TITLE: Differential forms as a sheaf on the site of all manifolds vs. sheaf on an individual manifold
QUESTION [6 upvotes]: The de Rham complex can be viewed as a sheaf $\Omega$ on the entire site $\mathsf{Man}$ of smooth manifolds via the usual pullback of differential forms $(f: M \to N) \mapsto (f^*: \Omega(N) \to \Omega(M))$. So, the de Rham complex is a smooth space. Diffeological spaces, for example, are concrete sheaves on the site of smooth manifolds, but I don't know enough about sheaves to say whether or not the de Rham complex is a diffeological space.
We can also consider the category of open sets of $M$ as a site $O(M)$, where the arrows are inclusions. Then we have the sheaf $\Omega_M \in \mathsf{Sh}(M)$ of differential forms on $M$. Likewise, we have the sheaf $\Omega_N \in \mathsf{Sh}(N)$ of differential forms on $N$.
Now, as I understand it, there are morphisms $i_M: \mathsf{Man} \to M$ and $i_N: \mathsf{Man} \to N$ of sites, given by the respective inclusion functors $i_M^t: O(M) \to \mathsf{Man}$ and $i_N^t: O(N) \to \mathsf{Man}$. 
Can we describe the sheaves of differential forms $\Omega_M$ and $\Omega_N$ on $M$ and $N$ in terms of these morphisms of sites? I figure they should somehow be induced by the sheaf $\Omega$ on all of $\mathsf{Man}$, but I am not very experienced with sheaves on sites yet. Moreover, given a smooth map $f: M \to N$, there should a relationship between the inverse image functor and the pullback of forms.

REPLY [4 votes]: The sheaf $ \Omega^k $ ($k>0$) on the site $ \mathsf{Man} $ is not a diffeological space, because it is not a concrete sheaf: Obviousely, 
$ \Omega^k(\mathbb{R}^k)\ni\alpha\mapsto \underline{\alpha}:\mathrm{hom}(\mathbb{R}^0,\mathbb{R}^k)\rightarrow\Omega^k(\mathbb{R}^0)=0 $
is not one-to-one.
Sheaves $ \Omega_M $ and $ \Omega_N $ are the restriction of the sheaf $ \Omega $ to the sites $ O(M) $ and $ O(N) $, respectively.
For a smooth map $f:M \rightarrow N $, the induced inverse image functor commutates with the restrictions.
The pullback induced by $ f $ is just the global section map.