# On the weak homotopy type of a differentiable (Chen) space

Suppose that $$M$$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ).

Assume that $$M$$ also has the structure of a topological space and that the two structures are compatible in the sense that every plot is a continuous map.

To distinguish the structures, let $${}_SM$$ denote the differentiable structure and let $$M_T$$ denote the topological one. There is an adjoint pair $$L: \text{Differentiable Spaces} \leftrightarrows \text{Topological Spaces}: R$$ in which $$L(X)$$ is $$X$$ as a set with the topology which defines a subset $$O\subset X$$ to be open iff its preimage with respect to every plot is open. The functor $$R$$ gives a space $$Y$$ a smooth structure in which every continuous map $$U \to Y$$ from a convex set in euclidean space is declared to be a plot.

With $$M$$ as above, we have a continuous map $$L ({}_{S}M)\to M_T$$.

Hypothesis: The map $$L ({}_{S}M)\to M_T$$ is a weak homotopy equivalence.

(I.e.: every continuous map from a sphere to $$M_T$$ can be deformed to a smooth map and every continuous map from a disk to $$M_T$$ which is smooth on its boundary can be deformed rel boundary to a smooth map.)

Let $$\text{plot}_M$$ be the category of plots of $$M$$. An object of this category is a plot $$\phi: U \to M$$. A morphism $$(U,\phi) \to (V,\psi)$$ is a $$C^\infty$$ map $$f: U\to V$$ such that $$\psi \circ f = \phi$$. Then there is an evident map $$\underset{(U,\phi)}{\text{colim } } U \to M_T .$$

Question: Under what reasonable conditions can we conclude that this map is a weak equivalence?

Note: I really do want the colimit here (not the homotopy colimit).

Here is a some evidence: Let $$g: S^k \to M_T$$ be continuous. By my hypothesis, we can assume that $$g$$ is smooth, i.e., we can assume $$g: S^k \to {}_SM$$ is a smooth map (i.e., it preserves plots where $$S^k$$ is given its usual differentiable structure). Then we have a commutative diagram $$\require{AMScd}$$

$$\begin{CD} \underset{\phi: U \to M}{\text{colim } } U @>>> M \\ @AAA @AAA \\ \underset{\phi: U \to S^k}{\text{colim } } U @>>> S^k \end{CD}$$

But the bottom map is a retraction up to homotopy: use a smooth triangulation of $$S^k$$ to construct a section up to homotopy. This shows that top horizontal map is a surjection on homotopy.