de Silva, Munch, and Patel (2015) observed that \(\gamma \circ \lambda\) is closely related to another endofunctor on topological spaces over \(M\), the Reeb space.
Definition. Given a continuous map \(f \colon X \rightarrow M\) and \(x \in X\) let \(\pi_f (x)\) be the connected component of \(x\) in \(f^{-1}(f(x))\). In this way we obtain a function \(\pi_f \colon X \rightarrow 2^X\) and we endow \(\pi_f (X)\) with the quotient topology[1]. By the universal property of the quotient space there is a unique continuous function \(\tilde{f} \colon \pi_f (X) \rightarrow M\) such that \(\tilde{f} \circ \pi_f = f\) and we define \(\rho_f = \tilde{f}\).
With this definition \(\rho\) forms an endofunctor on topological spaces over \(M\) and \(\pi\) a natural transformation from \({\operatorname{id}}\) to \(\rho\). Given a continuous map \(f \colon X \rightarrow M\) for a locally connected topological space \(X\) the universal property of the quotient space induces a unique map \(\phi_f \colon \pi_f (X) \rightarrow {\operatorname{dis}}\lambda_f\) such that \(\phi_f \circ \pi_f = \eta_f\) and thus in particular \(\rho_f = (\gamma \circ \lambda)_f \circ \pi_f\), hence we have the following commutative diagram \[ \xymatrix{ & {\operatorname{id}}\ar[dl]_{\pi} \ar[dr]^{\eta} \\ \rho \ar[rr]_{\phi} & & \gamma \circ \lambda } \] in the category of endofunctors on locally connected topological spaces over \(M\).
Proposition. The natural transformation \(\lambda \circ \phi\) from \(\lambda \circ \rho\) to \(\lambda \circ (\gamma \circ \lambda)\) is an isomorphism.
Proof. We apply \(\lambda\) to the previous diagram and obtain \[ \xymatrix{ & \lambda \ar[dl]_{\lambda \circ \pi} \ar[dr]^{\lambda \circ \eta} \\ \lambda \circ \rho \ar[rr]_{\lambda \circ \phi} & & \lambda \circ \gamma \circ \lambda . } \] Since \(\lambda \circ \pi\) is an isomorphism, it suffices to show that \(\lambda \circ \eta\) is an isomorphism. Given \(f \colon X \rightarrow M\) we apply the inverse bijection induced by the adjunction \((\lambda \dashv \gamma, \eta)\) to the diagram \[ \xymatrix{ f \ar[dr]^{\eta_f} \ar[dd]_{\eta_f} \\ & (\gamma \circ \lambda)_f \\ (\gamma \circ \lambda)_f \ar[ur]_{{\operatorname{id}}} } \] and obtain \[ \xymatrix{ \lambda_f \ar[dr]^{{\operatorname{id}}} \ar[dd]_{(\lambda \circ \eta)_f} \\ & \lambda_f \\ (\lambda \circ \gamma \circ \lambda)_f , \ar[ur]_{(\varepsilon \circ \lambda)_f} } \] hence \((\lambda \circ \eta)_f\) is the inverse to \((\varepsilon \circ \lambda)_f\).
Corollary. \(\phi\) and \(\eta \circ \rho\) are naturally ismorphic as functors from the category of topological spaces over \({\mathbb{R}}\) to the category of homomorphisms in the category of topological spaces over \({\mathbb{R}}\).
Example. Let \(f \colon X \rightarrow {\mathbb{R}}\) be a proper Morse function, then the critical points of \(f\) are isolated and since \(f\) is proper, it’s critical values are isolated as well. Hence for each \(r \in {\mathbb{R}}\) there is an \(\varepsilon_r > 0\) such that for all \(0 < \delta \leq \varepsilon_r\) the inclusion of \(f^{-1} (r)\) into \(f^{-1} ((r - \delta, r + \delta))\) is a homotopy equivalence and thus \(\phi_f\) is a homeomorphism.
de Silva, Munch, and Patel (2015) provide a self-contained treatment of the above when \(\lambda\) and \(\gamma\) are restricted to full subcategories of topological spaces over \({\mathbb{R}}\) respectively cosheaves on \({\mathbb{R}}\). When \(\phi\) is restricted to this subcategory of topological spaces over \({\mathbb{R}}\) referred to as constructible \({\mathbb{R}}\)-spaces, then \(\phi\) is a natural isomorphism. Further the authors provide a geometric description of the resulting subcategory of cosheaf spaces over \({\mathbb{R}}\). They refer to this category as Reeb or as the category of \({\mathbb{R}}\)-graphs.
[1] This is in line with the previously made assumption, since quotient spaces of locally connected spaces are again locally connected.