# On the proof of “Mapping space is a Chen space”

According to the page 5 in the paper Convenient Categories of Smooth Spaces https://arxiv.org/pdf/0807.1704.pdf by Baez and Hoffnung, Chen space is defined as follows:

(Note:I used different notations from the paper Convenient Categories of Smooth Spaces https://arxiv.org/pdf/0807.1704.pdf)

A Chen space $$X$$ is defined as a set $$X$$ equipped with , for each convex set $$U$$ there exists a collection $$\lbrace \phi_{i}: U \rightarrow X \rbrace_{i \in I}$$ of set maps called plots in $$X$$ satisfying the following properties:

(A convex set $$U$$ is defined as a convex set (with non-empty interior) in a Euclidean space $$\mathbf{R^n}$$ where $$n$$ can be any arbitrary non-negative integer. We call $$n$$ the dimension of $$U$$.A map $$f: U' \rightarrow U$$ from convex set $$U'$$ to convex set $$U$$ is called smooth function if $$f$$ has continuous derivatives of all order.)

1. If $$f: U' \rightarrow U$$ is a smooth function from convex set $$U'$$ to convex set $$U$$ and if $$\phi: U \rightarrow X$$ is a plot in $$X$$ then $$\phi \circ f$$ is also a plot in $$X$$.

2. Let $$U$$ be a convex set of dimension $$n$$. Suppose a collection of convex sets $$\lbrace U_j \subset U \rbrace_{j \in J}$$ forms an open cover of $$U$$ with respect to the subspace topology of $$\mathbf{R^n}$$. Let $$\lbrace I_j: U_j \rightarrow U \rbrace_{j \in J}$$ be the collection of inclusion maps. Let $$\phi: U \rightarrow X$$ be a set map. Now if each $$\lbrace \phi \circ I_j \rbrace_{j \in J}$$ are plots in $$X$$ then $$\phi$$ is also a plot in $$X$$.

3. Every function from the one point of $$\mathbf{R^0}$$ to X is a plot in $$X$$.

In page 6 they defined a set map $$f: X \rightarrow Y$$ to be smooth if for any plot $$\phi: U \rightarrow X$$ in $$X$$ the set map $$f \circ \phi: U \rightarrow Y$$ is a plot in $$Y$$.

In page 15 they mentioned that the mapping space $$C^{\infty}(X, Y)= \lbrace f:X \rightarrow Y: f$$ is smooth$$\rbrace$$ is a Chen space (where $$X, Y$$ are Chen spaces) whose plots are declared as those functions $$\phi:U \rightarrow C^{\infty}(X, Y)$$ such that the corresponding function $$\tilde{\phi}:U \times X \rightarrow Y$$ is smooth defined as $$(\zeta, x) \mapsto \phi(\zeta)(x)$$ (Note that there is a natural Chen space structure on both convex sets and Products ).

I was verifying $$C^{\infty}(X, Y)$$ is indeed a Chen space.

Property 1 and Property 3 are verified easily.

But I am not able to verify the property 2.

According to the definition of plot to verify Property 2, I need to show that if $$\tilde{\phi \circ I_i}:U_i \times X \rightarrow Y$$ is smooth for each $$i$$ then $$\tilde{\phi}: U \times X \rightarrow Y$$ is smooth. (where $$U_i \subset U$$ forms an open convex cover of $$U$$ and $$I_i$$ are inclusion maps). For that I need to show that if $$\psi:V \rightarrow U \times X$$ is any plot in $$U \times X$$ then $$\phi \circ \psi$$ is a plot in $$Y$$ .

I am not able to progess much after that!

I also note that there exist a collection of smooth maps $$I_i \times 1_X : U_i \times X \rightarrow U \times X$$ (where $$I_i$$ , $$1_X$$ are inclusion and identity map respectively).I feel somehow I need to use this fact also but not able to guess how.

I feel that I have to somehow express the plot $$\psi$$ in $$U \times X$$ in terms of plots of $$U_i \times X$$ so that I can use the smoothness of $$\tilde{\phi \circ I_i}$$. But I am not able to guess how!!

I feel it is some sort of local property of smoothness (as we have the analogue in case of finite dimensional smooth manifolds.)

I apologise priorly if this question is not upto the standard of MathOverflow. I am guessing I am mistaking or overlooking something.. But not able to guess what is that!!

Thank you.

I need to show that if $$\tilde{\phi \circ I_i}:U_i \times X \rightarrow Y$$ is smooth for each $$i$$ then $$\tilde{\phi}: U \times X \rightarrow Y$$ is smooth. (where $$U_i \subset U$$ forms an open convex cover of $$U$$ and $$I_i$$ are inclusion maps).
We have to show that a map $$U⨯X→Y$$ is a morphism of diffeological spaces if and only if its restrictions $$U_i⨯X→Y$$ are morphisms of diffeological spaces. To show that $$U_i⨯X→Y$$ uniquely glue to a morphism $$U⨯X→Y$$, consider some cartesian space $$S$$ and map it to both sides. We have to define a map on $$S$$-points natural in $$S$$: $$U(S)⨯X(S)→Y(S).$$ Fix an element in the left side, i.e., a smooth map $$f\colon S→U$$ and an element of $$X(S)$$. Fix a cover $$\{S_i=f^*U_i\}_{i∈I}$$ of $$S$$. The map $$f|_{S_i}$$ factors through $$U_i$$. Thus, we have an $$S_i$$-plot of $$U_i⨯X$$, which maps to an $$S$$-plot of $$Y$$ via the map $$U_i⨯X→Y$$. The constructed $$S_i$$-plots of $$Y$$ are compatible and glue to a unique $$S$$-plot of $$Y$$. This means that $$\tilde\phi$$ is a plot, as desired.
• Sir, According to the definition of Chen Space I mentioned above there is no assumption of topology on it. So how can you consider open sets of $U_i \times X$ and $U \times X$? Also a map $F:P \rightarrow Q$ between Chen spaces is defined to be smooth if for any plot $\phi:U \rightarrow P$ in $P$ , $F \circ \phi$ is a plot in $Q$. So how does this follow from your Answer? – Adittya Chaudhuri Jun 21 '20 at 3:32