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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.

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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.

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  • $\begingroup$ 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? $\endgroup$ – Adittya Chaudhuri Jun 21 '20 at 3:32
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    $\begingroup$ @AdittyaChaudhuri: I added a few more details. $\endgroup$ – Dmitri Pavlov Jun 21 '20 at 12:54
  • $\begingroup$ Thank you Sir for the edit. I am trying to understand the proof in detail. $\endgroup$ – Adittya Chaudhuri Jun 22 '20 at 6:15

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