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.



On the proof of "Mapping space is a Chen space"

> 
> 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.


