nLab Chen space

1973

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:

1. Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.

2. 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.

3. Each map $\{0\} \to X$ is a plot.

1975

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:

1. Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.

2. 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.

3. 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:

1. Let $\phi: U\to X$ be a continuous map and let $\{\theta_i: U_i \to U\}$ be a family of $C^\infty$-maps, $U$, $U_i$ being convex regions, such that a function $f$ on $U$ is $C^\infty$ if and only if each $f\circ \theta_i$ is $C^\infty$ on $U_i$. If each $\phi \circ \theta_i$ is a plot of $X$, then $\phi$ itself is a plot of $X$.

1977

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.

…

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:

1. Every plot is a map of the type $U\to M$, where $\dim U$ can be arbitrary.

2. 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.

3. Every constant map from a convex set to $M$ is a plot.

4. 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.