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).

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.

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

  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.

  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.

[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.