ U}\) and this implies the claim.

Now let \(g \colon Y \rightarrow {\mathbb{R}}\) be another continuous function.

  1. Corollary. The interleavings of \(\mathcal{C} \mathcal{E} f\) and \(\mathcal{C} \mathcal{E} g\) are in bijection to those of \(\pi^2_* \mathcal{C} \mathcal{E} f\) and \(\pi^2_* \mathcal{C} \mathcal{E} g\).