#### Ascending Spaces

Later we will make the ascending cosheaf $${\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f$$ for a continuous function $$f$$ the cosheaf version of the join tree associated to $$f$$. As an intermediate step we show that we can obtain this cosheaf not only by post-composing $$\lambda$$ with $${\operatorname{id}}^{-1} {\operatorname{id}}_*$$ but also by pre-composing $$\lambda$$ with another functor, the epigraph. This use of the epigraph in defining the join tree[2] is due to Morozov, Beketayev, and Weber (2013).

• Definition. Let $$f \colon X \rightarrow {\mathbb{R}}$$ be a continuous map, it’s epigraph is $${\operatorname{epi}}f := {\{(x, y) \in X \times {\mathbb{R}}~|~ y \geq f(x)\}}$$. Further we define $$\iota_f \colon {\operatorname{epi}}f \rightarrow {\mathbb{R}}, (x, y) \mapsto y$$ and $$\kappa_f \colon X \rightarrow {\operatorname{epi}}f, x \mapsto (x, f(x))$$.

With these definitions $$\iota$$ defines a functor on topological spaces over $${\mathbb{R}}$$ with $$\kappa$$ a natural transformation from $${\operatorname{id}}$$ to $$\iota$$.

1. Definition. A function $$f \colon X \rightarrow {\mathbb{R}}$$ is ascending if for all $$r \in {\mathbb{R}}$$ there is a continuous map $$H_r \colon X \times [0, 1] \rightarrow X$$ such that $$H_r (x, t) = x$$ for all $$0 \leq t \leq 1$$ and $$x \in X$$ with $$f(x) \geq r$$ and such that $$f (H_r (x, t)) = r + t(f(x)-r)$$ for all $$0 \leq t \leq 1$$ and $$x \in X$$ with $$f(x) \leq r$$.

2. Lemma. For any continuous function $$f \colon X \rightarrow {\mathbb{R}}$$ the projection $$\iota_f \colon {\operatorname{epi}}f \rightarrow {\mathbb{R}}$$ is ascending.

• Proof. For $$r \in {\mathbb{R}}$$ we set $$H_r \colon {\operatorname{epi}}f \times [0, 1] \rightarrow {\operatorname{epi}}f, ((x, y), t) \mapsto (x, \max \{r + t(y-r), y\})$$.

1. Lemma. For any ascending function $$f \colon X \rightarrow {\mathbb{R}}$$ the cosheaf $$\lambda_f$$ is ascending as well.

• Proof. Given $$-\infty \leq a < r < b \leq \infty$$ we proof that the maps from $$\Lambda(f^{-1} ([r, b)))$$ to $$\Lambda(f^{-1} ((a, b))) = \lambda_f ((a, b))$$ respectively $$\Lambda(f^{-1} ((-\infty, b))) = \lambda_f ((-\infty, b))$$ induced by the inclusions are bijections[3]. From this our claim follows. Since inclusions as maps of spaces always commute the two bijections commute with $$(\eta' \circ \lambda)_f$$ as well, hence $$(\eta' \circ \lambda)_f$$ is a bijection as a map from $$\lambda_f ((a, b))$$ to $$\lambda_f ((-\infty, b)) = {\operatorname{id}}_* \lambda_f ((-\infty, b)) = {\operatorname{id}}^+ {\operatorname{id}}_* \lambda_f ((a, b)) = {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f ((a, b))$$. And since the open intervals of $${\mathbb{R}}$$ form a basis, the lemma follows.

Given any point $$x \in f^{-1} ((a, r))$$ the map $$t \mapsto H_r (x, t)$$ defines a continuous path in $$f^{-1} ((a, b))$$ from $$H_r (x, 0) \in f^{-1} ([r, b))$$ to $$x$$, hence induced map from $$\Lambda(f^{-1} ([r, b)))$$ to $$\Lambda(f^{-1} ((a, b))) = \lambda_f ((a, b))$$ is surjective. Now suppose $$x, y \in f^{-1} ([r, b))$$ lie in the same connected component $$C$$ of $$f^{-1} ((a, b))$$, then $$H_r (C, 0)$$ is connected since $$H_r$$ is continuous. Further $$x, y \in H_r (C, 0)$$, hence the induced map from $$\Lambda(f^{-1} ([r, b)))$$ to $$\lambda_f ((a, b))$$ is injective. The induced map from $$\Lambda(f^{-1} ([r, b)))$$ to $$\lambda_f ((-\infty, b))$$ is a bijection by a similar argument.

• Remark. The previous result remains valid if instead of $$\lambda$$ we consider the pushforward of another cosheaf on $$X$$ that maps inclusions of open sets in $$X$$ that are homotopy equivalences to bijections of sets.

1. Lemma. Given a continuous map $$f \colon X \rightarrow {\mathbb{R}}$$ the homomorphism $${\operatorname{id}}_* (\lambda \circ \kappa)_f$$ from $${\operatorname{id}}_* \lambda_f$$ to $${\operatorname{id}}_* (\lambda \circ \iota)_f$$ is an isomorphism.

• Proof. Given $$b \in {\mathbb{R}}\cup \{\infty\}$$ we show that $$\kappa_f (f^{-1} ((-\infty, b)) = \{(x, f(x))\}_{\{x \in X | f(x) < b\}}$$ is a strong deformation retract of $$\iota_f^{-1} ((-\infty, b)) = {\{(x, y) \in X \times (-\infty, b) ~|~ y \geq f(x)\}}$$. Then the result follows by a similar argument as the previous lemma. We define $$R \colon \iota_f^{-1} ((-\infty, b)) \times [0, 1] \rightarrow \iota_f^{-1} ((-\infty, b)), ((x, y), t) \mapsto (x, f(x) + t(y-f(x)))$$, then $$R((x, y), 1) = (x, y)$$ and $$R((x, y), 0) = (x, f(x))$$ for all $$(x, y) \in \iota_f^{-1} ((-\infty, b))$$.

1. Proposition. The natural transformations $$\eta' \circ \lambda$$ and $$\lambda \circ \kappa$$ are isomorphic as objects in the category of functors from topological spaces over $${\mathbb{R}}$$ to cosheaves on $${\mathbb{R}}$$ under $$\lambda$$. In particular $${\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda$$ and $$\lambda \circ \iota$$ are naturally isomorphic.

• Proof. Given $$f \colon X \rightarrow {\mathbb{R}}$$ we have the commutative diagram $\xymatrix@C+2pc{ \lambda_f \ar[r]^{(\lambda \circ \kappa)_f} \ar[d]|-{(\eta' \circ \lambda)_f} & (\lambda \circ \iota)_f \ar[d]|-{(\eta' \circ \lambda \circ \iota)_f} \\ \, {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f \ar[r]_{{\operatorname{id}}^{-1} {\operatorname{id}}_* (\lambda \circ \kappa)_f} & \, {\operatorname{id}}^{-1} {\operatorname{id}}_* (\lambda \circ \iota)_f . }$ By lemma 5 and lemma 6 the homomorphism $$(\eta' \circ \lambda \circ \iota)_f$$ is an isomorphism. And by lemma 7 we have that $${\operatorname{id}}^{-1} {\operatorname{id}}_* (\lambda \circ \kappa)_f$$ is an isomorphism.

[2] Though join trees are referred to as merge trees in the cited paper.

[3] The space $$f^{-1} ([r, b))$$ may not be locally connected. However we won’t need this property.