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.