I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable space" structure:


Let $N$ be a smooth manifold. A continuous map $\alpha: N \to PM$ (where $PM$ has the compact open topology) is said to be *smooth* if the adjoint map $\tilde{\alpha}: N \times I \to M$ defined by $(n,t)\mapsto \alpha(n)(t)$ is smooth in the usual sense. The smooth map $\alpha$ is a *plot* if $N$ is an open convex subset of $\mathbb{R}^n$ for some $n$. Thus, we are modeling $PM$ locally with plots of varying dimension.


Chen defines a differential n-form $\omega$ on $PM$ as a rule which assigns to every plot $\alpha: U \to PM$ a differential form $\omega\_{\alpha} \in \Omega^n(U)$. We define $(d\omega)\_{\alpha}=d\omega\_{\alpha}$. It turns out that for Chen's purposes one does not need to develop more calculus tools on $PM$. He shows a De Rham type result: the cohomology of the complex $\Omega^\*(LM)$ (where $LM$ is the free loop space) is isomorphic to the real singular cohomology of $LM$.


However, for other purposes it is useful to consider forms as alternating tensors and to do this in this context we need a notion of vector fields on $PM$. I've always thought of a tangent vector at $\gamma \in PM$ as a vector field $T\_\gamma$ along $\gamma$ on $M$, so a vector field on $PM$ assings each point $\gamma \in PM$ a vector field along $\gamma$. However, following Chen, the natural way to define vector fields to make it compatible with his notion of differential forms is as follows: a vector field $T$ on $PM$ is a rule which assings to each plot $\alpha: U \to PM$ a vector field $T\_{\alpha}$ on $U$. 


How do we reconcile these two notions of vector fields on $PM$? Are they equivalent? 



Vector fields on path spaces
Good question!


There is a fairly well developed theory of *diffeological spaces*, with many recent references. The only difference between diffeological spaces and Chen's *smooth spaces* is that the plots of a diffeological space have *open* subsets $U \subset \mathbb{R}^n$ as their domains, while the ones of a smooth space have *convex* subsets. For questions concerning differential forms and tangent vectors, this seems to be irrelevant, so I'll continue with diffeological spaces.


As Dimitri pointed out in his comments, a *differential form* on a diffeological space $X$ is *not* just a form $\psi\_c$ on the domain $U\_c$ of each plot $c: U\_c \to X$. It must be required for two plots $c\_1: U\_1 \to X$ and $c\_2:U\_2 \to X$ such that one has a smooth map $f: U\_1 \to U\_2$ with $c\_2 \circ f = c\_1$, the forms satisfy 
$$
\psi\_{c\_1} = f^{\*}\psi\_{c\_2}.
$$ 
As also remarked by Dimitri, vector fields can neither be pulled back nor be pushed forward. Hence there is no analogous definition of vector fields. 


Yet the concept of a tangent space to a general diffeological space $X$ can be defined. A good reference is the paper "Diffeological spaces" by Martin Laubinger (Proyecciones Journal of Mathematics, Vol. 25, No2, pp. 151-178). Roughly, the tangent space at a point $x\in X$ is the sum of all tangent spaces $T\_0U\_c$ over all plots $c:U\_c \to X$ centered at $x$, i.e. $0 \in U\_c$ and $c(0)=x$. Then there is an equivalence relation that enforces the chain rule for maps.


Smooth manifolds form a full subcategory of diffeological spaces, by regarding a smooth manifold $M$ as a diffeological space with plots all smooth maps $c:U \to M$, for all open subset $U \subset \mathbb{R}^n$ and all $n\in \mathbb{N}$. Thus, for a smooth manifold $M$ we have now the classical tangent space, and the diffeological tangent space. Laubinger proves that both tangent spaces are isomorphic. 


Apparently less well-known is a theorem of M. Losik showing that *Fréchet* manifolds also form a full subcategory of diffeological spaces (with the plots defined in the very same way as for smooth manifolds). Thus, for a Fréchet manifold we also have two concepts of a tangent space, the classical one and the diffeological one. I don't know a reference for the statement that both concepts coincide, as in the smooth case, but I have looked at Laubinger's proof and I don't see why the same proof should not work for Fréchet manifolds. 


Now let's look at the path space $PM$ or the loop space $LM$ or at any other mapping space of a compact smooth manifold into a smooth manifold $M$. I continue with $LM$ for simplicity. There are, a priori, two diffeologies on the loop space $LM$. The first one is the one from the theorem of Losik, i.e. the one where the plots are the Fréchet-smooth maps from open sets $U$ to $LM$. The second one is the one described in the question, it is called the *functional diffeology*, with plots those maps $c: U \to LM$ whose adjoint $U \times S^1 \to M$ is smooth (demanding continuity for $c$ is superfluous). Luckily, the two diffeologies coincide (see Lemma A.1.7 in my paper [Transgression to Loop Spaces and its Inverse, Part I](http://arxiv.org/abs/0911.3212)).


And here is the result: classical Fréchet theory says that the tangent space of $LM$ at a loop $\tau$ consists of vector fields along $\tau$. The earlier claimed generalization of Laubinger's theorem shows that it coincides with the diffeological tangent space, and the coincidence of the diffeologies on $LM$ shows that it can be defined via plots $c$ whose adjoints $U \times S^1 \to M$ are smooth maps. This is, I think, the best one could hope for.


