# Differential Forms in Infinite Dimensions

In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of the bundle ${L}^k_{\mathrm{alt}}(TM, M \times \mathbb{R})$. This is the bundle whose vector space over a point $p\in M$ is the space of continuous, alternating, $k$-linear maps from $T_p M$ (the kinematic tangent bundle) to $\mathbb{R}$. Moreover, they distinguish this definition from several other possible definitions as the only model, where Lie derivative, pullback and exterior differential are all well-defined.

However, there is another possible definition for differential forms, using the definition of sheaves. Namely, the sheaf of differential $k$-forms on the site $\mathrm{Man}$ of smooth manifolds is the sheaf $\Omega^k$ that over a parameter manifold $S$ is simply given by $\Omega^k(S)$. For any (pre-)sheaf $\mathfrak{F}$ on $\mathrm{Man}$, one can then define $\Omega^k(\mathfrak{F})$ as the set of sheaf morphisms $\mathfrak{F} \rightarrow \Omega^k$, which naturally has the structure of a vector space.

Now any (finite or infinite) manifold $M$ can be considered as a sheaf $\mathfrak{F}_M$ by setting $\mathfrak{F}_M(S):= C^\infty(S, M)$. Since pullback works for the differential forms in the sense of Kriegl and Michor, one obtains $$\Omega^k(M) \subseteq \Omega^k(\mathfrak{F}_M),~~~~~~~ \theta \longmapsto \phi_\theta$$ where a differential form $\theta \in \Omega^k(M)$ induces a sheaf morphism $\phi_\theta$ by defining $\phi_\theta(f) := f^*\theta$ for $f \in C^\infty(S, M)$. It is easy to see that this defines an injective linear map, and it is clear that this is an isomorphism for $M$ finite-dimensional, by choosing $S=M$.

Question: Is this linear map always an isomorphism? That is, do we always have $\Omega^k(M) = \Omega^k(\mathfrak{F}_M)$? If not, what assumptions do we have to make on the manifold $M$ in the case that $M$ is infinite-dimensional in order for this to be true?

• I would guess it works up to Fréchet manifolds, which embed fully faithfully into diffeological spaces and even the Cahiers topos, I believe, which is where your sheaf should live. After that, I imagine what you say works when taking the convenient tvs approach to infinite-dimensional manifolds as K&M do. – David Roberts Aug 14 '17 at 17:47