#### The Reeb Space

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 $$(\varepsilon \circ \lambda)_f$$.

1. 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.