Let $U$ denote an open subset of $\mathbb{R}^n$ and
be the space of chains on $U$.
Let $X$ be a smooth space. For any plot $\phi:U\to X$, and any chain $c\in C(U)$, the push forward
is a chain in $X$. The collection of all plots allow us to push forward all such spaces of chains to obtain the space of chains on $X$, i.e.
Using plots $\phi:U\to X$ on chains on the domains of plots $C(U)$ gives chains $C(X)$ on $X$. Once we have chains on $X$, we can define cochains on $X$ to be the formal duals of chains, i.e.
For example, if $\alpha\in C^*(X)$ we can define its value on $\phi_* c\in C(X)$ via
A differential form $\alpha\in\Omega^p(X)$ on $X$ is a cochain such that its pull back via any plot is a differential form on the domain of that plot, i.e.
Let $\phi_i:U_i\to X$ denumerate a collection of plots such that any chain $c\in C(X)$ can be expressed as
for some chain $c_i\in C(U_i)$. Then we can define integration on $X$ via
Eventually the following will be a commented list – promised.
John Baez and Alexander Hoffnung, Convenient Categories of Smooth Spaces (arXiv, blog)
Patrick Iglesias-Zemmour, Diffeology (pdf)
Matthias Kreck, Stratifolds and differential algebraic topology (pdf)
William Lawvere, Taking categories seriously (pdf)
David Spivak, Quasi-smooth derived manifolds (pdf)
Andrew Stacey, Comparative Smootheology (arXiv)
Martin Laubinger, Differential Geometry in Cartesian Closed Categories of Smooth Spaces (pdf)
Alexander Hoffnung, Smooth spaces: convenient categories for differential geometry, (pdf slides)
Alexander Hoffnung, From Smooth Spaces to Smooth Categories, (pdf slides)
There are also Hofer’s polyfolds.
Concerning smooth ∞-stacks there is useful material in
Dual to generalized smooth spaces are generalized smooth algebras of functions on them, according to the general lore of space and quantity.
We had extensive discussion of generalized smooth spaces at the $n$-Café: