#### The Join Tree

The following definition[4] is from (Morozov, Beketayev, and Weber 2013).

• Definition. Let $$f \colon X \rightarrow {\mathbb{R}}$$ be a continuous map we define it’s join tree to be the continuous map $$(\rho \circ \iota)_f$$ from $$(\pi \circ \iota)_f ({\operatorname{epi}}f)$$ to $${\mathbb{R}}$$.

With this definition $$\rho \circ \iota$$ is an endofunctor on topological spaces over $${\mathbb{R}}$$. Given a continuous map $$f \colon X \rightarrow {\mathbb{R}}$$ we have $$(\pi \circ \iota)_f \circ \kappa_f = (\rho \circ \kappa)_f \circ \pi_f$$, so in somewhat sloppy notation $$(\pi \circ \iota) \circ \kappa = (\rho \circ \kappa) \circ \pi$$ is a natural transformation from $${\operatorname{id}}$$ to $$\rho \circ \iota$$. Similarly we have the function $$(\gamma \circ \lambda \circ \iota)_f$$ defined on the display space $${\operatorname{dis}}(\lambda \circ \iota)_f$$ of $$(\lambda \circ \iota)_f$$. And just as with $$\rho$$ we have the natural transformation $$(\eta \circ \iota) \circ \kappa = (\gamma \circ \lambda \circ \kappa) \circ \eta$$ from $${\operatorname{id}}$$ to $$\gamma \circ \lambda \circ \iota$$. The two constructions are related via the commutative diagram $\xymatrix@C+1pc@R-1pc{ & \rho_f \ar[r]^{(\rho \circ \kappa)_f} \ar[dd]|-{\phi_f} & (\rho \circ \iota)_f \ar[dd]^{(\phi \circ \iota)_f} \\ f \ar[ur]^{\pi_f} \ar[dr]_{\eta_f} \\ & (\gamma \circ \lambda)_f \ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} & (\gamma \circ \lambda \circ \iota)_f }$ given a function $$f \colon X \rightarrow {\mathbb{R}}$$. In the section on the Reeb space we considered the left triangle which suggests to replace the Reeb graph functor $$\rho$$ and the natural transformation $$\pi$$ by $$\gamma \circ \lambda$$ and $$\eta$$ respectively. Now $$\rho \circ \kappa$$ yields a nice and classic map from any Reeb graph to the corresponding join tree, so our replacement of the Reeb graph functor $$\rho$$ by $$\gamma \circ \lambda$$ is only complete, if also we can replace the join tree functor $$\rho \circ \iota$$ and the natural transformation $$\rho \circ \kappa$$ and if we can extend $$\phi$$ to a natural transformation from $$\rho \circ \kappa$$ to it’s replacement. And here the commutative square on the right hand side, suggests we may take $$\lambda \circ \gamma \circ \iota$$ as a replacement for the join tree functor $$\rho \circ \iota$$ and to take $$\gamma \circ \lambda \circ \kappa$$ as a replacement for $$\rho \circ \kappa$$, since then we can extend $$\phi$$ by $$\phi \circ \iota$$ to a natural transformation from $$\rho \circ \kappa$$ to $$\gamma \circ \lambda \circ \kappa$$. We further note that by corollary 2 in that section the natural transformation $$(\phi, \phi \circ \iota)$$ from $$\rho \circ \kappa$$ to $$\gamma \circ \lambda \circ \kappa$$ is isomorphic to the natural transformation $$(\eta \circ \rho, \eta \circ \rho \circ \iota)$$ from $$\rho \circ \kappa$$ to $$\gamma \circ \lambda \circ \rho \circ \kappa$$, so our choice of replacements is the same as if we applied $$\gamma \circ \lambda$$ to the upper row in the diagram. And by proposition 8 we have a natural isomorphism from $$(\gamma \circ \lambda \circ \iota)$$ to $$\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda$$ that commutes with $$\gamma \circ \lambda \circ \kappa$$ and $$\gamma \circ \eta' \circ \lambda$$ sothat we can use the following

• Proposition. $${\operatorname{id}}_* \lambda$$ and $$\gamma {\operatorname{id}}^{-1}$$ form a pair of adjoint functors $${\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}$$ with unit $$(\gamma \circ \eta' \circ \lambda) \circ \eta$$ and whose counit is an isomorphism.

• Proof. The first statement follows from theorem 1, proposition 3 and the general statement that the two pairs of adjoint functors, when composed in the same way as in our claim, form again a pair of adjoint functors with the unit described as in the claim, see for example https://en.wikipedia.org/wiki/Adjoint_functors#Composition. And for the counit of this composed adjunction we have the formula $${\operatorname{id}}_* \varepsilon {\operatorname{id}}^{-1} \circ \varepsilon'$$. By theorem 1 $$\varepsilon$$ is an isomorphism, hence $${\operatorname{id}}_* \varepsilon {\operatorname{id}}^{-1}$$ is an isomorphism and by proposition 3 $$\varepsilon'$$ is an isomorphism and thus our claim follows.

