Kuo Tsai Chen described several categories of generalised smooth spaces in his works. Four are reproduced below. His initial motivation appears to be extending the theory of differential forms from ordinary manifolds to path spaces.
By a convex $n$-region (or, simply a convex region), we mean a closed convex region in $\mathbb{R}^n$. A convex $0$-region consists of a single point.
Definition. A differentiable space $X$ is a Hausdorff space equipped with a family of maps called plots which satisfy the following conditions:
Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.
If $U'$ is also a convex region (not necessarily of the same dimension as $U$) and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi\theta$ is also a plot.
Each map $\{0\} \to X$ is a plot.
By a convex region we mean a closed convex set in $\mathbb{R}^n$ for some finite $n$.
Definition. A predifferentiable space $X$ is a topological space equipped with a family of maps called plots which satisfy the following conditions:
Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.
If $U'$ is also a convex region (not necessarily of the same dimension as $U$) and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi\theta$ is also a plot.
Each map $\{0\} \to X$ is a plot.
Remark. in [1973], a predifferentiable space is called a “differentiable space”. We propose to amend the definition of a differentiable space by adding the following condition:
The symbols $U$, $U'$, $U_i$, … will denote convex sets. All convex sets will be finite dimensional. They will serve as models, i.e. sets whose differentiable structure is known.
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Definition 1.2.1 A differentiable space $M$ is a set equipped with a family of set maps called plots, which satisfy the following conditions:
Every plot is a map of the type $U\to M$, where $\dim U$ can be arbitrary.
If $\phi: U \to M$ is a plot and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi \circ \theta$ is a plot.
Every constant map from a convex set to $M$ is a plot.
Let $\phi: U\to M$ be a set map. If $\{U_i\}$ is an open covering of $U$ and if each restriction $\phi | U_i$ is a plot, then $\phi$ is itself a plot.