nLab
Chen space

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.

1973

By a convex nn-region (or, simply a convex region), we mean a closed convex region in n\mathbb{R}^n. A convex 00-region consists of a single point.

Definition. A differentiable space XX 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 ϕ:UX\phi: U \to X, where UU is a convex region.

  2. If UU' is also a convex region (not necessarily of the same dimension as UU) and if θ:UU\theta: U' \to U is a C C^\infty-map, then ϕθ\phi\theta is also a plot.

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

1975

By a convex region we mean a closed convex set in n\mathbb{R}^n for some finite nn.

Definition. A predifferentiable space XX 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 ϕ:UX\phi: U \to X, where UU is a convex region.

  2. If UU' is also a convex region (not necessarily of the same dimension as UU) and if θ:UU\theta: U' \to U is a C C^\infty-map, then ϕθ\phi\theta is also a plot.

  3. Each map {0}X\{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 ϕ:UX\phi: U\to X be a continuous map and let {θ i:U iU}\{\theta_i: U_i \to U\} be a family of C C^\infty-maps, UU, U iU_i being convex regions, such that a function ff on UU is C C^\infty if and only if each fθ if\circ \theta_i is C C^\infty on U iU_i. If each ϕθ i\phi \circ \theta_i is a plot of XX, then ϕ\phi itself is a plot of XX.
1977

The symbols UU, UU', U iU_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 MM 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 UMU\to M, where dimU\dim U can be arbitrary.

  2. If ϕ:UM\phi: U \to M is a plot and if θ:UU\theta: U' \to U is a C C^\infty-map, then ϕθ\phi \circ \theta is a plot.

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

  4. Let ϕ:UM\phi: U\to M be a set map. If {U i}\{U_i\} is an open covering of UU and if each restriction ϕU i\phi | U_i is a plot, then ϕ\phi is itself a plot.

Remarks

  • In 1986, Chen gave a definition equivalent to the last.
  • It seems clear from the context that in the 1975 paper Chen was first recalling the definition from the 1973 paper. However, his recollection was not completely accurate as the underlying object was now a topological space rather than a Hausdorff space.
  • The forcing condition on the maps in the 1975 paper is actually stronger than that in the 1977 paper.
  • The final structure is of sheaves on a site. This is the definition used in, for example, Baez and Hoffnung [0807.1704].

References

Revised on July 11, 2013 23:19:09 by Urs Schreiber (82.113.106.241)