• Corollary. The category of cosheaves on $$\bar{{\mathbb{R}}}$$ is equivalent to the reflective subcategory of ascending cosheaf spaces over $${\mathbb{R}}$$.

• Proof. By Gabriel and Zisman (1967 Proposition 1.3 or http://ncatlab.org/nlab/show/reflective+subcategory#characterizations) the category of cosheaves on $$\bar{{\mathbb{R}}}$$ is equivalent to the reflective subcategory of those spaces $$f \colon X \rightarrow {\mathbb{R}}$$ over $${\mathbb{R}}$$ for which $$(\gamma \circ \eta' \circ \lambda)_f \circ \eta_f$$ is an isomorphism. Now suppose this is the case for $$f$$, then $$f$$ is isomorphic to $$\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda$$ which is in the image of $$\gamma$$ and thus a cosheaf space, hence $$\eta_f$$ is an isomorphism. From this it follows that $$(\gamma \circ \eta' \circ \lambda)_f$$ is an isomorphism as well, hence by proposition 8 $$(\gamma \circ \lambda \circ\kappa)_f$$ is an isomorphism. Now we consider the commutative diagram $\xymatrix@C+1pc{ f \ar[r]^{\kappa_f} \ar[d]_{\eta_f} & \iota_f \ar[d]^{(\eta \circ \iota)_f} \\ (\gamma \circ \lambda)_f \ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} & (\gamma \circ \lambda \circ \iota)_f . }$ Hence we have the retract[5] $$R := \eta_f^{-1} \circ (\gamma \circ \lambda \circ \kappa)_f^{-1} \circ (\eta \circ \iota)_f$$ from $$\iota_f$$ to $$f$$. By lemma 5 $$\iota_f$$ is ascending, so given $$r \in {\mathbb{R}}$$ there is a map $$H_r$$ as in definition 4. Now let $$\tilde{H}_r \colon X \times [0, 1] \rightarrow X$$ be defined by $$\tilde{H}_r (x, t) = r(H_r(\kappa_f(x), t))$$ then $$\tilde{H}_r$$ inherits the properties needed in order for $$f$$ to be ascending. Conversely if $$f$$ is an ascending cosheaf space over $${\mathbb{R}}$$, then $$\eta_f$$ is an isomorphism since $$f$$ is a cosheaf space. And by lemma 6 $$\lambda_f$$ is ascending, hence $$(\eta' \circ \lambda)_f$$ is an isomorphism.

In conclusion $$(\gamma \circ \lambda \circ \iota)_f$$ is an ascending cosheaf space over $${\mathbb{R}}$$ given a function $$f$$. It’s cosheaf of connected components $$(\lambda \circ \gamma \circ \lambda \circ \iota)_f$$ is isomorphic to $$(\lambda \circ \iota)_f$$ by theorem 1. By lemma 5 and lemma 6 $$(\lambda \circ \iota)_f$$ is ascending, and thus we have an associated cosheaf $${\operatorname{id}}_* (\lambda \circ \iota)_f$$ on $$\bar{{\mathbb{R}}}$$ via the adjunction $${\operatorname{id}}_* \dashv {\operatorname{id}}^{-1}$$ by proposition 3. By lemma 7 this cosheaf is isomorphic to $${\operatorname{id}}_* \lambda_f$$ which is the cosheaf on $$\bar{{\mathbb{R}}}$$ associated to $$\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f$$ via the adjunction $${\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}$$. Now applying $${\operatorname{id}}^{-1}$$ to $${\operatorname{id}}_* \lambda_f \cong {\operatorname{id}}_* (\lambda \circ \iota)_f$$ recovers $$(\lambda \circ \iota)_f$$, hence $$(\gamma \circ \lambda \circ \iota)_f$$ and $$\gamma {\operatorname{id}}^{-1} {\operatorname{id}}_* \lambda_f$$ are isomorphic and thus a posteriori $${\operatorname{id}}_* \lambda_f$$ is the cosheaf on $$\bar{{\mathbb{R}}}$$ associated to the ascending cosheaf space $$(\gamma \circ \lambda \circ \iota)_f$$ via the adjunction $${\operatorname{id}}_* \lambda \dashv \gamma {\operatorname{id}}^{-1}$$. (Here the author allowed himself some redundance repeating the proof of proposition 8.)

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

[5] By a retract we mean a homomorphism $$R$$ in the category of topological spaces over $${\mathbb{R}}$$ from $$\iota_f$$ to $$f$$ such that $$R \circ \kappa_f = {\operatorname{id}}$$.