Is the "sheaf of diffeologically-smooth real-valued functions" functor faithful?

Question

Let $\tilde{X} \in \mathrm{DiffeologicalSpaces} \in \mathrm{Set}^{\mathrm{Diff}^{\mathrm{op}}}$ be a diffeological space (per the nLab definition).

• Let $X \in \mathrm{DTopologicalSpaces}$ be the underlying D-topological space of $\tilde{X}$ (cf. the above nLab page).

Note $X$ is the topological space with underlying set $X$ of $\tilde{X}$, equipped with the final topology induced by all the maps $f\in\tilde{X}(M) \subset \mathrm{Hom}_{\mathrm{Set}}(M,X)$ for all $M \in \mathrm{Diff}$.

According to nLab, the $X$ above is then a $\Delta$-generated topological space, or D-topological space for short.

• Let us define a presheaf $\mathcal{F}_{\tilde{X}}$ on $X$ via $$ \mathcal{F}_{\tilde{X}}(U) = \left\{f : U \rightarrow \mathbb{R}\,\big|\, \forall M \in \mathrm{Diff} , \forall g \in \tilde{X}(M) \subset \mathrm{Hom}_{\mathrm{Set}}(M,X) ,\, \bigl(\,g^{-1}(U)\xrightarrow{f\circ g} \mathbb{R} \text{ is smooth} \,\bigr) \right\} $$ as a (unital) $\mathbb{R}$-algebra subpresheaf of the sheaf of continuous $\mathbb{R}$-valued functions on $X$.

• If I'm not mistaken, the presheaf $\mathcal{F}$ is a sheaf and has local stalks, making $X$ a locally ringed space. Let us call $\mathcal{F}_{\tilde{X}}$ the "sheaf of diffeologically-smooth functions" on $X$.

Thus we have a mapping sending a diffeological space $\tilde{X}$ to the locally ringed D-topological space $(X,\mathcal{F}_{\tilde{X}})$.

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Going further, I think the above should also produce a functor $\mathrm{DiffeologicalSpaces} \rightarrow \mathrm{LocallyRingedSpaces}$.

My question is: is this functor faithful?

Although the functor $\mathrm{DiffeologicalSpaces} \xrightarrow{\mathrm{Dtopology}} \mathrm{DTopologicalSpaces}$ is (I think) non-faithful, I was wondering if including the extra data of the sheaf described above would result in a faithful functor.

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Update: per the answer below, apparently the $\mathrm{Dtopology}$ functor is already faithful; and since this functor factors through the above one, the above one must be faithful.

Answer 1

The forgetful functor $\mathrm{Dtopology}$ is already faithful. Indeed, suppose $\tilde{F}:\tilde{X}\to\tilde{Y}$ is a morphism of diffeological spaces with underlying set map $F:X\to Y$, and suppose $a:U\to X$ is a plot. For each $p\in U$, writing $i_p$ for the inclusion $\{*\}\to U$ with value $p$, we must have $$\tilde{F}(a)i_p=\tilde{F}(ai_p)=\tilde{F}(i_{a(p)})=i_{F(a(p))}.$$ Thus, $\tilde{F}(a)$ must be the function that maps each $p\in U$ to $F(a(p))$, so $\tilde{F}$ is completely determined by $F$.